A Primer of Real Analytic Functions. 2 Ed. - Krantz - Parks. (Birkhauser). 1992

BirkhauserAdvanced Texts

.cvcn U. niaiiLc

Harold R. Parks

A Primer ofReal Analytic Functions

Second Edition

Birkhauser Advanced TextsBasler Lehrbiicher

Edited byHerbert Amann, University of ZurichSteven G. Krantz, Washington University, St. LouisShrawan Kumar, University of North Carolina at Chapel Hill

Steven G. KrantzHarold R. Parks

A Primer ofReal Analytic Functions

Second Edition

BirkhauserBoston Basel Berlin

Steven G. Krantz Harold R. ParksWashington University Oregon State UniversityDepartment of Mathematics Department of MathematicsSt. Louis, MO 63130-4899 Corvallis, OR 97331-4605U.S.A. U.S.A.

Library of Congress Cataloging-in-Publication Data

A CIP catalogue record for this book is available from the Library of Congress,Washington D.C., USA.

AMS Subject Classifications: Primary: 26E05, 30B10. 32C05; Secondary: 14P15, 26A99, 26B10,26840, 26E10, 30B40, 32C09, 35A10, 54C30

Printed on acid-free paper

02002 Birkhguser Boston Blfkhauser01992 BirkhNuser Verlag, First Edition

4))Ifl:

All rights reserved. This work may not be translated or copied in whole or in part without the writtenpermission of the publisher (Birkhauser Boston, do Springer-Verlag New York, Inc., 175 FifthAvenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews orscholarly analysis. Use in connection with any form of information storage and retrieval, electronicadaptation, computer software, or by similar or dissimilar methodology now known or hereafterdeveloped is forbidden.The use in this publication of trade names, trademarks, service marks and similar terms, even if theyare not identified as such, is not to be taken as an expression of opinion as to whether or not they aresubject to proprietary rights.

ISBN 0-8176-4264-1 SPIN 10846987ISBN 3-7643-4264-1

Reformatted from the author's files by TEXniques, Inc., Cambridge, MA.Printed and bound by Hamilton Printing, Rensselaer, NY.Printed in the United States of America.

987654321A member ofBertelsmannSpringer Science+Business Media GmbH

To the memory of Frederick J. Almgren, Jr. (1933-1997),teacher and friend

Contents

Preface to the Second Edition ix

Preface to the First Edition xi

I Elementary Properties 1

1.1 Basic Properties of Power Series 1

1.2 Analytic Continuation .. . 11

1.3 The Formula of Fail di Bruno .. . 16

1.4 Composition of Real Analytic Functions . . . 18

1.5 Inverse Functions . 20

2 Multivariable Calculus of Real Analytic Functions 252.1 Power Series in Several Variables . . . 25

2.2 Real Analytic Functions of Several Variables . 292.3 ................ .The Implicit Function Theorem . . 35

2.4 A Special Case of the Cauchy-Kowalewsky Theorem . 422.5 The Inverse Function Theorem . 47

2.6 Topologies on the Space of Real Analytic Functions ... . 502.7 .................. .Real Analytic Submanifolds 54

2.7.1 Bundles over a Real Analytic Submanifold . 562.8 The General Cauchy-Kowalewsky Theorem .. . 61

viii

3

Contents

Classical Topics 67

3.0 Introductory Remarks ........................ 673.1 The Theorem of Pringsheim and Boas . 68

h's Theore m3.3 Whitney's Extension and Approximation Theorems . 753.4 The Theorem of S. Bernstein . 79

4 Some Questions of Hard Analysis 834.1 Quasi-analytic and Gevrey Classes ................. 834.2 Puiseux Series .. . 954.3 Separate Real Analyticity . . . 104

S Results Motivated by Partial Differential Equations 1155.1 Division of Distributions I . 115

5.1.1 Projection of Polynomially Defined Sets .. . 1175.2 Division of Distributions II .. ........ . ....... ... 1265.3 The FBI Transform . 135

5.4 The Paley-Wiener Theorem .................... 144

6 Topics In Geometry 1516.1 The Weierstrass Preparation Theorem ..... . . .. ... . .. 151

6.2 Resolution of Singularities . . . .... .... . . ........ 1566.3 Lojasiewicz's Structure Theorem for Real Analytic Varieties . 1666.4 The Embedding of Real Analytic Manifolds . 171

6.5 Semianalytic and Subanalytic Sets . 1776.5.1 Basic Definitions . 177

6.5.2 Facts Concerning Semianalytic and Subanalytic Sets . . . 1796.5.3 Examples and Discussion ................. 1816.5.4 Rectilinearization . . 184

Bibliography 187

Index 203

Preface to the Second Edition

It is a pleasure and a privilege to write this new edition of A Primer of Real Ana-lytic Functions. The theory of real analytic functions is the wellspring of mathe-matical analysis. It is remarkable that this is the first book on the subject, and wewant to keep it up to date and as correct as possible.

With these thoughts in mind, we have utilized helpful remarks and criticismsfrom many readers and have thereby made numerous emendations. We have alsoadded material. There is a now a treatment of the Weierstrass preparation theorem,a new argument to establish Hensel's lemma and Puiseux's theorem, a new treat-ment of FaA di Bruno's formula, a thorough discussion of topologies on spaces ofreal analytic functions, and a second independent argument for the implicit func-tion theorem. We trust that these new topics will make the book more complete,and hence a more useful reference.

It is a pleasure to thank our editor, Ann Kostant of Birkhauser Boston, for mak-ing the publishing process as smooth and trouble-free as possible. We are gratefulfor useful communications from the readers of our first edition, and we look for-ward to further constructive feedback.


May, 2002

Preface to the First Edition

The subject of real analytic functions is one of the oldest in mathematical anal-ysis. Today it is encountered early in one's mathematical training: the first tasteusually comes in calculus. While most working mathematicians use real analyticfunctions from time to time in their work, the vast lore of real analytic func-tions remains obscure and buried in the literature. It is remarkable that the mostaccessible treatment of Puiseux's theorem is in Lefschetz's quite old AlgebraicGeometry, that the clearest discussion of resolution of singularities for real ana-lytic manifolds is in a book review by Michael Atiyah, that there is no compre-hensive discussion in print of the embedding problem for real analytic manifolds.

We have had occasion in our collaborative research to become acquainted withboth the history and the scope of the theory of real analytic functions. It seemsboth appropriate and timely for us to gather together this information in a sin-gle volume. The material presented here is of three kinds. The elementary topics,covered in Chapter 1, are presented in great detail. Even results like a real ana-lytic inverse function theorem are difficult to find in the literature, and we takepains here to present such topics carefully. Topics of middling difficulty, suchas separate real analyticity, Puiseux series, the FBI transform, and related ideas(Chapters 2-4), are covered thoroughly but rather more briskly. Finally there aresome truly deep and difficult topics: embedding of real analytic manifolds, sub-and semi-analytic sets, the structure theorem for real analytic varieties, and reso-lution of singularities are discussed and described. But thorough proofs in theseareas could not possibly be provided in a volume of modest length.

xii Preface to the First Edition

Our aim, therefore, has been to provide an introduction to and a map (a primerif you will) of the subject of real analytic functions. Perhaps this monograph willhelp to bring to light a diverse and important literature.

It is a pleasure to thank Richard Beals, Edward Bierstone, Brian Blank, HaroldBoas, Ralph Boas, Josef Siciak, Kennan T. Smith, David Tartakoff, and MichaelE. Taylor for many useful comments and suggestions. Of course the responsibilityfor all remaining errors remains the province of the authors.


1992

A Primer of

Real Analytic Functions

Second Edition

1

Elementary Properties

1.1 Basic Properties of Power Series

We begin with power series on the real line R. A formal expression

00

Eaj(x-a)tj=0

with the aj's being either real or complex constants and with a E R, is calleda power series. It is usually convenient to take the coefficients a j to all be real;there is no loss of generality in doing so. Our first task is to determine the natureof the set on which a power series converges.

Proposition 1.1.1 Assume that the power series

00

Eaj(x -a)jj=o

converges at the value x = c. Let r = Ic -al. Then the series converges uniformlyand absolutely on compact subsets of I = (x : Ix - aI < r).

Proof. We may take the compact subset of I to be K = [a - s, a + s] for somenumber 0 < s < r. It holds that

00 00 Ix-alaj(x-a)jI

laj(c-a)jl c-aj=0 j=0

2 1. Elementary Properties

In the sum on the right, the first expression in absolute values is bounded by someconstant C (by the convergence hypothesis). For X E K, the quotient in absolutevalues is majorized by L = s/r < 1. The series on the right is thus dominated by

00

EC-L3.j=O

This geometric series converges. By the Weierstrass M-Test, the original seriesconverges absolutely and uniformly on K. 0An immediate consequence of the proposition is that the set on which the powerseries

00

>aj(x -ar)jj--0

converges is an interval centered about a!. This interval is termed the intervalof convergence. The series will converge absolutely and uniformly on compactsubsets of the interval of convergence. The radius of the interval of convergenceis defined to be half its length. Whether convergence holds at the endpoints of theinterval will depend on the particular series. Let us use the notation C to denote theopen interval of convergence. While we have seen that a power series is uniformlyconvergent on compact subintervals of C, it is an interesting and nontrivial fact thatif the series converges at either of the endpoints, then the convergence is uniformup to that endpoint. This fact is a consequence of the following lemma due toAbel (see [AN 26]).

Lemma 1.1.2 Let uO, u I .... be a sequence of reals, and set

0

If

and if

then

Sn = E uj, for n = 0,1,....

j=0

a < sn < A, for n = 0, 1, .. .

...>En>0,

n

EOa < > Ej u j < co A, for n = 0, l .... .j=0

Proof. One can write

UO = S0, UI = 5 1 5 0 . . . . . Un = Sn - Sn_I, ....

1.1. Basic Properties of Power Series 3

Hence

EOUO + E1u1 + ... + EnUn = EOSO + EI(Sl - SO) + ... + En(Sn - Sn-1)

= (EO - EI)SO + ' + (En-1 - EnSn-l +EnSn (1.1)

We also have

(e1 -Ej+l)a <(Ej -Ej+l)Sj :5 (6j -Ej+1)A,

for j = 0, 1, ... , andEna < EnSn < EnA .

Adding up these inequalities and using (1.1), we obtain the result.

Proposition 1.1.3 Assume that the power series

00

E aj(x - a)tj_-O

has the bounded interval of convergence C. If p is an endpoint of C and if thepower series converges at the value x = p, then the series converges uniformlyon the closed interval bounded by a and p.

Proof. We may assume that C = (-1, 1) and that the series converges at x = 1.We take Ej = xi, u j = a j and consider summation from j = m to j = m + n,with m large. The assertion is then immediate from Lemma 1.1.2.

0

Remark 1.1.4 The procedure exhibited in Lemma 1.1.2 and its proof is oftenreferred to as "summation by parts." Indeed, the usual integration by parts proce-dure in calculus may be verified by applying summation by parts to the Riemannsums for the integral.

On the interval of convergence C, the power series defines a function f. Sucha function is said to be real analytic at a. More precisely, we have the followingdefinition.

Definition 1.1.5 A function f, with domain an open set U S R and range eitherthe real or the complex numbers, is said to be real analytic at a if the functionf may be represented by a convergent power series on some interval of positiveradius centered at or

00

f (x) = E aj (x - a)j.j=O

The function is said to be real analytic on V C_ U if it is real analytic at eachaEV.

4 I. Elementary Properties

Remark 1.1.6 It is true, but not obvious, that the function which a convergentpower series defines is real analytic on the open interval of convergence. Thisassertion will be proved in the next section. A consequence is that the set V in thepreceding definition may as well always be chosen to be open.

We need to know both the algebraic and the calculus properties of a real analyticfunction: is it continuous? differentiable? How does one add/subtract/multiply/di-vide two such functions?

Proposition 1.1.7 Let

00 00

Eaj(x -a)j and Ebj(x -a)jj=0 j=0

be two power series with open intervals of convergence C1 and C2. Let f (x) bethe function defined by the first series on C1 and g(x) the function defined by thesecond series on C2. Then, on their common domain C = C1 fl C2, it holds that

(1) f(x)±g(x)=Ej°__o(aj±bj)(x-a)t;

(2) f(x) - g(x) = E M00 j+k=m(aj . bk) (x -a)m;

Proof. Let

N N

AN = E aj (x - a)t and BN = b j (x - a)jj=0 j=0

be, respectively, the Nth partial sums of the power series that define f and g . IfCN is the NO' partial sum of the series

00

1:(a j ± bj)(x - a)1,j=0

then

00

f(x)±g(x) = lira AN± lim BN = lim [AN ± BN] = >(aj±bj)(x-a)J ,N- N-oo N-oo

j=0

proving (1).For (2), let

N 00

DN=1: (aj'bk)(x-a)m and RN = bj(x-a)t.m=Oj+k=m j=N+1

1. 1. Basic Properties of Power Series

We have

DN = aOBN +al(x -a)BN_I +...+aN(X -a)NBO= ao(g(x) - RN) +al(x -a)(g(x) - RN-1)

+ ... + aN (x - a)N(g(x) - Ro)N

= g(X)Jaj(x-a)jj=0

-[BORN +al(x -a)RN_1 +...+aN(x -a)NR01

ClearlyN

g(X) > a j (x - a)jj=0

converges to g(x) f (x) as N approaches oo. It will thus suffice to show that

IaORN +aI(x -a)RN_I +... +aN(X -a)NROI

converges to 0 as N approaches oo.Consider X E C to be fixed. We know that

00

E aj(x - a)j

j-0

is absolutely convergent so we may set

00

A=EIajIIx-al'.j=o

Given E > 0, we can find No so that N > No implies I RN I < E. So we have

IaoRN +al(x -a)RN_1 +...+aN(x - a)NRolIaoRN + ... + aN_N0(X - a)N-NO RNOI

+ IaN-No+l (x - a)N-N0+1 RNO_I + ... + aN (x - a)NRot

< A + IaN-N0+I (x - ,)N-No+1 RNO_ I + + aN (x - a)N ROI

5

By holding No fixed and letting N approach oo we obtain the result.

Next we turn to division of real analytic functions. Proving the analyticity ofthe quotient of analytic functions is more delicate than doing so for the sum orthe product. This endeavor will be facilitated by the following lemma and itscorollary.


Lemma 1.1.8 For the power series

00

E aj(x - a)t

j=0

define A and p by

and

A = lim sup Ian l'n- oo

0 if A = oo,p= 1/A if0<A<oo, (1.2)

00 ifA=O.Then p is the radius of convergence of the power series about a.

Remark 1.1.9 Equation (1.2) is called the Hadamard formula for the radius ofconvergence of a power series.

Proof. Observing that

limsup lan(x -a)nllj" = Alx -al,n-oo

we see the lemma is an immediate consequence of the root test.

Corollary 1.1.10 The power series

eo

Eaj(x -a)jj=0

has radius of convergence p if and only if for each 0 < R < p, there exists aconstant 0 < C = CR such that

Ianl <C

(1.3)

We will need the following elementary lemma.

Lemma 1.1.11 For -oo < M < oo and j = 1, 2,..., it holds that

j-l1+ME(1+M)t=(1+M)t.

t-0

Proof. The result is easily proved by induction on j.

If f and g are real analytic functions at a point a and if g does not vanish on anopen interval containing a, then we would like to show that f/g is real analytic ata (it certainly is a well-defined function) and we would like to be able to calculateits power series expansion at a by formal long division. These facts are what thenext result tells us.



00 00

Eaj(x - a)i and >bj(x - a)jj=o i=o

be two power series with open intervals of convergence C1 and C2. Let f (x) bethe function defined by the first series on C1 and g(x) the function defined by thesecond series on C2. If a E CI n C2, and g does not vanish at a, then the function

h(x) = ,f(X)g(x)

is real analytic at a. Moreover the power series expansion of h at a may beobtained by formal long division of the series for g into the series for f. That is,the zeroth coefficient co of the series for h is

co = ao/bo,

and the higher order coefficients cj, j = 1, 2, ..., are given recursively by

! 1

ci = 1 I ai - E btci-t/ (1.4)b° \ t=1

Proof. By (1.4), the coefficient o

00 00

f x' in

g(x) h(x) = (bk(x_a)k)(j--o

cj(x - a)t)

is

n n

E btcn-t = bocn - btcn-tt=o C=1

an

so the equation g h = f holds at the level of formal power series. Thus if we canshow that the power series

00

Ecj(x - a)ij=o

converges on an open interval about a, then the result on multiplication of seriesin Proposition 1.1.7 yields this proposition.

There is no loss of generality in assuming that a = 0 and bo = 1. Further, bydilation or contraction, it is also no loss of generality to assume that

(-1-c, 1+e)ccinc2,for some e > 0.


By the assumptions we have just made, the radius of convergence of each of theEOOseries o ajxi and E0o b jxi exceeds 1, so we conclude by Corollary 1.1.10

that there is a constant 0 < M < oc such that la j I < M, 1b11 < M hold forj=0,1,....

- -We will prove by induction that Icjl < M(1 + M)i holds for j = 0, 1, ....

Recalling the assumption bo = 1, we see that Icol = laol < M holds. Nowconsider j > 1. Using (1.4) and Lemma 1.1.11, we estimate

Icji -< lajl +EIbtllcj-tlt=1

iM + M2 L(1 + M)t-t

t=1

i-1= M+M2E(1+M)t

t-o= M(1+M)j.

It now follows from Lemma 1.1.8 that the radius of convergence of the theseries o cj (x - a)j is at least 1/(1 + M). 0

We conclude this section by obtaining continuity and differentiability resultsfor real analytic functions. From the power series

00

Eaj(x -a)jj=0

it is natural to create the derived series

CO

E jaj(x - a)ij=1

(1.5)

using term-by-term differentiation. We want to explore how the derived series isrelated to the derivative of the function defined by the original series.

Proposition 1.1.13 The radius of convergence of the derived series is the sameas the radius of convergence of the original power series.

Proof. For notational simplicity we assume that L = lim Ian I A exists and

0<L<oo.

By dilation or contraction, we may suppose L > 1.


We observe that

liminflnanIWT = liminf(Inanl")> lim Inan I

lim n I lim Ia.I n

= L.

On the other hand, for any choice of I > 1, we have

n

limsuplna,,I" = limsup(Inanl")MT

llm(Inanl! n)a

(limn I Iim ]an I")k

= L1.

Since L > 1 was arbitrary, we have

lim Ina,IWT = L,

and the result follows from the Hadamard formula.


0

00

Eaj(x - a)jj =0

be a power series with open interval of convergence C. Let f (x) be the functiondefined by the series on C. Then f is continuous and has a continuous derivativewhich is real analytic at a. Moreover, the derived series (1.5) has the same inter-val of convergence C and the derivative f' is the function defined by that derivedseries.

Proof. On each closed subinterval of C, f is the uniform limit of a sequenceof continuous functions: the partial sums of the power series representing f. Itfollows that f is continuous on that closed subinterval and thus on C. Since theradius of convergence of the derived series is the same as that of the original series,it also follows that the derivatives of the partial sums converge uniformly on anyclosed subinterval of C to a continuous function. It then follows (see [KS 91,Theorem 9] or [RW 79, Theorem 7.17]) that f is differentiable and its derivativeis the function defined by the derived series.

By induction we obtain the following corollary.


Corollary 1.1.15 Let00

>ai(x -a)ii=0

be a power series with open interval of convergence C. Let f (x) be the functiondefined by the series on C. Then f is continuous and has continuous derivativesof all orders which are real analytic at a.

We can now show that a real analytic function has a unique power series repre-sentation:

Corollary 1.1.16 If the function f is represented by a convergent power serieson an interval of positive radius centered at a,

f(x) _ Eai(x -a)t , (1.6)00

i=o

then the coefficients of the power series are related to the derivatives of the func-tion by

f(n) (a)art =

n!

Proof. This result follows readily by differentiating both sides of (1.6) n timesand evaluating at x = a, as we may by Proposition 1.1.14 and Corollary 1.1.15.

0

Remark 1.1.17 If a power series converges at one of the endpoints of its inter-val of convergence then, by Abel's Lemma 1.1.2 above, we see that the functiondefined by the power series is continuous on the closed interval including thatendpoint. On the other hand, the function defined by a power series may extendcontinuously to an endpoint of the interval of convergence without the series con-verging at that endpoint. An example is the series

00

E(-x)ij=1

which converges on (-1, 1), equals 1/(1 + x), and does not converge at x = 1even though the function 1/(1 + x) extends continuously, even analytically, to(-1,00).

Finally, we note that integration of power series is as well behaved as differen-tiation.

Proposition 1.1.18 The power series

00

>ai(x -a)iJ=a

1.2 Analytic Continuation 11

and the seriesccEat

(x - a)j+1j+1obtained by term-by-term integration have the same radius of convergence, andthe function F defined by

00F(x)jT . +

1(x - )j+1

on the common interval of convergence satisfies

F'(x)=Eaj(x-a)t.00

j=0

The proof is left as an easy exercise. It is similar in spirit to the argument fordifferentiation.

1.2 Analytic Continuation

Notation 1.2.1 A function on an open interval I is called k times continuouslydifferentiable if the first k derivatives of f exist on I and are continuous. We oftenwrite f e Ck(I) to denote this property. If derivatives of all orders exist (andhence are automatically continuous) then we say that f is infinitely differentiableon 1 and write f e C°O(I). In case f is real analytic on I we write f E C'(I).

We will need a result regarding summation of certain series.

Lemma 1.2.2 For each nonnegative integer n and each -1 < x < 1, we have

00(m)nxm_n

= n'm =n (1 x)n+1'

where we use the notation

W.WO

m(m- 1)(m-2)...(m-n+1),1.

Proof. This result is proved by differentiating the geometric series

m=0


Suppose the power series00

aj(x -a)jj=Q

has positive radius of convergence p and thus defines a real analytic function fon (a - p, a + p). If fl is a point with la - f I < p, then we can certainly definea power series

00

J bj (x - f)Jj=o

by setting

bt =f(i)ifi)

(1.7)jThe following proposition shows that this new power series is well behaved.

Proposition 1.2.3 Let the notation be as above. The power series

00

Ebj(x-f)3j=o

with coefficients bj defined by (1.7) has positive radius of convergence at leastr = p - la - 6I, and on the interval (f - r, f + r) it converges to f.

Proot We have00

f(n)(0) = E(m)nam(f - a)m-nm=n

By Corollary 1.1.10, we also know that, for any R < p. there is a constant C suchthat

Iajl < RCombining these facts and using Lemma 1.2.2. we see that

00lf(n)(f)l CE(m)nLO-alm-n

Rmm=nC= E(m)n (l#R" m=n

_ C n

- Rn (- ,Q )n+l1

n!D(R - I# -a I)n'

1.2. Analytic Continuation 13

where D = RCR

a . Since R 2bj(x-f)t.j=o

By Taylor's theorem, we know that

- bf (n+t) O

n+lj (x - ).f (x)(x - p)=

j=o (n + 1)!

where t; is a point between ,6 and x. But similar estimates hold for f ('+')(1;) asfor f (n) (P), so it follows that g(x) = f (x). U

The next corollary is an immediate consequence of the proposition.

Corollary 1.2.4 Let

00

Eaj(X -ot)jj=o

be a power series with open interval of convergence C. Let f (x) be the functiondefined by the series on C. Then f is real analytic at every point of C.

Corollary 1.2.5 If f and g are real analytic functions on an open interval U andif there is a point xo E U such that

f(j)(x0) = g(j) (XO), for j = 0, 1, ...,

then

f (x) = g(x), for all x E U.

Proof. The functions f and g clearly agree at xo and on the interval about xowhere the power series with coefficients

b =fU)(xo) - g(j)(xo)

j! j!converges. But that interval of convergence need not include all of U.


We set

V = U fl (x : f (')(x) = g(')(x), for j = 0, 1, ... }.

By continuity, V is closed in the relative topology of U, while by the propositionV is open. Thus, by the connectedness of U, we conclude that U = V.

The next corollary is an immediate consequence of the preceding one.

Corollary 1.2.6 If f and g are real analytic functions on an open interval U andthere is an open set W C U such that

f (x) = g(x), for all x E W,

thenf (x) = g(x), for all x E U.

In fact, the hypothesis of the preceding corollary can be weakened substantially:

Corollary 1.2.7 If f and g are real analytic functions on an open interval U andif there is a sequence of distinct points x j , x2, ... in U with xo = limn,00 Xn E Uand such that

f (xn) = g (xn ), for n

then

f (x) = g(x), for all x E U.

Proof. Without loss of generality assume that

x1

We will prove inductively that for each j = 0, 1, ... ,

fj)(xo) = g(j)(xo) (1.8)

and that there exists a sequence xi, j < x2, j < . . . in U with xo = lim,00 xn, jsuch that

f(j)(xn.j) = g(j)(xn.j), for n = 1, 2, .... (1.9)

To begin the induction, note that we have f (xo) = g(xo) and that we can setxn,o =xn forn = 1,2.....

Suppose that, for a specific integer j > 0, (1.8) has been proved and a se-quence xi, j < X2. j < . satisfying (1.9) has been constructed. Apply the meanvalue theorem to f (j) - g(j) at xn, j and xn+i, j to obtain a point xn, j+t suchthat f (J +t) (xn, j+l) = g(j+') (xn. j+i ). Thus we have (1.9) with j replaced byj + 1. It follows that xo = limn_00 xn, j, and consequently, (1.8) also holds withj replaced by j + 1.

This corollary now follows from Corollary 1.2.5.

In the next definition we find it convenient to think of a function with domain aset A C R and range in R as a collection of ordered pairs of real numbers:

1.2. Analytic Continuation 15

Definition 1.2.8 Given a real analytic function f defined on an open interval U,we see from the preceding corollary that

U {g : g is a real analytic function on an open interval V D U ]

is a well-defined analytic function called the analytic continuation of f.

Another corollary of Proposition 1.2.3 is the following:

Corollary 1.2.9 If f E C'(1) for some open interval I then, for each a E 1,there are an open interval J, with a E J C 1, and constants C > 0 and R > 0such that the derivatives off satisfy

If'i'(x)I <C Ri , Vx E J. (1.10)

In fact, the converse of Corollary 1.2.9 is also true.

Lemma 1.2.10 If f E C' (1) for some open interval I and if for each of E 1,there are an open interval J, with a E J C 1, and constants C > 0 and R > 0such that the derivatives off satisfy (1.10), then f E C°'(1).

Proof. The Root Test and the inequality (1.10) show that

00E fv'(a) (x - a)ii=o

!i

'

converges at least on the interval K = (a - R, a + R). Taylor's theorem and theinequality (1.10) show that the power series

00E fti)' a) (x - a)ii=o J"

converges to f on J n K.

Remark 1.2.11 It is interesting to note that, in the reference [TM 71], a general-ization of this result is proved in which plain differentiation (in several variables)is replaced by a suitable elliptic differential operator.

Together Corollary 1.2.9 and Lemma 1.2.10 provide a useful characterizationof real analytic functions that will be applied in many of the sections that follow.

Proposition 1.2.12 Let f E C°O (1) for some open interval 1. The function f isin fact in C°'(1) if and only if for each a E 1, there are an open interval J, witha E J C I, and constants C > 0 and R > 0 such that the derivatives off satisfy

i

If(j)(x)I <C. R. , VX E J.


Remark 1.2.13 We note that it follows from the results of this section that if areal analytic function B(x) satisfies B(O) = 0, but does not vanish identically,then it may be written in the form

B(x) =xNB(x),

for some positive integer N, where B(x) is also real analytic and B(0) 34 0.Likewise, if A(x) and C(x) are real analytic and C does not vanish identically,then A(x)/C(x) may be written

A(x)/C(x) = xMD(x),

for some integer M, where D(x) is real analytic and D(0) 96 0. 0

1.3 The Formula of Faa di Bruno

Next we would like to show that the composition of real analytic functions is realanalytic, but for this purpose we will need to develop the formula of Faa di Bruno.Since that formula is of interest in its own right, we treat it in some detail here.The composition of real analytic functions is treated in Section 1.4.

Leibniz's formula for the higher derivatives of a product of two functions iswell known. Less known is the formula for the higher derivatives of a compo-sition of two functions. The formula for the single variable case, due to Faa diBruno [FB 57], was not published until 1857.1 Limited multivariable formulasfor derivatives of compositions of functions appeared in 1871 in [HR 71] and[MR 71 ]. A full multivariable formula only appeared in 1986 in [GH 86] and witha more accessible form for the coefficients in 1996 in [CS 96].

In this section, we will give a simple proof of the one-variable formula of Fad diBruno. We will use the fact that a polynomial in several variables (real or complex)is either identically zero or it is nonvanishing on an open dense set (as one canshow by induction on the number of variables). We also will make use of themultinomial theorem. The multinomial theorem is well known, but for the reader'sconvenience we cite as a reference the treatise of Aigner [AM 79].

Theorem 1.3.1 (Multinomial Theorem) For positive integers k and n, it holdsthat

(xl +x2+...+xn)k = (k,k Jx1 x22xn"

k2 ... k,,

I The paper in which Fa3 di Bruno's formula appears was written in French. The author's name isgiven there in the form Fa3 de Bruno.

1.3. The Formula of Fa3 di Bruno 17

where the sum is taken overall kl, k2, ... , kn for which k1 +k2 + + kn = kand where the multinomial coefficient is defined by

k _ k!

G1 k2 ...j

k1! k2! ...kn!. (1.12)

Theorem 1.3.2 (The Formula of Faa di Bruno) Let I be an open interval in Rand suppose that f E C°O(1). Assume that f takes real values in an open intervalJ and that g E Coo (J). Then the derivatives of h = g o f are given by

h(n)(t) =n!

g(k)(f(t)) (f(I)(t))kI( f(21(t))kZ 'kl'

.k....kn'

. 1. 2! n. J

where k = k1 + k2 + + kn and the sum is taken over all k1, k2, ... , kn forwhichkl =n.

Proof. First note that it is clear by induction and the chain rule that

h(n)(t)

_ E C(n, k1, k2, ... , kn) 9(k) (.f (t)) (f(l)(t))k, (f(2) (1))k2 ... (f (") (t))kn

holds for some nonnegative integer coeffients C(n, k1, k2-., kn) that are inde-pendent of the choice of the functions f and g, where, as above, k = k1 +k2+ +kn and the sum is taken over all k 1 , k2, ... , kn for which k1 +2k2 + +nkn = n.Our job is to calculate C(n, k1, k2, ..., kn).

Fix a value of k and a value of n. Set

g(x) = Xk ,f(t) = alt+a2t2+...+antn,

where the ai are arbitrary real coefficients. By the multinomial theorem, takingh = g o f, we see that

h(t) _ k

-1: (ki k2... kni-l

where the sum is taken over all k1, k2, ... , kn for which k1 + k2 + + kn = k,so we have

h(n)(p) k

C

lk1 k2 ...kn

l n

i=I


Here the sum is now taken overall kl, k2, ... , k for which k1 +k2+- +kn = kand for which kt + 2k2 + + nkn = n. On the other hand, we have

h(n)(0) = E C(n, kt, k2, ... , kn)k! (1!)k' (2!)k2 ... (n!)kn

where this sum is also taken overall kt, k2, ... , k for which kt +k2+ +kn = kand for which kt + 2k2 + + nkn = n because g(t)(0) is nonzero only whene = k. We conclude that

C(n, kt, k2, ..., kn) k! (1!)kl (2!)k2 ... (n!)k"

k "n ak1(k1k2...kn 11

I I )(1.14)

holds for all choices of at, a2, ..., a,,, where both sums are taken over all kt, k2,...,kn forwhichkt+k2+ - -+k, n.Recognizing (1.14) as a polynomial equation in the variables at, a2, ..., an, wesee that the only way that (1.14) can hold for all choices of at, a2, ..., an is forthe corresponding coefficients on both sides to be equal, i.e., we have

Gk n!

t k2 ... k (1!)k- (2!)k2 ... (n!)kn

n! 1

kt ! k2! ... kn! (1!)kl (2!)k2 ... (n!)k.

proving the result. O

Our proof of Theorem 1.3.2 is similar to that given by Charles-Jean de la ValleePoussin in [VP 30]. An alternative proof can be found in [RS 80].

1.4 Composition of Real Analytic Functions

To apply the formula of Faa di Bruno to showing that the composition of real an-alytic functions is real analytic, we will need the following combinatorial lemmathat follows from a particular application of the formula (1.13).

Lemma 1.4.1 For each positive integer n and positive real number R,

k! Rk = R(1 + R)n-t

kt!k2!...kn!

holds, where k = kt + k2 + + kn and the sum is taken over all kl, k2, ... , knforwhich kt =n.

1.4. Composition of Real Analytic Functions 19

_ It is immediate that h(t) _Proof We take f (t) = and g(x) = 1 1.

go f (t) _ _(R+i 1 . But all these functions are also available as geometric series:

00

f(t) _ EtJ.j=o00

g(x) _ E RJ(x - I)J,j=o

h(t)_ 1 - t

(I-(R+1)t 1-(R+1)t

((1 + R)t tj - D1 + R)JtJ+tj=0 j=0

00

1+>R(1+R)J-ttJ.j=1

Evaluating f and h at t = 0 and g at x = 1, we find that f(J)(0) = j!,g(k)(f (0)) = k! Rk, and h(")(0) = n! R(1 + R)"-t, from which the lemma fol-lows by substituting this data into the formula of Faa di Bruno (1.13).

We now apply Lemma 1.4.1 together with Proposition 1.2.12 on the rate ofgrowth of derivatives to study compositions of real analytic functions.

Proposition 1.4.2 Let I be an open interval in R and suppose that f E C°'(1).Assume that f takes real values in an open interval J and that g E C`- (J). Theng 0 f E C°'(1).

Proof. Suppose a E I and 6 = f (a) E J. By Proposition 1.2.12, we mayassume that there are constants C, D, R, S such that, for x near enough to a andy = f (x), the inequalities

ifli)(x)I <C' R-iliand

Ig(j)(Y)1 < D . sj

hold.Now, by the formula of Faa di Bruno (1.13), the nth derivative of h = g o f is

given by

h(") (x)

=n! g(k)(Y) (fw(x)Y1 !

1. 1\ 2! n.


where k = kt + k2 + + kn and the sum is taken over all kt, k2, ... , k forwhich kt + 2k2 + + nkn = n. So we can estimate

jht' (a)I

with

`n t//1+S)

C

ktlk2n!kn!DSk

(R1)k'

(R2)k2...

\\

(Rn)kft

D k! CkRn Ekt!k2!...kn! $k

_ CD / C YE

S(1+C/S)and T=R(1+S) .

Thus h(x) satisfies the estimates of Proposition 1.2.12 that guarantee it to be areal analytic function.

Remark 1.4.3 We would be remiss not to point out that one natural way to provethat the composition of real analytic functions is real analytic is to complexify andthen notice that the composition of holomorphic functions is holomorphic (by thechain rule). However the spirit of the present monograph is, as much as possible,to prove all results by real methods. Moreover, the techniques using the formulaof of Fait di Bruno have considerable intrinsic interest.

1.5 Inverse Functions

It is natural to inquire whether the inverse of a univalent real analytic function isalso real analytic. This is too much to hope for: the function f (x) = x3 is realanalytic and univalent in a neighborhood of the origin, yet its inverse f -t (x) =X 1/3 is not even differentiable at 0. An additional hypothesis (nonvanishing of thefirst derivative) is required for the desired result to be true. These matters are bestunderstood in the context of the inverse function theorem. We now turn to thattopic.

Again we will need an identity which follows from a specific application of theformula of Fait di Bruno. First, we recall

Lemma 1.5.1 (Newton's Binomial Formula) For any real numbers a, and t with-1 < t < 1, the equation

00

jE (Oj)tj

1.5. Inverse Functions 21

holds, where

(a) a(a-1)...(a-j+j!j

and

for positive integers j

Lemma 1.5.2 For each positive integer n,

1) k. I g kZ

ik,

2(n + 1)

ktk2!k..kn!(,)k,

1(2) ...(n) (n1holds, where k = kI + k2 + + k and the sum is taken over all k1, k2, ... , knfor which k1 =n.Proof. We take f (t) = 1 - 1 --2t and g(x) = T'X. It is immediate that

h(t)=gof(t) --1

1 - 2t

= f'(t),and, hence, that

Also, we have

f(n+l)(t) = h(n)(t).

001 - (1 + (-2t)) l = -iE (j) (-2t)j,f (t)

00

E xi.

J=0

Using these series, we find that (for j > 1)t

f(i)(0) = -j! (j) (-2)t and g(k)(f(o)) = k!.

By the formula of Faa di Bruno, we have

1

-(n + 1)! (n + 1)(-2)n+l

f(n+1) (0)

= h(n)(0)n!k!kk!((I))kl

n2)"F,

k

(_()_E

(-1)kk! k, kp k(-2)nn!Ek1!k2!...kn!(',

1) (2) ...(n)


where k = kt + k2 + + kn and the sum is taken over all kt , k2, ... , k forwhich kt + 2k2 + - + nkn = n. Dividing this equation by n! (-2)', we obtainthe lemma. 0

Theorem 1.5.3 (Real Analytic Inverse Function Theorem) Let f E C' (1) forsome open interval I C R. If a E I and if f'(a) # 0, then there is a neigh-borhood J of a and a real analytic function g defined on some open interval Kcontaining f (a) such that g o f (x) = x for x E J and f o g(x) = x for allxEK.

Proof. Observe that the usual inverse function theorem of advanced calculusguarantees that a C°O inverse function g for the given f exists in a neighborhoodof a. Our job is to estimate the growth of the derivatives of g at points y nearp=f(a).

The function g satisfies the differential equation

g'(y) = hfg(y)],

where1

.f'(x)is known to be real analytic in an open interval about a. By Proposition 1.2.12,we may thus choose constants C > 0 and R > 0 such that

Ihci)(x)I 5 C . Ri

i

holds for all x sufficiently near a. By the C°O inverse function theorem, for ysufficiently near to P, g(y) will be sufficiently near to a that the estimates forhti)(x) will hold when x = g(y). Fix such a y and x = g(y). We claim that, forpositive integers j,

Iglil(y)I 5 j! (-W-1(,') (2C)'j Ri-I

(1.15)

holds. We prove this assertion by induction on j. Note that the case j = I isimmediate from

g '(y) = h[g(y)] and Ih(x)I < C.

Also, note that (- 1)i-t is positive. Supposing that (1.15) is valid for j = 1, 2,n, we estimate

Ign+t (y)I = I(h o g)cnl(y)Il k1 k

-5 n! E k! C (('' 1(2C) ... _kt!k2!...kn!Rk 1 n Rn-t

1.5. Inverse Functions 23

.(2C)n C (-1)k k! 1

k2 kn

1)nRn kl!k2!...kn!(1' ) (2) (n)

n!(-1)n(2

(n + 1)! (-1)n(fl+

) (2C)nlR"

which proves (1.15) for all positive integers j. Finally, it is easy to verify, from(1.15), that

IgU)(y)I _ D -

holds, where D and S depend only on C, R, and Ig(y)I. Thus g is analytic byProposition 1.2.12.

Remark 1.5.4 An alternative way to prove the real analytic inverse function the-orem is to complexify and then to use the complex analytic inverse function theo-rem (which can be found in many standard texts-see [KS 92a]). As was the casein the consideration of the composition of real analytic functions, we continue, asmuch as possible, to prove all results by real methods.

2

Multivariable Calculus ofReal Analytic Functions

2.1 Power Series in Several Variables

Definition 2.1.1 Set Z+ = 10, 1, 2, ...1. A multiindex µ is an element of (Z+)m;we will write

A(m)=(Z+)m,

but often the size m of a multiindex will be understood from the context.

We now recall some standard multiindex notation:

Notation 2.1.2 For

µ=(A1,A2,...,Am)EA(m)andx=(X1,x2,...,Xm)ER',

set

µ!

IPI

Xµ

ltl!, 2!...Am!,

Al +1L2+..+µm,Al K2 A,"

XI X2 ...Xm

]XI I'" IX2IA2 ... IXm I1m

aµl a122 altm

axµ' aXµ2 ' - axA1 2

26 2. Multivariable Calculus

We also extend the notation introduced in Lemma 1.2.2 by setting

m

fl(Xj)µjj=1

m

11 [Xj(Xj-1)...(Xj-Aj+1),,j=1

(AX)(A! .

For

lL _ (AI, , µ2, ..., µm) E A(m) and v = (v1, v2, ... , vm) E A(m),

we write

µ < v if µj <vjforj=1,2,...,m,µ < v ifµj<vjforj=1,2,...,m.

Lemma 2.1.3 For integers 1 < in and 0 < n and a real number -1 < t < 1, wehave

(1)

LL,,``\\ m-1 1l\m

(2)

n -1card[A(m) fl {µ : IAI = n)]

+ m= M-

(3)

00 t(n+j)!tj= do rn

Ejj j! dtn (1 -t)

We sketch the proof and leave the details as an exercise: The first conclusion isproved using the identity

G t 1/+\j/-\t j1which holds for any real t and any integer j and which should be familiar fromthe special cases occurring in Pascal's Triangle. For instance, when n = m = 2,we have

\2/ -

(11)

+

(21)

-

(11)

+

(21)

+

(22)

-

(11)

+

(21)

+

(11)

+

(21)

2.1. Power Series in Several Variables 27

and the last binomial coefficient equals 0.The second conclusion is proved using induction on m and the first conclusion,

beginning with the observation that, when m = 1, the left-hand side clearly equals1 = (a). The third conclusion is obtained by writing 1/(1-t) as a geometric seriesand then differentiating tens-by-term.

A formal expressionE a(x - a)µ ,

ILEA(m)

with of E Rm and aµ E R for each A, is called a power series in m variables.

Definition 2.1.4 The power series

E a,, (x - a)µµEA(m)

is said to converge at x E Rm if some rearrangement of it converges. More pre-cisely, the series converges if there is a function 4 Z -). A(m), which isone-to-one and onto, such that the series

00

a4,(j)(x - a)l' 0)j=0

converges.

Notation 2.1.5 For a fixed power series EA au.(x - a)µ, we denote by B the setof points x E Rm for which Iaa I Ix - a l µ is bounded. It is clear that if the powerseries converges at x, then x E B.

Definition 2.1.6 For x = (XI, x2, ... , xm) E R', define the silhouette, s(x), ofx by setting

s(x) = {(rlxt, r2x2, ..., rx) : -1 < rj < 1, j = 1, 2, ... , m).

Proposition 2.1.7 (Abel's Lemma) If the power series a,,,xµ converges ata point x, then it converges uniformly and absolutely on compact subsets of s(x).

Proof. Let K be a compact subset of s(x). Choose 0 < p < 1 such that Ik j I <plxjI holds for all k E K and for j = 1, 2, ..., m. Since X E B, we know thatthere is a constant C such that Ia,,IIxI" < C. So we have C p1µ1, Itfollows that

N

la,JIkI" _ L L lallklµ1µl__<N j--01µl=j

+<EC pj=0

m - 1


00

< EC, mm i i 1)Pij=o `

JJ

< C (m+j-1)!PiE (m - 1)!j --0

dm-1 Pm-1Cdpm-1 1-p

Since the upper bound is independent of k and N, the result is proved.

Definition 2.1.8 For a fixed power series E. aµ (x - a)A, we set

allIy-xI<r}r>0 A J

The set C is called the domain of convergence of the power series.

Remark 2.1.9 It should be clear from the proof of Abel's Lemma that 8 c C. Itis, of course, trivial that C C 9, so we have C = A.

Lemma 2.1.10 If a pointy is in the domain of convergence C of the power series

E a,,xI ,

µeA(m)

then there exists a constant C and an e > 0 such that

la I <

holds for each I.t. E A(m).

Proof. Since y E C implies y E 9, there exists an c > 0 such that

(IYII+E, Iy21+e,..., lyml+E) E B.

Since the series converges at (lyl I + E, Iy21 + E, . . . . Iym I + E), the terms of theseries must be bounded by some C. This gives us the desired inequality.

Definition 2.1.11 For a set S C Rm, we define log ]IS 11 by setting

log IISII = ((log IS1 1,1og Is21, ... , log Ism 1 ) : s = (sl , s2, ... , Sm) E S).

The set S is said to be logarithmically convex if log 11 S II is a convex subset of Rm.

2.2. Functions of Several Variables 29

Proposition 2.1.12 For a power series F_µ aµxµ, the domain of convergence Cis logarithmically convex.

Proof. Fix two points y, z E C and 0 0 so that

la I< - Cµ !!j I(IyjI+E)µJ

and

la I

" - nrj I(IzjI+E)"iboth hold for p. E A(m). Note that, since y, z, and ,l are fixed, we can chooseE' > 0 so that

(IYjI+E)a > Iyjla+E' and

hold for j = 1, 2, ..., m. Then we can choose a > 0 so that

(IYjI'`+E')(IzjII-z+E')? IYjlaIzjII-t`+a

holds for j = 1, 2, ..., m. We conclude that

CIaµI=laµl'Llaµl'-L <

H' i(a+IYjlalzjll-,)µj

Consequently, (lye I1Izt I "',1y21XIz2I'-A, , Iym h l zm l'-) E [ = C, or

X(log IYl I , ... ,1og IYm l ) + (1 - A)(1og Iz! I, ... , log Izm I) E log IICII . 11

2.2 Real Analytic Functions of Several Variables

Definition 2.2.1 A function f, with domain an open subset U c R'n and rangeR, is called real analytic on U, written f E Cw(U), if for each a E U the functionf may be represented by a convergent power series in some neighborhood of a.

Since, on compact subsets of its domain of convergence, C, a power seriesof several variables is uniformly and absolutely convergent, we conclude that areal analytic function is continuous. With the aid of Lemma 2.1.10, it is alsostraightforward to modify the proofs from Section 1.1 to prove the following:

Proposition 2.2.2 Let U, V C Rm be open. If f : U -+ R and g : V --> R arereal analytic, then f + g, f - g are real analytic on U fl V, and f/g is real analyticonUflVf1{x:g(x)96 0}.


Let v be a multiindex. Recall the notation (t: )v defined for multiindices in No-tation 2.1.2. If the power series

E aµ (x -a)"AEA(m)

is differentiated term-by-term with respect to the operator 8°/axv, we obtain thederived series

0()l' = E (lt + v)val'+v(x - a)µ .v<l' l'EA(m)

As in Section 1.1, we use the derived series to show that a real analytic functionis differentiable:

Proposition 2.2.3 Let f be a real analytic function defined on an open subsetU C Rm. Then f is continuous and has continuous, real analytic partial deriva-tives of all orders. Further, the indefinite integral off with respect to any variableis real analytic.

Proof. Let f be represented near a by the power series

E aµ(x - a)µ .

/tEA(m)

We can choose T > 0 such that the series converges at a + t, where t =(T, T_., T) E Rm. But then we see that there is a constant C such thatIal' IT1µl < C holds. Choose 0 < p < 1, and consider x with Ixj - aj I < pT forj = 1, 2..., m. For the derived series we can estimate (recall the notation (x)vdefined for multiindices in Notation 2.1.2)

µ < Ix -all'CEE(µ+v)v la,+vl Ix -aI (lt+v), Tlµl+Ivl

_ C °OE Ix - all'+( )TIvI vlt v TIµ1

1=01µl=1

-<

C 00

T 1 Di + IvDlvl 1 m 1 P'm 1

I=0

and the last series is seen to converge by the ratio test.A similar argument can be used to show that any indefinite integral of f is

represented by a convergent power series. O

Remark 2.2.4 We can now relate the coefficients of the power series representinga real analytic function to the partial derivatives of the function. By evaluating thederived series at a, we find that

a1µ1

axµf (a) = 11 al' .


It is interesting to verify that a function f defined by

f (x) = E aµ(x - a)µ$LEA(m)

for x in the domain of convergence C of the power series is, in fact, real analyticon C. To this end we will need the Taylor Formula for functions of m variables(see [SK 83; p. 285]).

Theorem 2.2.5 If f : IP' --). R is CN+I at each point of the line segment from yto z, then there is a point l; on this segment such that

181AI 1 alAl

P z) f (Y)(z - Y)'` + F - axa f ( )(z - Y),lµ +l

We will also need to know that certain special series converge.

Lemma 2.2.6 If a and b are real numbers with J a l + I b I < 1, then

(1)

(i+J)ibJ00 00

i-oj-o J 1-a-b'

(2)

EE (Ii+v)IMIbIvI(i)m

LEA(m)EA(m) 1- a- b

Proof. For any integer n, we have

()aki,= (a + b).

It follows that00 n

1

X:En=Ok-o k 1-a-b'but this is just a rearrangement of the series in (1).

Conclusion (2) follows easily from (1) and the fact that

(µV V1 =

Jn(isi i

vt.


Proposition 2.2.7 LetE aµ(x -a)A

AeA(m)

be a power series and C its (nonempty) domain of convergence. If f : C -- R isdefined by

f(x)= E a,(x-a)µ,$4EA(m)

then f is real analytic.

Proof. We may assume that a = 0. Let X E C be arbitrary. For simplicity ofnotation, we will suppose that xj 54 0 for all j. We can choose 0 < R so that

(1+R)xEC.

Then there exists a constant C such that la, II(1 + R)xlµ < C. Set

1 31"I

bV = --f(x),v! axv

and observe that

_ (k + v)V ubV - a,+vxVI

(2.1)

(2.2)

Choose 0< p < R. Consider y E W' with ly j- x j l< p I x j I for all j. We thenestimate

IbvIIY - x1°

C, [ : (A + y)v PIVIL+V" U

v! (1 + R)IA+vl

1 8IVI

V!I a- f(x)IIY-xI°

+v)°Ia,,,,Ilx11ly - xlvV

µV!

C, (1 + R)"R - p

Finally we note that, for some i; on the line segment from x to y,

) - b ( )V =f( - F 1-

vY Y x ,v<N IvI=N+1

(I+ + y)vvIv[=N+1 it


So we can estimate

If (Y) - E bl,(Y -x)v1 _<

(µ Viy)vlau+vIlxl`ÌY -xlvv<N IvI=N+1 i

< C . (lk + v)v Tlvl

v! (1 + T)IA+vl'Ivl=N+l A

and observe that the last series approaches 0 as N approaches oo.

Next, we show that the composition of real analytic functions is real analytic.

Proposition 2.2.8 If fl, f2, ..., fm are real analytic in some neighborhood ofthe point a E Rk and g is real analytic in some neighborhood of the point(fl (a), f2(a), ... , fm (a)) E Rm, then g[fl (x), 12(x)..... fm(x)] is real ana-lytic in a neighborhood of a.

Proof. We may and shall assume that a is the origin in Rk and that 0 = fl (0) _f2(0) = ... = fm (0). Suppose g (y) is given for y = (Yl, y2, ... , Ym) near0 E Rm by the power series

g(Y) = bs ys .PEA(m)

For i = 1, 2, ..., m, suppose fi (x) is given for x = (XI, x2, ... , xk) near 0 E Rkby the power series

fi (x) = ai, n x"vEA(k)

151.1

where the summation is restricted to multiindices I < laI because of the assump-tion that fi (0) = 0.

By Abel's Lemma 2.1.7, we can find Y = (YI, Y2, ... , Ym) with 0 < Yj, forj = 1, 2, ... , m, and 0 < N < oo such that

E IbsIYS <NPEA(m)

holds. Similarly, we can find X = (X1, X2, ... , Xk) with 0 < Xj, for j =1,2,...,k,and0 < M < oosuch that

Ia1 I X°f < MaeA(k)

151.1

holds for i = 1,2,...,m.Set p = min{ 1, Y1 1M, Y2/M , ... , Ym /M }. Then, for i = 1, 2, ... , m and

f o r x = (XI, X2, ... , xk) E Rk with Ixj 1 < pX j, for j = 1, 2, ..., k, we have

E Iai,al IxI° <- E Iai,a1 pI°I xaaeA(k)

151.1,EA(k)

151a1

< pM <Yi.


It follows that, for any positive integers A and B we have

mm

lbfli 11 ( E lai.a,I Ixl° N,6EA(n1 i=1 1 ajEA(k)

16158 I5jo, I5A

so the series

m [E b, L aj, a, xafEA(m) i=1 jEA(k)

151-i l )

fli

converges absolutely and can be rearranged as a power series in x that is conver-gent in a neighborhood of 0 E Rk. O

As our last result in this section, we will obtain a characterization of real ana-lytic functions using an estimate on the derivatives. For this purpose, we need thefollowing lemma.

Lemma 2.2.9 For V E A(m) and X E RM with -1 < xi < 1, for i = 1, 2, ... , m,we have

aivi ( - vi )L1 +V)VXIA v f J .

,LEA(m)aX i=1 1 - Xi

P r o o f . For i = 1, 2, ... , m, apply Lemma 2.1.3 with n replaced by vi and treplaced by xi and multiply the resulting equations. 0

Proposition 2.2.10 Let f E C°O(U) for some open U E Pm. The function f isin fact in C'(U) if and only if, for each a E U, there are an open ball V. witha E V C U, and constants C > 0 and R > 0 such that the derivatives of fsatisfy

R ,`dx E V. (2.3)8xµ (x)I

C -L.

Proof. If f is real analytic at a, then it is given by a power series, F-,, a, (x -a)A.convergent in a neighborhood of a. From Lemma 2.1.10 we obtain the estimatel aµ I < C r-1 Z1 for appropriate constants C and R > 0. Using the analogues at aof (2.1) and (2.2), we see that

dIzIf

aXA(x)

p1µ1

Du + v)v Rla+vl

holds f o r x E U with lxi - a; I < p, f o r i = 1, 2, ... , m. Fixing p < R, choosingV small enough so that each x E V satisfies the conditions Ixi - ai I < p, fori = 1, 2, ..., m, and replacing C by a larger constant, we see that (2.3) followsfrom Lemma 2.2.9.

2.3 The Implicit Function Theorem 35

Conversely, if the estimates (2.3) are satisfied, then we easily see that the serieswith coefficients aµ defined by µ! a,,, = is convergent in a neigh-borhood of a and it follows from the proof of Proposition 2.2.7 that the functiondefined by this series agrees with f near a.

2.3 The Implicit Function Theorem

Arguably the most significant theorem of multivariable calculus is the implicitfunction theorem. The basic form of the implicit function theorem is the asser-tion that a system of in equations in n variables, of sufficient smoothness, andsatisfying an appropriate nondegeneracy condition, can be used to define in of thevariables as functions of the other n - in variables. It is natural to inquire as tothe minimum smoothness required. The well-known answer is that the equationsmust be at least C 1. The usual proofs are based on that hypothesis.

On the other hand, one may also inquire as to what happens in very smoothcases. Typically, the Ck or CO0 category is considered and the result is that theimplicit functions are as smooth as the equations. Less often is the C' categorydealt with. Here we will consider the implicit function theorem in that real analyticcategory. The interested reader can find a wide range of information about theimplicit function theorem in the monograph [KP 02].

The proof of the theorem below uses Cauchy's method of majorization. Thismethod is also the principal tool used in proving the celebrated Cauchy-Kowa-lewsky theorem for existence and uniqueness of solutions to partial differentialequations with real analytic data (see Section 2.4). In fact the implicit functiontheorem can be obtained as an application of the Cauchy-Kowalewsky theorem,as shown in Section 2.5.

In Theorem 2.3.1, we will consider power series expansions for a functionV(x, y), where x e RN and y E R. So we will have powers xa, for a a mul-tiindex, and yk, for k a nonnegative integer. We shall write such a power seriesexpansion as

W(x, y) = >aa.kxayka,k

It will be understood in this context that a ranges over all multiindices and kranges from 0 to oo. If we write ao,o, it will therefore be understood that the first0 is an N-tuple (0, ..., 0) and the second 0 is the single digit.

Theorem 2.3.1 Suppose the power series

F(x, y) = Eaa,kxayk (2.4)a,k

is absolutely convergent for lxl < R1, lyl c R2. If

ao,o = 0 and ao, 1 34 0, (2.5)


then there exist 0 < ro and a power series

.f (x) = F, Caxa

lal>0

such that (2.6) is absolutely convergent for Ix I < ro and

(2.6)

F(x,f(x)) = 0. (2.7)

Proof. It will be no loss of generality to assume ao,l = 1, so that (2.4) takes theform

F(x, y) = y + (aa,o + aa.IY)xa + E aa,kxaYk (2.8)laI>0 lal2:O,k>2

Introducing the notation b0,0 = bo,l = 0 and, for all other a and k, ba,k =-aa,A, we can rewrite the equation F(x, y) = 0 as

y = (ba,o + ba, 1Y)xa + E ba,kxa yk (2.9)lal>o lal2:0.k?2

or y = B(x, y), where

B(x, Y) = >2 (ba,O + ba.l Y)xa + E ba,kxa yk . (2.10)lal>0 lal?O,k>2

By Lemma 2.1.10, there exist finite, positive numbers C and R such that

Iba,kI < C Rlal+k (2.11)

holds for all multiindices a E A(N) and all k = 0, 1, ....Substituting y = f (x) into (2.9) with f (x) given by (2.6), we obtain

E Caxa ba.Oxa+ba,lcpxa+p

I0I>0 lal>O,IpI>0k

+ rL. ba,kxa cpxp (2.12)lal_O.k_2 1p

If all the series in (2.10) are ultimately shown to be absolutely convergent, thenthe omier of summation can be freely rearranged Assuming that absolute conver-gen-e., We can equate like powers of x on the left-hand and right-hand sides of(_.1_2) and obtain the following recurrence relations:

C" = be,0. (2.13)

W bere

ci = (1.0.0....,0.0),

= tt).QO.....0.1).FN

2.3. The Implicit Function Theorem 37

This first recurrence allows us to solve for each cei. Next we indicate how eachcoefficient ca of higher index may be expressed in terms of the by, j and indices cpwith index of lower order. In point of fact, let us assume inductively that we haveso solved for ca, for all multiindices a with lal ¢ p. Then, fixing a multiindex awith lal = p and identifying like powers of x, we find that

ca+e1 ba+ej,o + E by,1cpIpI>O.IYI>0.p+y -+rj

+ Y' M(P......IYI>O.k>2.

Here M(fl...... $k) is a suitable multinomial coefficient and the superscripts onthe Ps are not exponents-they are only for identification purposes. Observe thateach such M is positive. We can see by inspection that all of the multiindices oncoefficients c that occur on the right-hand side have size less than or equal to p.This is the desired recursion.

While the recurrence relations (2.13) uniquely determine the coefficients c j inthe power series for the implicit function, it is also necessary to show that (2.6) isconvergent. The easiest way to obtain the needed estimates is by using the methodof majorants described in the following definition:

Definition 2.3.2 Consider two power series in the same number of variables:

O(XI,X2,...,Xp)00

Oi1,j2,...,jpX1'X22...Xpp, (2.14)

*(XI,X2,...,Xp)00

il j2 jp'`j,,j2,...,jpx1 X2 2 ...Xpi1 12,...,jp=0

(2.15)

We say that 41(xl, x2, ... , xp) is a majorant of (Nx1, x2, ... , xp) if

l0n,/2..... i,, l :5 'Gj1,12,....ip (2.16)

holds f o r all jl , j2, ... , jp.

Resuming the proof of the theorem, we note that, because all the coefficientsM(fi'.... , #k) in (2.14) are positive, it follows that if

G(x, y) _ ga,kxlYk,ja I>0,k>0

(with go,o = go,I = 0) is a majorant of


and if

B(x, y) = 1: (ba,0 + ba, I y)x" + E ba,kxa yk

lal>0 lal_0,k>2

h(x) = E h"x" 2.17)

solves

Ial>I

h(x) = G[x, h(x)], (2.18)

then h(x) will be a majorant of f (x). Consequently, if the series (2.17) for h(x)is convergent, then the series (2.6) is convergent and its radius of convergence isat least as large as the radius of convergence for (2.17).

We take

G(x,y) = -C -CRy + E C Rlal+k xa yk (2.19)a,k

N

= -C-CRy+C(1-Ry)-' fl (1-Rxj)-t. (2.20)j=1

By (2.11), (2.19), and the fact that bo,0 = b0,I = 0, we see that G(x, y) is amajorant of B(x, y). By (2.20), we see that, for this choice of majorant, y = h(x)is a solution of the equation

N

(1-Ry)[C+(I+CR)y]=Cfl(1-Rxj)-t. (2.21)j=t

The equation (2.21) is quadratic in y, so it can be solved explicitly. The solutionis clearly analytic at x = 0.

Theorem 2.3.1 covers just the special case of one dependent variable and ar-bitrarily many independent variables. To give the complete theorem in the realanalytic case we apply an inductive method due to Dini (see [DU 07]).

Our proof of the general implicit function theorem will be simplified notation-ally if we use the following lemma from linear algebra.

Lemma 2.3.3 Let A be an n x n real matrix. Then there exists an invertible realmatrix U such that UA is upper triangular.

Proof. The matrix A can be reduced to echelon form by a sequence of elementaryrow operations. A square matrix in echelon form is necessarily upper triangular,and each elementary row operation can be accomplished via left multiplication byan invertible matrix. The result follows.

Now we describe a preliminary simplification that we will use in the generalimplicit function theorem.


Notation 2.3.4 Suppose that we are given a set of equations

ft(XI,x2.....xt;Y1,y ....,Y.)=0, i = 1,2,...,n, (2.22)

where the functions ft, f2, ... , f, are real analytic, and suppose also that (p; q) =(P1. P2, , pt; qt, q2, . q") is a point at which all the equations (2.22) holdand at which we have

iiYf2

detY. Yi

0.

2A Y- 2Ad y, ri are

(2.23)

In this situation, we can think of the functions fi(p; ) as giving a mappingF : R' -> R" defined by

y r-> F(y) = (ft (p; y), f2(p; y), . f. (p; y)). (2.24)

Following this function by the linear transformation corresponding to left multi-plication by the invertible matrix given by Lemma 2.3.3 with A = DF(p; q), butwithout changing notation, we can assume that

af;(p; q) = 0 whenever i > j. (2.25)

ay;

After this preliminary modification, we have

ay; 9Y 0 a

rl n ay. ay.

and, consequently,

(2.26)

i=1 aYi0 (2.27)

at the point (p; q).


Theorem 2.35 (Real Analytic Implicit Function Theorem) Let a system of realanalytic equations be given as in Notation 2.3.4. Then there exists a neighborhoodU C Rt of p and a set of real analytic functions Oj : U -> R, j = 1, 2, ... , n,suchthat4j(p) = qj, j = 1,2,...,n,and

h[x;01(x),02(x),...,On(x)] =0, i = l,2,...,n, (2.28)

hold for x E U.

Proof. We argue by induction on n. The case n = 1 is of course Theorem 2.3.1.Suppose now that n > 1 and that the theorem is true with n replaced by n - 1.

As indicated in Notation 2.3.4, we can assume that (2.27) holds, so we have

afvn (p; q) 0. (2.29)

Let us introduce the notation y' = (Y1, y2, ..., y1); then Theorem 2.3.1 isapplicable to the equation

fn (x; Y'; yn) = 0 (2.30)

at the point (p; q'; qn), where we are treating the variables x1, x2, ... , xt andyi, y2, ..., Yn-1 as independent and only the variable yn as dependent. Thus, byTheorem 2.3.1, there is a neighborhood V C Rt+rt-1 of (p; q') and a real analyticfunction 0, : V -> R such that *(p; q') = qn and

f [x;Y;0,(x;Y')]=0 (2.31)

holds for (x; y') E V.Notice that if (2.31) is differentiated with respect to yj, 1 < j < n - 1, then

we find thatafn afn a*+ -- = 0. (2.32)aYj ay" aYj

Evaluating (2.32) at x = p, y' = q', and using (2.25) and (2.29), we see that

a*(p; q') = 0ayj

holds for j = 1, 2, ... , n - 1.

(2.33)

Now, for each i = 1, 2, ..., n - 1, define the function h; by setting

hi(xi,x2,...,xt;Y1,Y2,...,Yn-1) = frlx;Y,*(x;Y)). (2.34)

Consider the system of equations

hi(xi,x2.....xt;y1,y2.....yn-1)=0, i = 1,2,...,n- 1.For j = 1, 2, ..., n - 1, by (2.33), we have

(2.35)

ahi(p; q) = afi (p; q'; qn) + afi (p; q'; qn)

a*(p; q') = afi (p; q)

ayj aYj ay" ayj aYj(2.36)


and, accordingly,

as ah ahlaye

a ...aye-l

ah Lhz ah2an ay.-I

ahnR ah,,-laye an ...

2Aay, aye

2Aay, aye

ay, aye

ay.-IMay, J

an.-I

Here the partial derivatives in the left-hand determinant are evaluated at (p; q')and those in the right-hand determinant are evaluated at (p; q).

By induction, there exist a neighborhood U' c Rt and real analytic functionscJ : U' --) R such that

hi [x; 01(x), 02(x), .... On-I (x)] = 0, i = 1, 2, ... , n - 1, (2.38)

hold for x E U'.Set 4 (x) = (x, 01(x), ..., On-1(x)) and

U=U'n(P-1(V)

and define ¢n : U -> R by setting

On (x) = 0[x; 01(x), 02(x), ... , On-I (x)]-

(2.39)

(2.40)

By the definition of the hi, i.e., (2.34), we see that the desired equations (2.28)hold.

The proofs above apply equally well in the complex setting because we haveshown the absolute convergence of the power series. Likewise, the real analyticimplicit function theorem can be obtained by complexifying and applying thecomplex analytic theorem. In the complex setting, an alternative proof can bebased on the Cauchy integral formula. We refer the reader to [KS 92a] for a de-tailed consideration of various kinds of analytic implicit function theorems in thecomplex setting.


2.4 A Special Case of the Cauchy-KowalewskyTheorem

The point of the Cauchy-Kowalewsky theorem is that, for a real analytic partial(or ordinary) differential equation with real analytic initial data, a real analyticsolution is guaranteed to exist. This result is arguably the most general theoremin the lore of partial differential equations. The original papers are [CA 92; pp.52-58] and [KS 75]. The technique used in the proof is called majorization: Onesets up a problem which is already known to possess an analytic solution and usesthe resulting convergent power series to show that the power series arising for theoriginal problem is smaller and thus is convergent (see Definition 2.3.2). We haveused this technique in previous proofs, for example, in the proof of the implicitfunction theorem. Our discussion will follow that of Courant and Hilbert [CH 621.

It is simplest to prove the theorem for a certain type of system of quasi-linearfirst order equations with initial data given along a coordinate hyperplane. Laterwe show how to generalize this result. Let the functions Fi, j,k be real analytic onsome neighborhood of the origin in Rn, and let the functions Oi be real analyticon some neighborhood of the origin in Rm, where i and j range from 1 to n and kranges from I to m. We also assume that the functions ¢i vanish at the origin. TheCauchy problem is to find functions, u 1, u2, ..., un, defined in a neighborhoodof the origin in Rm+1 and satisfying

au(n m

E Fauj

(2.41)ay ,

j=1 k=1aXk

ui(X,0) = Oi(x)- (2.42)

We call (2.42) the initial condition and the right-hand side of (2.42) the initialdata (or Cauchy data).

In this section, we will prove the following result:

Theorem 2.4.1 (Cauchy-Kowalewsky, Special Case) If the system of partialdifferential equations

aui n m au j_

syj=1 k=1 axk

and the initial conditions

with

(2.43)

ui (X, 0) = 4i (X) , (2.44)

00) = 0are real analytic at the origin, then there exist functions u 1, u2, ... , un that arereal analytic at the origin and satisfy the differential equations (2.43) and theinitial conditions (2.44).

2.4. Cauchy-Kowalewsky Theorem-Special Case 43

Proof. The proof will proceed in four steps.

Step 1: Computing the Coefficients of the SolutionThe plan is to write

ui (x, Y) = LThe Cauchy problem gives us enough data to compute the coefficients a. j uniquely.The difficulty is in showing that the series is convergent.

To see how the coefficients are determined, let the functions Fi, j,k and Oi berepresented by power series as

E i j.k d

$

Oi(W ) _ c' xrY

where in the first equation the multiindex P has n components and in the secondequation the multiindex y has m components. By hypothesis, we have co = 0.Note that by differentiating the initial data we find that

ui(x, 0) _

a0i W,a1 Iand this information, when substituted into the differential equations, gives us

a(x,0)=F_ EFi.j.k(01(x),...,On (x))a0j(x)axk

ayj=1k=1

Evaluating at x = 0, we see that

ao,0 =°i

,0 =

ca=0,ca, for lal = 1 ,

n mi i.j,k j

ao,1 = 0 cet ,j=1 k=1

where we have used the notation ek for the multiindex with lek I = 1 and with 1 asits kt° entry. The coefficients a'' j are obtained inductively as follows: We createm + 1 equations by differentiating the equation

auin m

auj(x, Y) = 1: 1: Fi,j.k(uI(x),...,un(x))-(x,Y)

ay axkj=1k=1

with respect to each of the variables xl, x2, ... , Xm, y. Likewise, we createm (m + 1)/2 independent equations by differentiating each of the equations

auiax . (x, 0) = aui (x),

I I


with respect to each of the variables x1, ... , xm. These are evaluated at x = 0,y = 0 to obtain the coefficients asp with lal = 2, the coefficients as 1 withlal = 1, and the coefficients a0 2. Subsequent differentiation and evaluation atx = 0, y = 0 gives the complete set of coefficients for the expansion of the uiabout (0, 0).

It will not be necessary for us to obtain the explicit formula for the variouscoefficients a,,, t ; instead it suffices to note that each a,, t is a polynomial func-

tion of the coefficients and c`c and each such polynomial has nonnegativecoefficients. We write

i q.r, csaa,t = Pai.t(6p

y

and we note that P.',, depends only on finitely many of the arguments bpq.r, C's.

We emphasize that the key facts are that the form of Pat is independent of thechoice of the functions Fp,q,, and ¢s and the coefficients of Pat are nonnegative(in fact nonnegative integers).

Step 2: A Majorizing ProblemTo make use of the observations in Step 1, we will use the method of majorants(see Definition 2.3.2). That is, we will find another problem

avi

Y

nn m avjEFGi.j,k(v1,...,un)a , (2.45)

vi(x,0) =j=1 k=1

ifi(x) (2.46)

for which the coefficients of the Gi, j,k exceed the absolute value of those forFi,j,k, for which the coefficients of * exceed the absolute value of those for 0i,and for which the problem (2.45)-(2.46) is known to have real analytic solutionsvi. The coefficients of vi will then exceed the absolute value of the coefficientsaa,t found above, and thus the series for each ui will converge.

Recall that by Lemma 2.1.10 there exist positive constants R and C such thatthe inequalities

Ibpj,kI RI61 < C,

Ic,I Rlrl < C

hold. Set

and

00

Gi,j.k(u) = G(u) = C (I - W)-1 = C (W)tR Rt-o

a v 00

Gi(x)=*(x)=CR (I R)CJ(a)'

Rt=1

2.4. Cauchy-Kowalewsky Theorem-Special Case 45

wheren m

W=EU , o=Ext.t=1 t=1

Note that the coefficients of G;,j,k exceed those of F;,j,k and the coefficients ofyi; exceed those of O;.

By symmetry, the solution of (2.45) will be of the form

V1 (X. Y) = V2(x, Y) _ ... = Vn (x, Y) = v(x, Y)

The function v must solve the equation

- = nG(v,... , v)axY k=1 k

and satisfy the initial condition

v(x, 0) = *(x) .

That is

aV nVl -l m aV= nC (1 - -1

ay R k-1 axk

a o -1v(x,0)=CR (1-R)

(2.47)

(2.48)

Step 3: Generating Solutions of a Differential EquationThere is a procedure for generating solutions of a partial differential equation ofthe form

m

av = E Ak(V) ex . (2.49)Y k=1 k

First one chooses functions H(v) and Bk(u) so that

m

H(v) = > Ak(V)Bk(V)k=1

holds. Then one introduces another arbitrary function W (v). If

M

W'(v) - H(v) - E Bk(v)xk :A 0

k=0

holds, then the implicit function theorem applies and we see that the functionV(X1, X2, ... , xm, y) can be defined implicitly by the equation

m

W(v) = H(v)y + E Bk(V)Xk . (2.50)k=0


By computing the partial derivatives of v implicitly, we easily verify that v is asolution of (2.49). (When we apply this method in the next part of the proof, itwill be possible to solve (2.50) explicitly for v without appealing to the implicitfunction theorem.)

In solving an initial value problem, the functions W(v), H(v), and Bk(v),k = 1 , 2, ... , m, cannot be chosen quite so freely. Nonetheless, the precedingapproach will allow us to solve the equation (2.47) while satisfying the initialcondition (2.48).

Step 4: Solution of the Majorizing ProblemApplying the method of Step 3 to the specific problem

for which

we may set

Bk(v) 1, fork = 1,...,m,_1

H(v) = mnC (1 - R)

We see that a solution can be defined by

provided

It is routine to see that

W(v)=mnC(1-R)_1

y+a

a 1

W C Ra

(1 - R) =a.

W(t) C+tand to then conclude that the solution is real analytic at the origin, as required. O

8v nv-l mm` 8va - nC(1-R

k=1) `8xk,

Y

v(x,0) CaR(1

or -1

R)

Ak(v)=nC(1- nRv)-1, fork=1....,m,

Remark 2.4.2 In Section 2.8 we will return to the discussion of the Cauchy-Kowalewsky theorem, after we have introduced the machinery needed to stateand prove a very general form of that theorem.

2.5 The Inverse Function Theorem 47

2.5 The Inverse Function Theorem

In this section, we give a proof of the multivariable inverse function theoremusing the special case of the Cauchy-Kowalewsky theorem proved in the previoussection.

Theorem 2.5.1 (Real Analytic Inverse Function Theorem) Let F be real ana-lytic in a neighborhood of a = (a1, ... , an) and suppose DF(a) is nonsingular.Then F-1 is defined and real analytic in a neighborhood of F(a).

Theorem 2.3.5 gives another approach to the implicit function theorem whichimplies the inverse function theorem. Here we are examining the inverse functiontheorem from a different point of view. The proof of Theorem 2.5.1 is by inductionon dimension; this is legitimate since we have already proved the inverse functiontheorem for real analytic functions of one variable (Theorem 1.5.3). The proof ofthe following special case contains the heart of the argument.

Proposition 2.5.2 Let n be a positive integer. Suppose that the real analytic in-verse function theorem is true for functions of n real variables. If F : R1+1 -+R"+1 i s real analytic near (0, ... , 0) w i t h F(0..... 0) = (0, ... , 0) and is suchthat DF(O,... , 0) is nonsingular and F(R" x {0}) C R" x {0}, then F-1 isdefined and real analytic near (0, ... , 0).

Proof. We are assuming that the real analytic inverse function theorem has beenproved for functions of n variables. Now let us treat the case of n + 1 variables. Letthe component functions of F be F1, ... , Fn+1. Define the function f : R" -+ R"by setting

f(x1,...,xn)=(F1(xi,...,xn,0),...,Fn(XI,...,xn,0))

There is thus a real analytic function g defined near 0 E R" such that

g(f(x))=x forxER".

By the usual inverse function theorem for CI functions, F-1 is defined in aneighborhood of (0, ... , 0) E Rn+1; let us write F-1 in terms of its componentfunctions as (u 1, ... , un+, ). We know that

i = 1,...,n+1,

and that

au; IdFpaYn+1 axq

hold, where Ai,n+1 is the algebraic function of the components of an (n + 1) x(n + 1) matrix which gives the entry of the inverse matrix in the is row and


(n+1)' column. Thus we see that the component functions u I, ...,u,,+1 of F-tsatisfy a real analytic system of partial differential equations with real analyticinitial data. Further, the initial value problem is of the restrictive type dealt within the previous section. Therefore the functions u l , ... , un+1 are real analytic ina neighborhood of (0, ... , 0).

Proof of Theorem 2.5.1. Now we can do the inductive step in the proof ofTheorem 2.5.1. Suppose the theorem is true for functions of n real variables andsuppose that F : H21+1 + 1EY1+1 is real analytic near a = (al, ... , an+I) and issuch that DF(a) is nonsingular. It is clearly no loss of generality to assume that ais the origin and F(a) is also the origin. By an orthogonal change of coordinatesin the domain, we may assume that

aFn+t(0) = 0, for l < i < n,

ax;

andaFn+l

(0)960.axn+l

Let the component functions of F be FI , ... , Fn+I and once again define thefunction f by setting

f(xl,...,xn)=(FI(xl,...,Xn,0),...,Fn(xl,...,xn,0))

Since the matrix of partial derivatives of components of f at the origin is thematrix M given by

Mi.j= ±!(0), 1<i,j<<n,i

we see by the inductive hypothesis that there is a real analytic function g definednear 0 E R' such that

g(f(xi,X2,...,Xn)) _ (XI,X2,...,Xn).

We now define F by setting

F(X) = (Ft (x), ... , Fn(x), Fn+l (x) - Fn+l (g[FI (x), ... , Fn(X)), 0)) .

Clearly we have

ax' (0) ax'(0), for 1 >

aFn+I 8F( aFn+I a9k aFi(°)axj (0) =

ax j(0) -

axk (0) aye(0)

aXk.1=1 j

2.5. The Inverse Function Theorem 49

= 0, for l < i < n,

aFn+1

°aFi °

0

aFn+t a9k aft0 ° °( ) - ( ) - ( ) ( ) ( )aXn+1 axn+1 k.t=1 aXk axn+1aYt

a F` (0),axn+1

So we see that det(DF(0)) = det(DF(0)) 96 0. Since we also have

F(R"x(0))cR"x{0},we may apply Proposition 2.5.2 to obtain a mapping G which is real analytic near0 E R"+1 and inverts F. But then, if one defines G by setting

G(u 1, ... , un+1) = G(u 1.... , un, un+1 - Fn+1(g[ul, ... , un], 0))

one sees that G is real analytic near 0 E Rn+1 and that

G(Fl(x), F2(x),..., Fn+1(x))

= G (Fi(x), ... , Fn(x), Fn+1(x) - Fn+1(g[F1(x),... , Fn (x)], 0))

= W w) = X,

so G inverts F.

In Section 2.3, we proved the real analytic implicit function theorem using themethod of majorants and the inductive argument of Dini. The implicit functiontheorem is often obtained as a corollary of the inverse function theorem. Usingthe usual proof of this implication (see [RW 79]), we have a second proof of thereal analytic implicit function theorem.

Theorem 2.53 (Real Analytic Implicit Function Theorem) Suppose thatF : R'+n -+ R" is real analytic in a neighborhood of (xo, yo), forsome xo E R'and some yo E R". If F(xo, yo) = 0 and then x n matrix with entries

aFi(xo, YO)ayj

is nonsingular then there exists a function f : R' --)- R" which is real analyticin a neighborhood of xo and is such that

F(x, f (x)) = 0

holds in a neighborhood of xo.

Remark 2.5.4 Using the machinery that we have developed, it is possible to for-mulate and prove a real analytic rank theorem (see [RW 79] or [KP 02]). We shallnot provide the details here.


2.6 Topologies on the Space of Real AnalyticFunctions

Let 0 C RN be a connected open set. Let C"(S2) denote the linear space of realanalytic functions on U. Here we consider topologies on this linear space.

In the study of holomorphic functions of one or several complex variables, itis natural to consider the topology of uniform convergence on compact sets-equivalently, the compact-open topology. Call that topology U0. If k = 1, 2, ... ,then we let Uk denote the topology of uniform convergence of the kth derivativeson compact sets. Because of the Cauchy estimates, the topology Uj and Uk areequal for any j and k.

Unfortunately, the situation described in the last paragraph does not obtain forreal analytic functions. There are no Cauchy estimates in this category (indeedone of the many rationalizations for complex analysis is that one can replace eachoccurrence of x by z in the power series expansion of a real analytic functionand thereby pass from the real analytic category to the much more fruitful com-plex analytic category), and it soon becomes clear that no elementary topologyon C°'(S2) will result in a complete, linear (Frechet) space. Certainly CI(Q) iscontained in Ck(n) for each k, but none of those topologies renders C°'(S2) as aclosed subspace.

The natural thing to do in these circumstances is to equip C' (S2) with eitheran inductive limit or a projective limit topology. The topology of uniform conver-gence on compact sets is already a projective limit topology. Our intention is togeneralize on that observation. We initially follow the exposition in [SE 81].

Let us recall the general setup for the two types of topologies just mentioned.We simplify the presentation here by only taking the direct and inverse limits oversequences of objects (rather than over a directed set of objects). This will sufficefor our purposes.

In our context, therefore, a direct system is a collection of sets {Sj } and a col-lection of mappings (f : Si -> Sk) for j < k such that

(i) The mapping fi = id.

(ii)Ifj<k<lthen fkofJ=fJ:Sj->St.

Then we consider the collection of objects, called dir{S), f ), consisting of in-dexed collections of mappings { fj : Si -+ Z}, for some set Z, such that fj =fk o if for j < k. The "morphisms" in this situation are mappings h : Z -> Z'such that h o fj = f when { fj : Si -). Z} and { f : Si -> Z'} are objectsin our category. A terminal object E in dir{S1, f } is called the direct limit orinductive limit of the system (Si, fjk}. In fact we can give an explicit realization

of this terminal object as follows: Let S be the disjoint union of the sets Si. De-fine an equivalence relation on S as follows: Suppose that x E Si C 9 and

2.6. Topologies on the Space of Real Analytic Functions 51

y E Sk C S. We say that x y if there is an index t such that e > j and t > k- -andfj(x)=fk(y).Set

If we let ii : Si --> S be the canonical injection map and 7r : S -' S/quotient map then we may define, for each j, the canonical map

the

As a result we have

fj='roii:S'-->E.

E=Ufj(Si). (2.51)

Observe that

hence

j

fkof; =fj

fj(S') c fk(Sk) 2.52)

whenever j < k. Thus we may think of the identity (2.51) on E as an increasingunion of the family { 1. Furthermore, if x, y E S with x E Si andy E Sk, then the relation

fj(x) = fk(Y) (2.53)

implies (by the definition of the canonical map) that f (x) = f (y) in E. In sum-mary, the given elements x, y E S for which (2.53) holds are indeed equivalent.Therefore, by the definition of our equivalence relation, there is an index t withf > j and t > k such that

fj(x)=fk(Y)See [MA 86] for further discussion of these ideas.

In the case that each Si is a topological space, we may induce a topology on Eas follows. Consider the diagram

fi+I

Si Sj+1 ...

fi ',, f fi+

We endow 6 with the finest topology that will allow every map f j to be continuous-see [MJ 921. This topology on E is called the direct limit topology on E.

Obversely, an inverse system is a collection of sets {Sj} and a collection ofmappings {fj : Sk - Sj} for j < k such that

(i) The mapping fi = id.

(ii) If j < k < t, then f 0 fk = fJ . St -+ Sj.


Then we consider the collection of objects, called inv(Sj, ff ), consisting of in-dexed collections of mappings {f j : X -> S j ), for some set X, such that f j =if o fk for j < k. The "morphisms" in this situation are mappings h : X - X'such that f o h = fj when { fj : X -* S} and { f : X' -> Sj } are objectsin our category. An initial object F in inv{Sj, f } is called the inverse limit or

projective limit of the system {Sj, f }. In fact we can give an explicit realizationof this initial object as follows: Consider the cartesian product

00

F1 Si.j=1

We define the following distinguished subset of S:

f-{x=(xj)ES:xj=

Let 7r j be the canonical projection map of S onto Sj. Then we have the mappings

fjnjlY: Sj.

In general the map f j is not onto. We find that

fj=f;ofkfor any j < k. See [MA 86] for further discussion of these ideas.

In the case that each S j is a topological space, we may induce a topology on Fas follows. Consider the diagram

11 +1Si - Sj+1fj fj+t

We endow .' with the coarsest topology that will guarantee that every map n j iscontinuous-see [MJ 92]. This topology on .F is called the inverse limit topologyon.F.

Now consider a fixed compact set F C RN C CN and let us fix a sequence ofopen sets Ui in CN ? RN, each of which contains F, such that

U'DU2D...JFand such that fl j Uj = F. Equip the space O(Uj) of holomorphic functions onUj with the topology of uniform convergence on compact sets. Then there arenatural mappings f : O(Uj) -+ O(Uk), whenever j < k, consisting simply ofrestriction. We claim that we now have a direct system, with the role of Si being

2.6. Topologies on the Space of Real Analytic Functions 53

played by O(U j) and the role of Z played by Z = C O(F). The map fj : Si _+ Zwill of course be a restriction. It is obvious that these Si and if form a directsystem, and we may form the direct limit O(F). Since every real analytic functionon F has an extension (just replace the x j's by z j's) to an open set V C_ C" suchthat V D F, we see that O(F) may be identified in a natural way with C°'(F).Thus the direct limit topology on O(F) gives a topology on CO'(F). In practiceit is convenient to choose this direct limit topology to be the finest locally convextopology that makes every f j in our construction continuous. See [MJ 92] formore on the choice of topologies.

If, in the last paragraph, we replace the compact set F C RN by an opendomain f2 C RN, then we obtain the direct limit topology on CO'(S2). Specifically,we fatten 0 C RN up into open sets Uj C CN with Uj fl RN = S2 and performthe direct limit construction described above.

For the inverse limit topology on CO'(S2), we exhaust 0 by an increasing se-quence of compact sets:

We let SJ = O(Kj), equipped with the direct limit topology developed in the lastparagraph but one. Then there are natural (restriction) mappings if : Sk -> Sj,whenever j < k. Then {Sj, f } forms an inverse system. The inverse limit canbe identified in a natural way with CO'(S2), and we thereby have equipped CO'(S2)with an inverse limit topology. In practice it is convenient to choose this inverselimit topology to be the finest locally convex topology that makes every f j in ourconstruction continuous. See [MJ 92] for more on the choice of topologies.

It is in fact possible to write down a more concrete rendition of the topologyon the space of real analytic functions. We do so by examining the direct limitdefinition. Fix a bounded domain 0 C RN and a decreasing sequence of opensets of CN, U1 D U2 D D S2, with fljUj = Q. Shrinking 92 and the Uj'sif necessary, we may assume that the analytic functions under consideration arebounded, with bounded derivatives, on each of these sets. Then the terminal object6 that we constructed may be thought of as a collection of elements {(hj, Uj)},where h j is holomorphic on Uj and hj I uk = hk when k > j. Fix a compact setK C 0 C Uj+t. Then a sub-basis element for the topology on Cm(S2) will be aset of the form

WK = [f real analytic on S2 : sup J(a"/ax") f (x)I < Cj[3j(K)]-10`1XEK

for all multi-indices a!} .

Here Cj is a constant that depends on the supremum of hj on Uj+t and aj (K) isthe Euclidean distance of K to the complement of Uj+t in CN. We leave it to thereader to verify that the topology so generated is equivalent to the direct limit thatwe defined earlier by a more abstract construction.


Now the big theorem in this subject is that the direct limit topology and theinverse limit topology on 0°'(S2) are equal. The details of this assertion may befound in [MA 66]. Further discussion appears in [DV 001.

A further important result is that the space of real analytic functions is closedin the specified topology. Again, we omit the details of this assertion. With thespecified topology, the space C°'(n) is a complete, ultrabornological, reflexive,nuclear, separable, webbed space and its dual is a nuclear LF-space.

2.7 Real Analytic Submanifolds

In the next section we shall state and prove a very general form of the Cauchy-Kowalewsky theorem which involves real analytic submanifolds of R". In thissection we give the basic definitions.

Definition 2.7.1 A set S C R" is called an m-dimensional real analytic subman-ifold if, for each p E S, there exists an open subset U C R' and a real analyticfunction f : U - R" which maps open subsets of U onto relatively open subsetsof S and which is such that

p E f (U) and rank [Df (u)] = m, Vu E U,

where Df (u) is the Jacobian matrix off at u.

Remark 2.7.2 Definition 2.7.1 requires a real analytic submanifold to be locallyparametrizable. Additionally, the condition that f map open subsets of U ontorelatively open subsets of S implies that the manifold is regularly embedded inR" (as is shown in Proposition 2.7.3).

Following [FH 69], we note that there are a number of equivalent definitionseach of which is useful in certain circumstances; we record them in the nextproposition.

Proposition 2.7.3 Let S be a subset of R". The following are equivalent:

(1) S is an m-dimensional real analytic submanifold.

(2) For each p E S there exist an open V with p E V C R", a real analyticdiffeomorphism a : V -> R", and an m-dimensional linear subspace L ofR" such that

o(SnV)=Lnima.

(3) For each p E S there exist an open V with p E V C R" and a real analyticfunction g : V -+ Rk, with k > n - m, such that

S n V = g-'[g(p)] and rank [Dg(v)] = n - m, du E V.

2.7. Real Analytic Submanifolds 55

(4) For each p E S there exist an open V with p E V C R', a convex openU C R', and real analytic maps ¢' : V --> U, , : U -- V such that

S n V = im ><r and $' o ,G is the identity on U .

(5) For each p E S there exist an open V with p E V C Rn and an orthogonalprojection n : R' --> R' such that

n I (S n V) is one-to-one, n (S n v) = 17 (V) is convex,

[n I (s n v)]-1 : n(V) -> Rn is real analytic,

D[n I (S n V)]-'n(p) is the adjoins of 11.

Proof (1) = (2) Let f be the function whose existence is guaranteed by Defini-tion 2.7.1. For i = 1, 2, ... , m and u E U set

v1(u) =of

(u).aui

Let ua be such that f (uo) = p. Then the set of vectors

{v1 (uo),v2(uo),...,vm(U.))

is linearly independent and can be enlarged to a basis for Rn by the addition ofvectors Vm+1, vm+2, , vn. Define a function F : U x Rn-` -+ Rn by setting

n-m

F(u, w) = f(u)+>2 WkVm+k,k=1

n-mU E U, w = (wt, w2, ... , wn_m) E R

By construction DF(ua, 0) is nonsingular, and the real analytic inverse functiontheorem may be applied to obtain (2).

It is trivial to see that (2) implies (3), while (3) implies (1) follows from the realanalytic implicit function theorem. Finally, it is easy to see that (2) (4) (5)

(4) (1).

It is essential to have a notion of what it means for a function defined on a realanalytic submanifold to be real analytic.

Definition 2.7.4 Let S be a real analytic submanifold of Rn, and let h : S -> R.We say that h is real analytic at p E S if, for f as in Definition 2.7.1 and for apoint uo such that f (uo) = p, the function h o f is real analytic at uo.

Remark 2.7.5 In Chapter 6, we shall consider an abstract real analytic manifold.By this is meant a paracompact Hausdorff space with a locally Euclidean structuresuch that the transition functions are real analytic. It turns out that there is no trueincrease in generality: Every abstract real analytic manifold can be embedded,by a real analytic embedding, in a Euclidean space of sufficiently high dimen-sion. However, this is a deep theorem. We shall discuss it, and related results, inSection 6.4.


2.7.1 Bundles over a Real Analytic Submanifold

It will be important to be able to define various real analytic vector bundles overS, starting with the tangent bundle and the normal bundle. To begin this process,we define the tangent space and normal space at a point.

Definition 2.7.6 Suppose S c R" is a real analytic submanifold. Associated witheach point p E S are two linear subspaces of R", the tangent space denoted byTSp and the normal space denoted by NSp. The tangent space is defined bysetting

TSp =span (v1,v2,...,vm),

where f is as in Definition 2.7.1, uo is such that f (uo) = p, and

vi =8f

(uo) .8ui

The normal space is the orthogonal complement of TSp in R".

In general, a bundle, is a triple (E, n, B) consisting of two topological spacesE and B and a continuous map tr : E -+ B, which also satisfy other conditions,to be discussed below. The space E is called the total space, the space B is calledthe base space, and the map rr is called the projection of the bundle. For eachp E B. n' 1 (p) is the fibre of the bundle over p.

For (E, n, B) to be a k-dimensional real vector bundle, each fibre n (p) mustbe a k-dimensional vector space over R and, for each p E B, there must exist anopen neighborhood W p of p in B and a continuous map

Op (2.54)

which is a homeomorphism onto its image and has the property that, for eachp' E U. Op restricted to p' x Rk is a vector space isomorphism onto rrt (p').Because the simplest bundle is a cartesian product and because Op maps n- (WP)homeomorphically to a cartesian product, the homeomorphisms 4p are said toestablish local triviality.

In the definition of the tangent and normal bundles, the following notation willbe used.

Notation 2.7.7 Suppose S C R" is a real analytic submanifold of dimension m.

(1) LetpbeapointofS.

(2) Let f be as in Definition 2.7.1 and let uo be such that f (uo) = p. Set

vi = 8ffor i = 1.2..... m

8ui


(3) Complete vi, v2, ..., vm to a basis for R" by adding vectors

(4) Let U be a small enough open neighborhood of u,, that

(u),Of ......2f-(u), Vm+l, Vm+2, ... un (2.55)

au l au Bum

is a basis for R" for all u E W.

(5) Let 0I(u),v2(u)....,u"(u) be the basis for R" obtained by applying theGram-Schmidt orthogonalization procedure to the vectors in (2.55). Note thatif the Gram-Schmidt procedure is applied to the vectors in (2.55) in the orderin which the vectors are given there, then the vectors

UI(u).v2(u), vm

will span the same subspace as do

8u (u), a 22 (u), ... . aum (u) .

We now define the tangent bundle over a submanifold.

Definition 2.7.8 Suppose S c R" is a real analytic submanifold of dimension m.The tangent bundle over S, (TS, rrT , S) is defined as follows:

(1) The base space of the tangent bundle is the submanifold S.

(2) The fibres of the tangent bundle are the tangent spaces TSP.

(3)

(4)

(5)

As a set, the total space, TS, of the tangent bundle is the disjoint union of theT SP, that is,

TS=((p,v): pES,

The projection 2rT of the bundle is the mapping that sends (p, v) to p.

To establish the local triviality of the tangent bundle, we use Notation 2.7.7as follows: Let WP be f (U) and define OPT : WP x RI -+ TS by setting

OT(PI,Cl,C2,...,Cm)= 1P,ciUi(./-1(P'))) . (2.56)i=1

The topology on T S is defined by requiring that the maps OPT be homeomorphismsonto their images. Also, equation (2.56) shows that TS is a real analytic subman-ifold of R"+m


The definition of the normal bundle over a submanifold is similar to that of thetangent bundle:

Definition 2.7.9 Suppose S C R" is a real analytic submanifold of dimension m.The normal bundle over S, (NS, nN, S) is defined as follows:

(1) The base space of the normal bundle is the submanifold S.

(2) The fibres of the normal bundle are the normal spaces NSp.

(3) As a set, the total space, NS, of the normal bundle is the disjoint union of theNSp, that is,

NS=((p,v): pES, vENSp).

(4) The projection of the normal bundle, nN, is the mapping that sends (p, v) toP.

(5) To establish the local triviality of the normal bundle, we use Notation 2.7.7 asfollows: let Wp be f (U) and define ON : Wp x RI-m -+ NS by setting

Op(P.C1,C2.....Cn_m)= (n'. C(f(P))J . (2.57)i-m+l

The topology on NS is defined by requiring that the maps ON be homeomor-phisms onto their images. Also, equation (2.57) shows that NS is a real analyticsubmanifold of Rn+(n-m)

The vi used above in defining the tangent and normal bundles will be used be-low is proving the less well-known characterization of real analytic submanifoldsgiven in the next theorem. For the theorem, we must agree that a 0-dimensionalreal analytic submanifold is a (discrete) set of isolated points.

Theorem 2.7.10 Suppose S is a connected subset of R". Then S is a real analyticsubmanifold if and only if there exists a real analytic map retracting some opensubset of R" onto S.

Remark 2.7.11 This theorem also holds in the C* category (see [FH 69; 3.1.20]).

Proof. First, let us suppose that there is an open set U and a real analytic mapm : U -> S retracting U onto S. To determine the dimension of the submanifold,set

m = sup(rank [ DO (x)] : x E U).

To use suggestive language. we say that the good points are those for which therank of the differential is m; set

G=Ul(x:rank (D4(x)]=m}.


Since the rank is the size of the largest square submatrix with nonvanishing deter-minant, we see that G is open, so s fl G is open relative to S.

In case m = 0, we see that 0 is constant on each component of G, but since Sis also connected, we see that S is a singleton.

We now suppose that m > 1. Since 0 o ¢ = ¢, we have

D4[4(x)] o D4(x) = D¢(x),

so, for x E G,

m = rank [D¢(x)] < rank [DO(O(x))] < m.

Thus ¢(G) c S n G, so S n G is nonempty.For X E S n G, we have Do (x) o Do (x) = Do (x) and rank [Do(x)] = m, so

Do (x) must be the identity map on its image. Thus, for an x E S n G, 1 is a rootof the characteristic polynomial with multiplicity m, and this is certainly a closedcondition. Thus S n G is also closed relative to S. Since S is connected, it followsthat S=SnG.

Suppose p E S = SnG. Letting { v1, v2, ... , vm } be them orthonormal eigen-vectors of D4 (p) associated with the eigenvalue 1, it follows that the function fdefined by

f(uI,u2,...,um) =O(P+Euivi)shows that S is a real analytic submanifold at p.

Conversely, suppose that S is a real analytic submanifold. Let p be a point ofS and let f : U -+ R" be as in Definition 2.7.1. Let ua be such that f (ua) = p.Proceeding in a manner similar to the first part of the proof of Proposition 2.7.3(2),set

vi(u)=af(u)aui

Then enlarge the set of vectors

{vl (uo), v2(uo), ... , vm(uo)}

to a basis f o r R" by the addition of vectors um+I , vm+2, , v.. In a neighborhoodof up, the set (v) (u), v2 (u), ... , vm(u), vm+I, , is a basis for R". We applythe Gram-Schmidt orthogonalization procedure to obtain an orthonormal basis{v1(u), .... V. (u)} which has the additional properties that

(i) {v1 (u), v2(u), ... , vm (u)} is an orthonormal basis for T S foul,

(ii) {vm+1(u), vm+2(u), ... , v (u)} is an orthonormal basis for NSf(u),

(iii) each vi (u) is a real analytic function of u.


Let F : U x ltn_m -+ Rn be defined by

n-mF(u, W) = f(u) + Wkuk+m, U E U, W = (W1, W2, ... , Wn-m) E R"-m .

k=1

Of course, DF(uo, 0) is nonsingular, so the real analytic inverse function the-orem may be applied. We conclude that the map 0 = f o lI o F-1, where11 : U x 1P"-` -+ U is projection on the first factor, is real analytic. Note thatin a sufficiently small neighborhood of p, ¢ coincides with the "nearest point"retraction. Since the nearest point retraction is well defined in a neighborhood ofS (see [KP 99; Lemma 1.2.5]), we obtain the desired real analytic retraction.

It is clear from the preceding theorem that a function is real analytic on a realanalytic submanifold if and only if it extends to a real analytic function on anopen subset of the ambient space.

The vector fields v1(u), u2(u), ... , vn(u) satisfying (i), (ii), and (iii) in theproof of the preceding theorem are useful in defining what it means for sections ofthe normal and tangent bundles over S to be real analytic. The term section of thetangent bundle simply means a function a : S -+ TS such that, for each p E S,a(p) E TSP. Equivalently, a section a satisfies n o a = id.

Definition 2.7.12 Let S be a real analytic submanifold of Ill" and let f be as inDefinition 2.7.1. A real analytic section of the tangent bundle, a, is a section suchthat each of the functions v ; (u) [a o f (u)] is real analytic f o r i = 1, 2, ... , m.Here denotes the usual inner product in R".

Similarly, one defines

Definition 2.7.13 A real analytic section of the normal bundle , Y1, is a sectionsuch that each of the functions v;(u) [q o f(u)] is real analytic for i = m +1,m+2,...,n.

Since the general Cauchy-Kowalewsky theorem involves higher derivatives,we will need to define structures suitable for the consideration of higher deriva-tives in the context of submanifolds. To this end, consider that the derivative ofa function is the best linear approximation to the function. If one is consistentin thinking of derivatives in this way, then higher derivatives must be multilin-ear functions, and in fact, higher derivatives are symmetric multilinear functions.Thus we are motivated to introduce the symmetric algebra of a vector space.

Briefly, the symmetric algebra of the vector space V is the quotient algebra ofthe tensor algebra of V,

®*V=9000®m V,

determined by dividing out by the two sided ideal generated by all commutators

x®y-y®x.

2.8 The General Cauchy-Kowalewsky Theorem 61

The symmetric algebra of V is a graded algebra denoted O. V, and inside thisgraded algebra is the span of the symmetric products of m elements of V denotedby Om V. We have

O.V=®m oOmV.

The dual space of Om V is denoted Om V.A function on the m-fold cartesian product Vm is said to be symmetric if its

value is unaltered by interchanging arguments of the function; the function is saidto be m-linear or multilinear if it is linear in each of its arguments. There is anatural linear isomorphism between Om V and the vector space of symmetric m-linear forms on the m-fold cartesian product Vm. A thorough discussion of thesematters can be found in [FH 69; Sections 1.9-1.10].

For the purpose of stating the general Cauchy-Kowalewsky theorem we need todefine the normal symmetric algebra bundle and sections of the normal symmetricform bundle. In the bundle definitions that follow, we will define the total spaceas a set, but we omit the detailed discussion of local triviality. For each p E Slet O.(NSp) = ®-0 Oi (NSp) denote the symmetric algebra of NSp, and letO'(NSp, Rm) = ®-0 O` (NSp, Rm) denote the algebra of symmetric forms onNSp with coefficients in R. Then the normal symmetric algebra bundle is

O.(NS) = {(p, v) : p E S, v E O.(NSp)},

and the normal symmetric form bundle with coefficients in pm is

O*(NS, Rm) = ((p, µ) : p E S, / E o*(NSp, Rm)}.

Definition 2.7.14 A real analytic section of the normal symmetric i-form bun-dle with coefficients in IFm is a function a : S O'(NS, Rm), with a(p) E0' (NSp, Rm), such that the functions

(a o f (u))rii (u), uh (u), ... , uil (u))

are real analytic for each choice of (j1, J2, ... , j; ) C_ {m + 1, m+2..... n}. Heref is the function the existence of which is guaranteed by Definition 2.7.1.

2.8 The General Cauchy-Kowalewsky Theorem

It will be useful for us to think of the kth derivative of a k-times continuouslydifferentiable function u : IIt" -+ pm as being, at each point p E R", a symmetric,multilinear function on k-tuples of elements of R' taking values in 11Pm. This ideais illustrated in the next example. The interested reader should also see [FH 69;3.1.11).


Example 2.8.1 Suppose u : R3 -a R is C2. The second derivative of u is oftenidentified with the Hessian matrix

f uxx uxy uxzH = uYx uYY uYz

uu uzY uzz

where the subscripts denote partial differentiation. The Hessian matrix can in turnbe used to define a function Bu : R3 x R3 -+ R by setting

where is the usual Euclidean inner product and Hw is the usual matrix product.Since BH is symmetric (by the equality of mixed partial derivatives) and bilinear,we know that if vt, v2, wt, w2 E R3 are such that vi 0 wt = v2 0 w2, thenBH (VI, wt) = B1 (v2, w2). Thus, we can consider BH to be a real-valued linearfunction on 02(R3, R). 0

The general situation is described by the next definition.

Definition 2.8.2 Let W c R" be open and let u : W -+ R' be a Ck function. Weexpress u in terms of its component functions by writing u = (U 1, u2, ... , um).Fix a point p E W and let k vectors vt, V2, ..., Vk E R" be given. For i =1 , 2, ... , k, we suppose the vector vi has the components Vi, 1, v 1 2 . . . . . vi,". Wedefine the pairing

(Dku(P); v1, v2, ... , vk) E R'

by setting its j'h component equal to

a u.

F ... F, axi, axi2 ... axik vt.ri U2.i2 ... yk,ik , (2.58)F,i1=1 i2=1 ik=1

where the partial derivatives are evaluated at p.

Remark 2.8.3 Because the right-hand side of (2.58) is linear in each of the visand is unchanged if the vis are interchanged, we can identify Dku(p) with anR' -valued symmetric, multilinear function on k-tuples of elements of R".

We can also identify Dku(p) with a linear function on Ok(R"). Specifically,given ut , v2, ... , vk E R", we set

(Dku(P)) (vt 0 V2 0 ... 0 Vk)

equal to the element of R' with ju' component given by (2.58). 0

2.8. The General Cauchy-Kowalewsky Theorem 63

The space of R"'-valued symmetric, multilinear functions on k-tuples of ele-ments of R" is denoted 0k(R", R'°) A differential equation of order k on R' canthus be thought of as an equation of the form

F(x, u(x), Du (x), ... , Dku(x)) = 0 E R9 ,

whereF:R" xR' xO(R",Rm)x... x0k(R",Rm)->RQ.

It is harder to describe the general initial data (also, called Cauchy data) for adifferential equation if the data is to be specified on a real analytic submanifold:This is the situation that we have in the general Cauchy-Kowalewsky theorem.We let S be a real analytic submanifold of R". Let Oo : S -' R. Then we canseek a solution u of the differential equation which also satisfies

u(p) _ fo(p), for p E S f1 U.

But for a differential equation of order k we should also specify the derivativesup to order k - 1 . W e do this as follows: For each i = 1, ... , k - 1, let 4'i be afunction such that, for each p E S, Oi (p) is a symmetric multilinear function oni -tuples of elements of N Sp with values in R. In the terminology of Section 2.7,these are sections of the normal symmetric form bundle with coefficients in R.We assume that each Oi is real analytic. To fully determine the ith derivative ofu, we must know not only the effect on vectors normal to S, but also on vectorstangent to S. Since the functions 4j, for j < i, are defined and differentiable onS, they can be used to obtain the needed information: For vt, ... , yr E T Sp, andw t, ... , Ws E NSp, with r + s = i, we require

(Dù(P); Vt,... , ur, wi, ... , ws) = ((Drss(P); V 1 ,- . , vr); wt, ... , ws) .

Likewise, much of the behavior of Dku(p) is similarly restricted if the initialconditions are to be satisfied. What is not determined is

(Dku(P); w t, ... , Wk)

when wt, ... , wk E NSp. Assume that S is a d-dimensional submanifold. ThenNSp is of dimension n-d. Simple combinatorial reasoning shows that the numberof unordered k-tuples of basis elements from NSp must then be

k+n-d-1n-d-1

Thus the dimension of the space of multilinear symmetric functions on k-tuplesof normal vectors is

7k+n-d-1m

n-d-1


Accordingly, one requires

1q = m .n-d-l+n-d-

and one would like to be able to solve F = 0, by the inverse function theorem,for the undetermined normal part of Dku(p). If this is possible we say that theequation is non-characteristic. Even after the normal part of Dku(p) has beenfound, it is still necessary to require that the equality of mixed partial derivativesholds for derivatives of order higher than k. If this condition is satisfied, then wesay that the equation is consistent.

Theorem 2.8A (Cauchy-Kowalewsky) Suppose S c R" is a real analytic sub-manifold of dimension d. Suppose 4o : S - R' is real analytic on S and fori = 1, ... , k - 1, 4i is a real analytic section of the normal symmetric formbundle O'(NS, R"'). If

F:R"xR'"xO(R",R'")x...xOk(R",R'"),R9,

with1k+n-d-1

q = mn-d-1

is real analytic, non-characteristic, and consistent, then there exists a function uwhich is real analytic in an open set U with S c U and satisfies

u(p) = Oo (p), for p E S,

Du(P) INS =P

01(P), for p E S,

D2u(P)IO2(NSp) 02(x), for p E S,

Dk-lu(P)IOk-I(NSp) = Ok-l(x), for p E S,

F[x, u(x), Du(x), ... , Dku(x)] = 0, for x E U.

Proof. The first step in the proof is to apply the characterization (2) from Proposi-tion 2.7.3 to rid ourselves of the various bundles and reduce the problem to a moreconcrete form: W e write R" = Rd x R"-d, so points in R" are (xl, ... , xd, yl . . ,

y,d), and after solving for the highest normal derivative, the differential equa-tion becomes

kau=Fa x y u au atu (2.59)ava

, .. . axday-T

2.8. The General Cauchy-Kowalewsky Theorem 65

The initial conditions become

au

aY1

u(x,0) = 004),

= 0(1'0'...'0)'

k-1 (x, 0) = W(0.....O,k-1) .

ayn-dTo be able to apply Theorem 2.4.1, the independent variables must not occur

in the differential equations. Certainly, the independent variables can occur in(2.59). To overcome this obstacle we introduce new (dependent) variables wi,,1,where i 1 E (1, 2, ... , d), and wi2,2, where i2 E 11, 2, ... , n - d). For each i 1 E

(1, 2, ... , d), the variable mi,,1 will play the role of xi, and, similarly, for eachi2 E (1, 2, ... , n - d), the variable wi2,2 will play the role of yi2. The equationsand initial conditions that these variables must satisfy are as follows: For i1 =1,2,...,d,i2 = 1,2,...,n -d,i3 = 1,2,...,n -d, we have

awi,,1

ayi2

awi 2

= 0,

3,

ayi2= ail i3 ,

wi1,I(x,0) = xi,wi3,1(x,0) = 0,

where Sit i3 is the Kronecker delta.To complete the reduction to the special case in Theorem 2.4.1, we continue in-

troducing additional (dependent) variables. These new variables will be wi, wherei E (1, 2, ... , m), and wi,a,p, where i E 11, ..., m} and where a and fl are mul-tiindices with I < lal + 1,61 < k and 101 < k - 1. These new w's satisfy thefollowing equations:

wi = Ui,awi

wi.(1,0,....0).(0.....0) =ax1

wi. (0,....0), (0,....0.1)

wi, (2,0,...,0), (0,....0)

ak-lu

awi

8Yn-d8w;.(1.O.....O),(0.....0)

ax1

a Wi,(0,...,0),(0.....O,k-1)

axd

66 2. Multivanable Calculus

f o r i = 1, ... , m. The above equations involving derivatives with respect to they's are appended to the system of differential equations to be solved, while equa-tions involving derivatives with respect to the x's are used as initial data (deter-mined by the initial data for u and the various derivatives of u).

Theorem 2.4.1 is applied with the w's defined above playing the role of theu's of Theorem 2.4.1. The solution is built up inductively. Begin by setting y- =Y3 = = Yn-d = 0 and applying Theorem 2.4.1 to extend the functions to aneighborhood of yj = 0. This provides the initial data to solve in a neighborhoodof y2 = 0 with Y3 = = Yn-d = 0. After n - d applications of Theorem 2.4.1,the real analytic w's will have been defined and, consequently, a real analyticfunction u is defined in a neighborhood in R".

Remark 2.8.5 The Cauchy-Kowalewski theorem has been influential in the the-ory of partial differential equations. Even in such modern developments as thetheory of analytic wave front sets (see [SJ 82]) one sees some of the ideas andtechniques that have been presented here.

3Classical Topics

3.0 Introductory Remarks

Prior to the middle of the nineteenth century, mathematicians thought about func-tions very much as do beginning calculus students today: A function is given bya formula. As an extreme example, Leonhard Euler (1707-1783) addressed oneof the great questions of the late eighteenth century-whether an arbitrary set ofdata for the wave equation (i.e., any function representing the initial position of avibrating string) has an expansion in terms of sines and cosines-as follows: Onepossible initial configuration for the string on the interval [0, 27r] is

xZ/n if 0<x<7r;OW 27r-x if n<x<27r.

However ¢ is not one function but two functions (reasoned Euler). Thus it couldnot possibly be expanded as a sum of sines and cosines (each a single function).See [LR 47] for more on this matter.

While from our modern perspective the argument of Euler is preposterous, it issobering to note that in his classic text published in 1928 Osgood felt compelledto point out that

10 if x <0,OW - e-l/xz

if x > 0

really is a function, and is therefore a legitimate example of a CO0 is not a real an-alytic function (see [OW 28; Chap. 3, §9]). Mathematicians of the late nineteenthand early twentieth century struggled hard to come to grips with the facts that

68 3. Classical Topics

(i) The power series of a C°O function, expanded about a point a, need notconverge except at a;

(ii) Even if the power series converges in an open neighborhood of a, it may notconverge to f.

Since the nineteenth and early twentieth centuries had been devoted in part to see-ing that the Fourier series of any reasonable function converges to that function,it came as quite a shock that nothing could be further from the truth for the powerseries of a C00 function. In fact one can use elementary considerations to see thatthe collection of real analytic functions on the interval (a, b) forms a set of firstcategory in C°O(a, b).

We devote this chapter to reviewing some of the results from the period 1890-1935. Many of the results and investigations from that time were either ill-advisedor have been superseded by modern insights. We shall give little space to those.(A charming treatment of some of the issues considered in those times appears in[PJ 05; pp. 214-219].) But a number of results that are quite striking have beenessentially lost to the modern mathematician. In order to give the flavor of theinvestigations that were made, we shall devote some detail to several of these andshall mention several others. There is no intention here to be complete. We striverather to provide the reader some guideposts to the classical literature.

3.1 The Theorem of Pringsheim and Boas

Much of the material in this chapter draws its inspiration from the lovely arti-cle [BR 89] by Ralph Boas. Although we shall attempt to cover a much largerterritory, Boas's article was our entry point to the topics discussed.

The example of the non-analytic function 0 in Section 3.0 has the property thatit is real analytic on both the left and right half-line. However the power seriesexpansion of ¢ about a point t > 0 has radius of convergence t. Thus the radiusof convergence shrinks to zero as t moves from the right toward the non-analyticpoint 0. In light of this example, we ask the following question: If a CO0 functiong on an interval (a, b) has the property that the radius of convergence of the powerseries of g about any t E (a, b) is at least d > 0, can we hope that g is real analyticon (a, b)?

A classical theorem of Alfred Pringsheim [PA 93] answers the question affir-matively. Forty years after Pringsheim's proof was published, R. P. Boas, whilestill an undergraduate, discovered that Pringsheim's proof was fallacious. Boasthen succeeded in finding a correct proof (see [BR 89] for details of this matter).Pringsheim's theorem was formulated in extremely old-fashioned language whichwould be inappropriate to the present book. We state it instead as follows:

3.1. The Theorem of Pringsheim and Boas 69

Theorem 3.1.1 (Pringsheim-Boas) Let f be a C°O, real-valued function on anopen interval I= (a, b). Let aj (t) = f t>> (t)/j ! be the ju` Taylor coeffcient of fattE1.

For each t E I, let

p(t) =1

limsupj.00laj(t)Illj

be the radius of convergence of the power series

00

E aj(t)(x - t)t

j=a

at t. If there is a 8 > 0 such that p(t) > 8 for all t E I, then f is real analytic on1.

Before proving the theorem, we consider a weaker result, the proof of whichillustrates the basic technique.

Proposition 3.1.2 With the same notation as in Theorem 3.1.1, if [c, d] C (a, b)with c < d and p(t) > 0 for each t E [c, d], then there is a nonempty opensubinterval of [c, d] on which f is real analytic.

Proof. Setting

Ft = [c, din {x : If(°)(x)I < (n!)t', n = 1, 2, ...}

f o r f = 1 , 2, ... , we note that each Ft is closed. By hypothesis we have

00

[c, d]=UFt,t-o

so by the Baire category theorem (see, for example, [RW 87; Chapter 5]) some Ftmust contain a nonempty open subinterval of [c, d]. But then, by Lemma 1.1.8,on that open subinterval we have exactly the estimate needed to show that f isreal analytic.

Corollary 3.13 With the same notation as in Theorem 3.1.1, if p(t) > O for eacht E (a, b), then f is real analytic on an open dense subset of (a, b).

The real usefulness of the lower bound on the radius of convergence is capturedin the following lemma. This is a variant of a lemma used by Hoffman and Katzin their proof of the Pringsheim-Boas theorem (see [HK 83]).

Lemma 3.1.4 With the same notation as in Theorem 3.1.1, if f is real analyticon (c, d) with a < c < d < b, if 0 < p(c), and if for some x E (c, d) it holds


that x - p(x) < c, then

00

f(t) = E aj(c)(t - c)j

j=0

holds for all t E [c, c + p(c)).

Proof. Fix such an x E (c, d). Setting

00

g(t) = E aj (x)(t - x)j ,j=o

we see that g is real analytic on (x - p(x), x + p(x)). Since f and g and all theirderivatives agree at x, they must be equal on

(c, d) fl (x - p(x), x + p(x)) = (c, min{d, x + p(x)}).

By continuity, we also have f (j)(c) = g(j)(c) for j = 0, 1, .... We know fromSection 1.2 that

9(i>' c) (t - c)j = E a j (c)(t - c)j00

j=o ' j=a

converges tog on (c - p(c), c + p(c)) n (x - p(x), x + p(x)) _ (a, P). Sinceg = f on [c, min{d, x + p(x)}) c (a, fi), the lemma is proved.

Remark 3.15 A similar result clearly holds for the right-hand endpoint of theinterval [c, d].

The proof of the theorem will require a second application of the Baire categorytheorem.

Proof of Theorem 3.1.1. Arguing by contradiction, suppose there are a and Awith a < a 0 is such that [a - 8, j6 + 8] c(a, b).

Let B denote the set of points in [a, fl] at which f is not real analytic. Then Bis closed and thus may be considered in its own right as a complete metric space.

SetFt=Bfl{x:If(')(x)I :5 (n!)f", n=1,2,...},

for e = 1, 2, .... Note that each Ft is closed. By hypothesis, we have

00

B = U Ft,t=o

3.1. The Theorem of Pringsheim and Boas 71

so by the Baire category theorem there must be some I and some open intervalI c (a, P) such that

00Bf11cFt.Since we can always replace I by a smaller interval around any of the points inB fl I, it will be no loss of generality to also assume that the interval I has lengthless than or equal to min(S, 1/(21)). Fix such a value of t and such an openinterval I.

Consider any point x E I \ B. There is some maximal open subinterval (c, d)of I which contains x. It is possible that c = a or d = !4, but not both becauseB fl 1 54 0. For definiteness, let us suppose a < c E B. Then the hypotheses ofthe previous lemma are satisfied, so

00

f (t) = c)ii=a

holds fort E (c, d). With the notation (j)n as in Section 1.2, we can use Lemma1.2.2 to estimate

If(n)(x)I00

If (j)(C) I1:(J)n 1 Ix - cIj-n

j=n '00

E(i)n fj Ix - cii-n

j=n00

< enE(I)n(LIx-CI)j-nj =n00

< to E(j)n (1/2)j -ni=n

= 2n+1 (n!) to

It follows that, for every x E 1, the estimate

If (n)(x)I < (n!)(4t)n, n = 1, 2, ...

holds, which suffices to show that f is real analytic on 1. This contradicts the factthat 00 Bn1.

In fact the argument presented here suffices to prove the following strictlystronger, but somewhat more technical, result:

Theorem 3.1.6 Let f be a COO, real-valued function on an open interval I =(a, b). Let a j (t) = f (J)(t)/j ! be the jth Taylor coefficient off at t E 1.

For each t E I let

1

P(t) = lim suPn-.00 Iai(t)I lji


be the radius of convergence of the power series of g at t. If for each point t E Iwe have p(r) > 0 and lim infx.i p(x)/Ix - t I > 1, then f is real analytic on I.

Due in some measure to the influence of Hardy and Littlewood, mathematiciansof the period described here did not study functions of several real variables. How-ever it is not difficult to see that the theorem of Pringsheim and Boas also holds inRN. (In fact as an exercise the reader may wish to use the separate real analytic-ity ideas in Section 4.3 to prove such an N-dimensional result.) As an intuitivelyappealing sufficient condition for real analyticity, Pringsheim and Boas's theo-rem is reminiscent of an important, but unfortunately rather obscure, "converseto Taylor's theorem" that we now record. We refer the reader to [KS 83] and thereferences therein for discussion and detailed proof.

Theorem 3.1.7 Let f be a function defined on an open a domain U c RN.Suppose that there is a C > 0 such that for each x E U there is a kt' degreepolynomial P., (h) with

If(x + h) - Px(h)I < CIhlk+t

for h small. Then f E Ck(U) and the Taylor expansion to order k off aboutx E U is given by Px(h).

One may view Pringsheim and Boas's theorem as the order-infinity analogueof this last result. The converse to Taylor's theorem has proved to be an importanttool in global analysis (see [AR 67]).

In the next section we consider the behavior of a real analytic function at theboundary of its domain of analyticity from another point of view (that of Besi-covitch). In the third section we present some work of Whitney which will bothunify and supersede that which went before.

3.2 Besicovitch's Theorem

An old theorem of E. Borel is as follows (see [HL 83]):

Theorem 3.2.1 Let (a j } -0 be any sequence of real or complex numbers. Then

there is a C°O function on the interval (-1, 1) such that f U) (0) = j ! a j .

In other words, the Taylor coefficients of a C°O function at a point may be spec-ified at will. The next theorem, due to A. Besicovitch [BA 24], gives a powerfulextension of Borel's result:

Theorem 3.2.2 Let (aj } ' and {b j } ' be sequences of real or complex num-bers. There is a C°O function f on the closed interval [0, 1] such that

(1) f is real analytic on the interval (0, 1);

3.2. Besicovitch's Theorem 73

(2) f(j)(0) = j! aj and f(j)(1) = j! bj

It is convenient, and correct, to think of the function f in the theorem as being therestriction to the interval [0, 1] of a function that is C°O on the entire real line. Theconclusion is not only that one may specify all derivatives of f at both endpointsof the interval, but that the function can be made analytic on the interior of theinterval.

By applying Besicovitch's theorem to both sides of the point 0 E R we mayobtain the following strengthening of E. Borel's theorem:

Corollary 3.2.3 Let {a j 0 be any sequence of real or complex numbers. Then

there is a CO0 function on the interval (-1, 1) such that f (J)(0) = j! aj and fis real analytic on (-1, 0) U (0, 1).

We shall now present the proof of Besicovitch's result. The heart of the matteris the following lemma:

Lemma 3.2.4 Let (a j ) be a given sequence of real or complex numbers. Thenthere is a function f that is C°O on [0, oo) and real analytic on (0, oo) and suchthat f(j)(0)=aj.

Proof. We may and shall assume the a j are all real. Formally define the series

_ COIx dt fx /11 dt

F(x)co+x +el

o c_1+ t+e2

0 Jo 2 t dtl+...

Here the numbers co, cl, ... are positive numbers to be specified. Also the num-bers co, E1, ... will each be specified later to take one of the values -1, 0, 1.

Fix an interval [0, A], A > 1. Notice that, for x E [0, A], the jth summand ofour series does not exceed

A" dtfA f1J_l fJ It dt2 dtj_2dtj_1 . (3.1)

The integral (3.1) equals

Aj-1/2 2Aj1 3 2'-1 (j - 1)!

Of course the series

E(j-1)!converges. We conclude that the series named F(x) converges uniformly on [0, A]regardless of the choice of the c's and c's.

°O '2A


A straightforward imitation of the argument just presented allows one to checkthat the formally differentiated series F'(x) converges uniformly, and likewise forall higher order derivatives. It follows that the series F defines a C°O function on[0, oo).

The simplest way to see that F is real analytic on (0, oo) is to think of x asa complex variable and verify directly that the complex derivative exists (the es-timates that we just discussed make this easy). Alternatively, one may refine theestimates in the above paragraphs to majorize the j1h derivative of F by an ex-pression of the form C C. Li j!. In any event, F is plainly analytic when x > 0.

It remains to see that the parameters cj, e j may be selected so that the deriva-tives of F take the predetermined values a j at x = 0. Differentiating F at 0 andsetting the ju' derivative equal to aj leads to the equations

ao = Eo(co)-112

Of = 2E0(c0)-3/2+El(cl)-1/2

0(2 =2

. 2 .EO(c0)-512 - 2E1(CI)-3/2 + E2(C2)-1/2

We may rewrite these equations as

Eo(c0)-112 = 010 (3.2)

EI (C1)-1/2 = a1 + 1 CO(co)-312 (3.3)

E2(C2)-112 = a2 -3

22 eo(co)-5/2 +21.61 (CI)-3/2 (3.4)

Now we reason as follows: If ao = 0, then we set co = 0 and the choice of cois moot; otherwise, set co = sign (ao) and co = (CIO)-2. Next we choose El to be-1, 0, or 1 according to whether the right-hand side of (3.3) is negative, zero, orpositive. In case El = 0 the choice of cl is again moot; otherwise (3.3) determinesthe value of cl from known data. We continue in this fashion, choosing the E j insuccession so that the equations are consistent with the signs of known data. 0

Lemma 3.2.5 Let (a j } be a given sequence of real or complex numbers. Thenthere is a function f that is C°O on [0, 11 and real analytic on (0, 1) and such thatfU)(0) = aj, and fU)(1) = 0, all j.

Proof Let h (x) be a nonnegative C00 function on R which is supported in [0, 1 ],real analytic in (0, 1), and satisfies f h(x) dx = 1. Set

H(x) = 1 -Ix

h(t)dt.00

3.3. THE THEOREMS OF WHITNEY 75

Then H is C1 on R, real analytic on (0, 1), and

H(0) = 1, H(1) = 0,

H(°)(0) = H(i)(1) = 0, for j = 1, 2, ... .

Choosing F according to the previous lemma so that F(i)(0) = ai for j =0,1,2.... and setting f = FH, we see that

f(J)(0) _ (FH)(i)(0)

((j)

F(i) (0) = ai,

f(i)(1) _ (FH)(i)(1)

(i)F(1(1)Hu-i(1) = 0. 0

Proof of Theorem 3.2.2. Let F be a function that is real analytic on (0, 1) andC°O on [0, 1] and such that F(i)(0) = j!a1 for every j and F(i)(1) = 0 for all j.Likewise, by symmetry, let G be a function that is real analytic on (0, 1) and C°Oon [0, 1] and such that G(i)(0) = 0 for every j and G0)(1) = j!bi for all j. Setf = F + G. It is now obvious that f satisfies the conclusions of the theorem.

In the next section we shall give Hassler Whitney's dramatic generalization ofthese results to N dimensions.

3.3 Whitney's Extension and ApproximationTheorems

When compared with higher dimensions the analysis of one real variable is rel-atively simple at least in part because any open set in R is the disjoint union ofcountably many open intervals. It was Hassler Whitney [WH 34] who discoveredthe correct multivariable analogue for this fact. He was able to exploit it to proveseveral important extension and approximation theorems. Even today Whitney'stheorems, and especially his techniques, exert a decisive influence over the direc-tions that real analysis has taken.

The key geometric result that plays the role for RN of the decomposition ofopen sets in R into intervals is the Whitney Decomposition Lemma which westate below.

0

Notation 3.3.1 We will use the notation Q to denote the interior of the set Q.


Lemma 3.3.2 (Whitney Decomposition) Let SZ be a proper open subset of RN.Then there are closed cubes Qk such that

(1) S2 = UQk;0 0

(2) Qj n Qk= 0 if j 0 k;

(3) For each k, 2 diam (Qk) < dist (Qk, `S2) < 4. diam (Qk) .

Notation 3.3.3 In what follows, when Q C RN is a cube with center xo andc > 0 we let cQ denote the set {x E RN : Xo + (1/C)(x - Xo) E Q}. In otherwords, cQ is the cube with center xo and with sides parallel to those of Q andhaving side-length c times the side length of Q itself.

Lemma 3.3.4 The Whitney decomposition of an open set SZ C RN can be takenso that no point of 0 is contained in more than 12N of the closed cubes.

The Whitney decomposition is generally applied in conjunction with the fol-lowing lemma:

Lemma 33.5 (Partition of Unity) Let Q C RN be an open set and { Q } aWhitney decomposition for 0. Then there exist C°D functions O j on RN satisfying

(1) 0 < Oj < 1 for all j;

(2) 4j(x) = l forx E Qj;

(3) O j (X) = 0 when x¢ (4/3) Q j;

(4) I(a°/8x°)Ok(X)I < K° (diam Qk)for any nudtiindex a;

(5) (x)= = I whenx E Q.

These lemmas are treated in considerable detail in [SE 70] and [KP 99]. Seealso the original paper of Whitney [WH 34]. We now present an elegant applica-tion to the theory of C°O functions:

Proposition 3.3.6 Let E C RN be any closed set. Then there is a C°O function fon RN such that {X E RN f (x) = 0) = E.

Proof. Let { Qj } be a Whitney decomposition of 0 = RN \ E and let 10j) be thecorresponding partition of unity. For each j let S j denote the diameter of Q3. Set

P x) = E2_2j/'J

O j (x).j

The series converges absolutely and uniformly on all of RN, because it is locallyfinite. Notice that the zero set off is precisely the complement S2, that is, E itself.It remains to check that f is infinitely differentiable.

3.3. The Theorems of Whitney 77

If a is a multiindex, then the series obtained by applying as/axa formally tothe series defining f has jth term that is majorized by

Ka-2-2j/6J . (Si)-lal. (3.5)

Now fix a point x in Q. If v is the distance of x to E, then x is contained inat most 12N of the cubes (2Qj,)k2

iand each of those cubes has diameter Stk.

Moreover Sjk < v < 48jk. Thus we may use (3.5) to see that, at this x,

asf(x) C . (12)N .2-21/' , m-lal

aXa

As v --). 0 we see that this last expression tends to zero. It from these estimates

that all derivatives of f exist on °E and that they tend to zero at points of CEtending to E. By the same token, all derivatives of f on a E are zero. Of course,on the interior of E, all derivatives off are zero by definition. It follows that f isa C°O function on all of RN.

The principal result of Whitney's classical paper [WH 34] is to characterize thecircumstances under which a smooth function on a closed subset E C RN can beextended to be C1 on all of RN in such a way that the extended function is realanalytic on the complement of E. We shall formulate and discuss, but not prove,this result. It is obviously a generalization of Besicovitch's theorem presented inthe last section: in that context, the role of the set E is played by just two points-the endpoints of the interval being studied.

Clearly there is an obstruction to formulating Whitney's theorem. If E is a trulyarbitrary closed set, then what do we mean by a "smooth" (or CO0) function onE? One possible definition is that a function f is smooth on E if it is obtained byrestricting to E a function that is smooth on all of RN. For some purposes such adefinition is satisfactory. However, when one is proving extension theorems sucha definition is inappropriate. Therefore we proceed as follows.

Definition 3.3.7 Let E C RN be a closed set and f a function on E. We say thatf is Ck on the set E if for each x E E there are numbers fx,a, defined for eachpoint x E E and for each multiindex a with 0 < Ja 1 < k, such that

f (x + h) = E fx.a'9 hp +Ra (x, h)la+flI<k 0'

holds when x + h E E. Here Ra (x, h) is a remainder term with the property that,if e > 0, then there is a S > 0 (independent of x) so that if Ih I < S then

IRa(x, h)I < e lhlk-1a1


It is not difficult to see that if E is a simple set like a closed half space, thenthe definition of Ck function just given is equivalent to any other reasonable def-inition. For pathological closed sets, there is no other reasonable definition ofsmooth function. See [KS 83] for more on these matters. Notice in passing thatthis definition of smooth function on a closed set is very much in the spirit of theconverse of Taylor's theorem that was presented at the end of Section 3.1.

Whitney's main theorem (see [WH 34]) is as follows:

Theorem 3.3.8 (Whitney Extension Theorem) Let E be a closed subset of R) v.Let f be a function on E that is Ck according to Definition 3.3.7. Then there is aCk function f on all of RN such that

(1) f1E = f;

(2) f is real analytic on the complement of E.

The proof of Whitney's theorem proceeds in two steps. First, we produce aCk extension F of f to all of RN. Then an approximation procedure (similar inspirit to the Weierstrass approximation theorem) is used to replace F by func-tions which (a) agree with f on "most" of E. (b) are real analytic off E, and(c) approximate F closely. The desired function f is then obtained by a limitingargument.

To see how Whitney's extension technique works, we let { Qj) be a Whitneydecomposition for 0 =_ È. Choose for each j an element pj E E such thatdist(pj. Qj) = dist(E, Qj). Set

P(x; pj) _ - (x - 0"lcrl<k

f(X) - ( f(X) if x E EFjOj(X)P(X: pj) if X E RN \E,

where the 0, are the functions in the partition of unity (see Lemma 3.3.5). It turnsout (we shall not prove this) that this defines a Ck function on all of RN thatagrees with f on E. It requires some extra work to obtain an extension operatorthat extends an f that is C°O on E to an f that is C°0 on all of RN, and we referthe interested reader to Whitney's original paper [WH 341 for this matter.

The necessary approximation result that allows one to arrange for the extensionof f to be real analytic on the complement of E is as follows:

Proposition 33.9 (Whitney Approximation Theorem) Let K c RN be a com-pact set. Let f be a function of class Ck on K. /f E > 0, then there exists a realanalytic function G on R ' such that

supif(X)-G(x)I<E.xEA:

3.4 The Theorem of S. Bernstein 79

In fact it is not difficult, given our modern perspective, to prove such a result.Let O(x) be a positive real analytic function of total mass 1 (the Gaussian kernel,suitably normalized, will suffice). For E > 0 set

fa(x) = g-NO(XIS).

We may use the Ck extension theorem above to extend f to an open set U thatcontains K. Let ' be a nonnegative cutoff function that is supported in U and isidentically equal to 1 on K. Define g(x) = fi(x) f (x). Now set

f6(x) = f Os(x - t)g(t)dt.

Then straightforward arguments show that fd -a f uniformly on K.In fact it can be shown that fs -+ f in the Ck topology of K. Now, as already

outlined, the approximation result can be used to make successive alterations tothe Ck extension theorem to arrange that the extended function is real analytic offthe set E.

It is interesting to note that there is no successful definition, analogous to Def-inition 3.3.7, for a real analytic function on an arbitrary closed set E. There is,however, an interesting generalization of (the spirit of) Definition 3.3.7, due toJ. Siciak [SJ 86]: Let f is a C°O function on an open domain Q. If X E Q thenlet r(x) be the radius of convergence of the Taylor series expansion off about x.Then we set

(i) A(f) = (a E S2 : f is real analytic in a neighborhood of a);

(ii) S(f)=S=S2\A;

(iii) D(f) = D = (a E S : r(a) = 0);

(iv) F(f) = F = (a E S : r(a) > 0) = the points of "false convergence".

It is straightforward to check that A is open, D is a Ga, and F is an Fo of the firstcategory. The theorem is

Theorem 3.3.10 Let S2 be an open domain in RN. Let S2 = A U D U F, whereA is open, D is a G8, and F is an Fo of the first category. Then there is a C°Ofunction f on 12 such that A = A(f), D = D(f), and F = F(f).

3.4 The Theorem of S. Bernstein

We conclude this chapter by presenting a curious and not well-known theorem ofSerge Bernstein that gives a sufficient, and easily checked, condition for a functionto be real analytic. For convenience we work on the real line, but there are obviousanalogues in several variables.


Theorem 3.4.1 Let f be a C°O function on an open interval 1 c R. If f andall its derivatives are nonnegative on the entire interval 1, then f is real analyticon 1.

Remark 3.4.2 The functions e2, ex2, x, xz, etc. on the interval (0, oo) certainlysatisfy the conditions of the theorem. Of course the functions sin x, cos x, andlogx do not, so the utility of the result is unclear. The theorem spawned, in itsday, a rash of work on the patterns of the signs of coefficients of real analyticfunctions. We refer the reader to [BL 67] and [PG 75] for more on these matters.

Proof. Let a E 1. Recall Taylor's theorem with remainder:

i(n-t)

(x-a)n- l +R.(x),f(x) = f(a)+f (a)(x-a)+ LL 2(a-) (x-a)2 +...(n

f(- 1)!

where

R. (x) = (n l 1)J

f(n)(t)(x - t)n-'dt.

This result is proved by integrating the fundamental theorem of calculus

f(x)-f(a)= Jf'(t)dta

by parts a total of n - 1 times. It is convenient to use two changes of variable torewrite R. as

1 s-°Rn(x) _ (n-1)!f f(")(u+a)(x-u-a)n-' du

(x - a); j f ((x - a)t +a)(1 - t)i dt .

In what follows we assume that b E I with a < b. For any x with a < x < b.we have

0 < Rn(x) < (x - a)"1

f(n)((b - aft + a)(1 - t)"-) dt.(n - 1)! o

Here we are using the fact that f ("+l) > 0 hence f (") is monotone increasing on1. The right-hand side of the last inequality is nothing other than

(x - a)"(b - a)"

Since Taylor's expansion tells us that

f"(a) z f("-t)f(b) = f(a)+f (a)(b-a)+ 2! (b-a) +... (n - 1)!(b-a)n-I +Rn(b).

3.4. The Theorem of S. Bernstein 81

and since all terms on the right but the last are positive, we conclude that f (b) >R,,(b). Combining our inequalities, we get

0< R.

(x) < (b - a) rt f(b)

Now letting n -- +oo, we find that 0. This shows that the Taylorexpansion converges, uniformly on compact subsets of (a, b), to f. Since a < bwere arbitrary in 1, we conclude that f is real analytic on 1.

We refer the reader to the book of Boas [BR 60] for further discussion of thephenomenon identified in Bernstein's theorem.

4Some Questions of Hard Analysis

4.1 Quasi-analytic and Gevrey Classes

In the theory of functions on R" there is a great chasm between the space of CO0functions and the space of real analytic functions. If, for instance, a real analyticfunction vanishes on a set of positive measure, then it is identically zero. [Thisis most easily proved by induction on dimension, beginning with the fact that indimension 1 we have the stronger result that if the zero set has an interior accu-mulation point, then the function is identically zero.] By contrast, any closed setis the zero set of a C°O function. In dimension 1 this is seen by noting that thecomplement of the closed set is the disjoint union of open intervals; it is straight-forward to construct a C°O function of compact support on the closure of an openinterval whose support is precisely that closed interval. In several real variablesthe Whitney decomposition serves as a substitute for the interval decompositionof an open set and, with more effort, allows a similar construction to be effected(see Proposition 3.3.6).

Real analytic functions have (locally) convergent power series expansions; C°Ofunctions, by contrast, generically do not. Locally supported C°O functions can bepatched together using a CO0 partition of unity; there is no similar construct in thecategory of real analytic functions.

Since both C°O functions and real analytic functions play an important role inthe regularity theory of partial differential equations (see [HL 83]), it is desirableto have a scale of spaces incrementing the differences between the space CO0 andthe space Cw. (An analogue of the scale one might wish for is the scale of spaces

84 4. Some Questions of Hard Analysis

Ck, 1 < k < oo spanning from C = CO, the continuous functions, to C°O, theinfinitely differentiable functions.)

Unfortunately, no such scale is known. However there are some very interestingand useful spaces that are intermediate between C°O and CW and that interpolatebetween the two extremes in a variety of precise senses. These are the quasi-analytic classes and the Gevrey classes. We shall discuss both of these types ofspaces and their interrelationships in the present section.

Before proceeding, we note that the classes of functions defined in this sectionare specified in terms of rate of growth of Taylor coefficients. For an arbitrary C°Ofunction the Taylor coefficients can be fairly unpredictable as the next theoremshows.

Theorem 4.1.1 (E. Bore[ [HL 83]) For each multiindex a of length N let as be areal number. Then there exists a C°O function on the unit ball B(0, 1) a RN withthe property that aaf

axa(0) = a! as

for every multiindex a.

This theorem is the multivariable analogue of Theorem 3.2.1. It may be provedeither by adding infinitely many small bump functions, each of which carries theinformation about one Taylor coefficient, or by a straightforward category argu-ment.

In fact considerable investigation was made in the late nineteenth and earlytwentieth centuries into the pathological nature of the Taylor expansion of a C°Ofunction. We discussed some of these ideas in Chapter 3.

Hassler Whitney considered to what extent the Taylor coefficients of a C°Ofunction may be specified on an arbitrary set E. His result, valid in any dimen-sion, is described in detail in [FH 69] or [HL 83]. See also our Section 3.3. Whit-ney's results are remarkable for the fact that their hypotheses are as weak as onecould possibly hope. We state the result here in self-contained form (see also The-orem 3.3.8):

Theorem 4.1.2 (The Whitney Extension Theorem) Let E be any compact subsetof RN. Let k be a positive integer and for each multiindex a, with Ian < k, let uabe continuous functions on E. If x, y E E are unequal then we define

Ua(X, Y) = I ua(x) - ua+p(Y)(x - Y)O Ix - ylla!-k.!S!<k-!a!

fl'

Also we set Ua (x, x) = O for x E E. If each Ua, la 1 < k, is a continuous functionon E x E, then there is a function u E Ck (RN) such that

axa(X) = Ua(X)

for all x E E and lad < k.

4.1. Quasi-analytic and Gevrey Classes 85

Now we turn to our subject proper. It is convenient in this section to do analysisneither on RN nor on R1 but rather on the unit circle. Equivalently, we do analysison the set T =- R/2nZ. We are in effect working on the interval [0, 27r] butidentifying the endpoints of the interval. This is useful because we shall then beable to use some elementary ideas from Fourier series. Fourier series are built upfrom the characters e'jt , where i = , and these functions are supported ina natural way on T. We use ordinary Lebesgue measure in doing analysis on T.(See [KY 76] for a detailed consideration of analysis on T.) In what follows welet f U) denote the ju' ordinary derivative of a function f on T.

Definition 4.1.3 If 0 < al < a2 < a3 < is a sequence of real numbers, thenwe say that the sequence is logarithmically convex if (log aj ) is a convex functionof j, that is, if whenever t < m < n, then

.log am < n - e log at + n - e log an

In some sense, a logarithmically convex sequence is more convex than an ordi-nary convex sequence. For example, the sequence (j2) is convex but not logarith-mically convex. On the other hand, (el) is logarithmically convex. Logarithmicconvexity is an important concept in analysis; it arises in the three lines theorem,in interpolation of linear operators, and in calculating domains of convergence ofthe power series for (real and complex) analytic functions of several variables.

Definition 4.1.4 Let M1, M2, ... be a monotone increasing, positive, logarithmi-cally convex sequence of real numbers. A C°O function f on T is said to belongto the class C((M j )) if there is an R > 0 such that

sup If(i)(x)I < Mj . R.T

Remark 4.15 The spaces C((Mj }), with some extra refinements, will be ourquasi-analytic classes.

Example 4.1.6 If M j = j!, each j, then it is not difficult to see that (Mi) isincreasing and logarithmically convex. The class C({Mj }) consists exactly of thereal analytic functions on T.

For the next example, we will need the following lemma of S. Bernstein.

Lemma 4.1.7 There is a constant K > 0 such that, if q (t) = E j _N a jeìr is atrigonometric polynomial of degree N, then

sup Igr(t)I < K N N. sup Iq(t)I0<t52tr 0<r<2tr

Proof. The kernelVN(t) = 2K2N+1(t) - KN (t) ,


where2

1

Kj(r)sin (41-t)

j + 1 sin (fit)

is the standard Fejer kernel of harmonic analysis (see [KY 76; pp. 12-17] or[ZA 59]), has the property that

1z,r

q(x) = 9 * VN(x) = 2n f q(t)VN(x - t)dt.0

Therefore

It follows that

2n

q'(x) 2a f q(t)dxVN(x-t)dt.0

sup Iq'(x)I _< sup Iq(x)I0<x<2rr 0:5x<2rr

Straightforward estimates show that

dx VN<K.N,

Li

ddx VN

L1

completing the proof.

Example 4.1.8 If M1 = 1 for all j then, by Bernstein's lemma, all trigonometricpolynomials lie in C({Mj}). The converse is true as well. It is a standard fact ofFourier analysis (see [KY 76j) that for p E Z one has

If(P)1 < (2rIPI)m supilf(m)II

for any 0 < m E Z. But for the specified class of Mj this gives

mIf(P)I < (2'IPI) R=

(p,)rn

If I p I is large enough so that the fraction in parentheses is less than 1 then lettingm -> oo yields that f (p) = 0. In other words, f is a trigonometric polynomial.

Example 4.1.9 If MJ = 22J, then the class C({MJ}) will contain functions thatare not real analytic. Certainly the function

f (x)exp(-1/('rx - x2)) if x 540, 2Jr

t 0 if x = 0, 2,r

will lie in C({Mj}) but it is not real analytic.


In the material that follows we shall develop a method for manufacturing func-tions in a given C({Mj}).

We begin with an important alternative definition of a quasi-analytic class interms of the L2 norm instead of the L°O norm.

Definition 4.1.10 Let MI, M2.... be a monotone increasing, positive, logarith-mically convex sequence of real numbers. A C°O function f on T is said to belongto the class C'({Mi}) if there is an R > 0 such that

Ilf(i)(x)IIL2(T) < Mi Ri

Remark 4.1.11 The two definitions of C({Mi}) and C"({Mi}) give rise to es-sentially the same spaces of functions in the following sense: First, since T is acompact measure space we have that

Ilf(j)IIL2 < C sup If(J)I .T

It follows that C((Mi)) e C"((MJ)) for any positive, monotone increasing, log-arithmically convex sequence Mi. For a near converse, notice that for j > 0 andf E CI(T) we have

f(i+t)(t)dt = fi(2n) - f(i)(0) = 0J0

by periodicity. Thus there is a point po E T such that f(i+l)(po) = 0. Hence forany x E T we have

If (i)(x)l = I f f(i+')(s)dxl,vo

and by Holder's inequality the expression on the right is bounded by a constanttimes Il f (J+') II L2. Hence

suplf(i)I <c IIfU+')IIL2T

In general, we cannot place an a priori bound on Mi+i /Mi, so the two spacesare not exactly the same. 0

Definition 4.1.12 A C°O function f on T is said to vanish to infinite order at apoint p E T if f(i)(p) = 0 for all j = 0, 1,2,... .

Definition 4.1.13 A set or class of C°O functions S is called quasi-analytic ifwhenever a function f E S vanishes to infinite order at a point p E T thenf=0.


Obviously the class of real analytic functions is a quasi-analytic class (hencethe name). The main result of this section will be the Denjoy-Carleman theo-rem which gives a complete characterization of quasi-analytic classes of the formC*({M j }). To this end we introduce a final piece of notation.

Definition 4.1.14 If {M j } is a positive, monotone increasing, logarithmically con-vex sequence of numbers, then we create from it a function on {R : R > 0} by

rM1(R) = r(R) __ inf Mj R-1. (4.1)j>0

Following Katznelson [KY 76], we refer to r as the associated function for thesequence {M1).

Theorem 4.1.15 (Denjoy-Carleman) Let {M1) be a positive, monotone increas-ing, logarithmically convex sequence of real numbers. The following three state-ments are equivalent:

(1) C*({Mj}) is a quasi-analytic class.

(2) J I1 }-r2) dr = -oo.

t

(3)Mj

_ +oo.Mj+1

We prove the Denjoy-Carleman theorem in three steps. Fix once and for alla positive, monotone increasing, logarithmically convex sequence (Mj) of realnumbers.

Step 1: Proof that (2) (1). Assume property (2) and let f E C#((Mj)). Totest for quasi-analyticity, we take (without loss of generality) p = 0 and assumethat f (1)(0) = 0 for all j. We shall prove that f =- 0. Define the Fourier-Laplacetransform

2n

*(z) =J

e-z' f(r)dt2n 0

where z is a complex variable unequal to zero. [For convenience here we do notput the factor of 2n in the exponent as was done with the Fourier transform else-where in this book. But we do put a factor of 1/27r in front of the integral.] Weintegrate by parts, using our hypotheses about f to eliminate the boundary term,to obtain

2

>G(z) = 2nz

rna-tz f'(t) dt .

Integrating by parts j - 1 more times yields

iir(z) = I f e-rz f(j)(t)dt.

27rzj o


Restricting attention to {z E C : Re(z) ? 0}, we have that

Ie-zf I < 1

hence, by Holder's inequality,

1 *(Z)1 5 Z .Ile-uii0(r).I1fv)(t)IIL2(T)

-Mj1

(--)j,2n Izl

I* (z)1 5 Mj(R Izl)!

. (4.2)

so

Letting lzl/R play the role of R in the definition of the associated function r, andtaking the infimum in this inequality over all j, allows us to conclude that

I*(z)I 5 r (IzI/R)

or, equivalently,

In conclusion, we have

log I'1r(z)l 5 log[r(Izl/R)J

00 logI*(is)Ids < j log[r(R)]ds.t 1+s2 t 1+s2

Using the fact that r(.) is a nonincreasing function and that r(s) = Mo, for0 < s < 1, we can see that the statement (2) implies

/ O° log[r()]f ds = -oo.1 + s2

This estimate provides a contradiction for the following classical reason: Observethat the function ' is holomorphic on the open right half plane and continuous onthe closed half plane. Moreover is bounded for z large by the estimate (4.2) andfor z small by inspection. Thus is in the function space H°O of G. H. Hardy.The classical inequality of Jensen for the location of zeros of such a function (see[KY 76; p. 114] or [KS 82] or [HK 62)) then yields that

°O log 1 *001 +s2

ds > -oo.

That is the required contradiction.

Before the next step in the proof of Theorem 4.1.15 we give an interesting

construction that provides examples of C°O functions in many of the classesCK({Mj }).


Lemma 4.1.16 Let [At) ' be positive numbers that sum to a number not exceeding 1. For each integer k, the infinite product

sin(µtk)v(k) = -

1100t=0 Atk

converges and, if we define

00

f (t) = E v(k)e'kik=-00

then f ; 0 is supported on [-1, 1) (mod 2tr), is infinitely differentiable, andsatisfies the estimates

Proof Notice, using Taylor's formula, that

sin(iLtk) µtk - (ptk)3/6 = 1 (Aek)2

Atk ptk 6

Thus

1 -sin(µtk)

Atk

certainly converges and therefore the infinite product defining v(k) converges.Moreover, v(k) tends to zero faster than any negative power of k (look at thedenominators in the infinite product) so that the series defining f converges uni-formly and absolutely. For the same reason, the series may be differentiated term-by-term so that f is infinitely differentiable. Finally, f has a nontrivial Fourierseries hence f is not identically 0 (see [KY 76]).

We do the final analysis on f by examining the partial products of the coeffi-cients v(k). By direct calculation, the sequence

J sin(itk)100i rttk J k=-oo

consists precisely of the Fourier coefficients of the function

re(t)trAi t if Itl < µi

l 0 if Itl > Aj

fort=0, 1,2,....Setsin(µtk)

Nvn+(k) =ptkt=o


and define00

fN(t) _ vN(k)eikrk=-oo

Then the formula (see for instance [KY 76])

tells us thatfN(t)=ro*rt *...*rN(r).

Since the support of I't lies in [-µt, µt], it follows that the support of IN lies in[- Eo At, o µt] (mod21r). Thus, since IN -> f uniformly, the support of flies in [-1, 1] (mod 2n).

Finally, we use Plancherel's theorem and the fact that (f (i))" (k) = (ik)j f (k)(again see [KY 76]) to see that

1/2

Ilf(j)IIL2 = (II(k)I2k2i)= VOO /2Iv(k)12k2j

k=-oo /We observe that

t

.

o µrµt) k-j-1Iv(k)I 5 1 1 1 = (nk o

Putting together (4.3) and (4.4), we see that

IIP)IIL2

_t0o j

k-i-t

2

k2i

1/2

Ek=-oo o

(k_2)1/22.0µl)-t#o \o /

(4.3)

(4.4)

This completes the proof of the lemma.

Remark 4.1.17 As the reader can easily see, Lemma 4.1.16 may be applied tothe situation at hand by setting At = (Mi)-' and µj = Mj_t/Mj. Notice thatthis yields

= Mi.Ri=o At

automatically. The condition that F_ At 5 1 may be arranged by scaling, as willbe noted below in the proof that (1) implies (3).


Step 2: Proof that (1) = (3) . We will prove the contrapositive of the statementthat (1) implies (3). Suppose that E Mj/Mj+i < oo. By replacing Mj by Mj*

M j Rj for R small we certainly shall not change the class C* but we may arrangethat EMj/Mj+i < 1/2. We define

AO = 1/4,Al = 1/4,

Aj = , forj>2.Mj

Then we have E A j < I and

.F1 µl t = 16-00

The lemma then provides us with a nonzero function f that is in C$((Mj)) andthat vanishes outside [-1, 1] modulo 21r. Thus the class C'({Mj}) cannot bequasi-analytic.

Remark 4.1.18 Notice that the construction above demonstrates that if a classC1({Mj}) is not quasi-analytic, then it contains nonzero CI functions of ar-bitrarily small support. This is a much stronger assertion than the definition ofquasi-analytic class suggests.

Step 3: Proof that (3) (2). Thus far we have not used the logarithmic con-vexity of the sequence {Mj} but now this property will prove to be important. Wemay as well assume that the sequence { M j } increases faster than Rj for everyR > 0; otherwise the class C((Mj)) is no different from the class defined withM j = 1 for all j and, as was shown in Example 4.1.8, that class consists only ofthe trigonometric polynomials.

With this assumption about the growth of the Mj, we see that the infimum in(4.1) is attained. Thus

r(R) = min MjR-J .j>o

Define Al = MI 1 and A j = Mj_i/Mj for j = 2, 3, .... Then the sequence(A j } is monotone increasing; for this assertion is equivalent to

Aj+t M2_ j <1

Aj Mj-IMj+i

which is true by logarithmic convexity. Clearly MjR-j = fi(pzR)-'. As aresult, we will minimize this expression by selecting j to be the last term (µt R)-


that is smaller than 1. In other words,

r(R) = fl (AtR)-1ltr R> 1

Let us define

M(R) = the number of elements At such that AtR > e

(here e is the base of natural logarithms). Then

-logr(R) = E log(NigR) ? log(µtR) ? M(R),a,R>I µtR>e

We conclude that, for k = 2, 3, ..., we have

Iek+ -logr(R)dR >

r`*+i M(gt)dR >

I1 Rz f et T+ 2e2k+z 20 ek

(4.5)

On the other hand notice that the number of At between e l -k and ez-k is M (ek) -M(ek-1). Hence we have

At (M(ek) - M(ek-I)\ . e2-k < 10.M(ek)

J _ (4.6):5eke l -t At <e2-k

Putting together (4.5) and (4.6), we get

rek+' _ log r(R)At <200.1+Rz dR.

eI-k<ytt<e2-k ek

Summing over k = 2, 3, ..., we obtain

r -log r(R)At <200J

1+RzdR.I

But this just says that (3) implies (2).

The implications (2) (1) (3) = (2) complete the proof of the Denjoy-Carleman theorem.

A somewhat different treatment of the Denjoy-Carleman theorem-one thatuses no complex analysis or Fourier analysis but is quite technical and difficult-appears in [HL 83; vol. 1, p. 23].

We now say a few words about another collection of spaces known as theGevrey classes. Following [HL 83], we define these as follows:


Definition 4.1.19 Let Lo, Lt, ... be a sequence of positive numbers with theproperty that there exists a C such that, for every k,

(4.7)

Thus the sequence grows at least arithmetically and at most exponentially. Wesay that afunction f E C°°(T) belongs to the Gevrey class G({L j }) if there is aconstant C such that, for every j, it holds that

sup If (i) (t) 1 < (C . Lj)j . (4.8)

It is easy to see that Definition 4.1.19 is just a variant of the definition of theclass C({Mj}). Some modern treatments of the material in the present sectionoften formulate the Denjoy-Carleson theorem in the language of the G((Lj))rather than the C*({Mj}) as we did in Theorem 4.1.15. A Gevrey class G({L j ))is quasi-analytic if and only if > 1/Lj diverges.

Each Gevrey class is closed under differentiation (exercise) and is preserved un-der real analytic mappings. Gevrey classes are in some ways more attractive thanquasi-analytic classes because they are localizable. That is because the growthrate of the derivatives of a typical cutoff function is swamped by the right-handside of the inequality (4.8).

One might hope to prove real analytic regularity theorems for a partial differen-tial operator L by first proving an estimate in each Gevrey class and then amalga-mating all this simultaneous information. The essential tool in such an approachis the following theorem of Bang [BT 46].

Theorem 4.1.20 The intersection of all of the non-quasi-analytic Gevrey classesconsists precisely of the real analytic functions.

Curiously, the intersection of all the Gevrey classes does not give the quasi-analytic functions or the real analytic functions as one might expect. Since thesematters are all quite technical, we refer the interested reader to [BT 46] or to[HL 83).

Just to give the interested reader the flavor of the types of questions one mightask about the classes of functions being discussed here, we briefly describe somework of Rudin [RW 62]. Recall that in classical analysis it is of interest to de-termine under what algebraic operations a class of functions is closed. Considerthe operation of taking the reciprocal of a nonvanishing function f. It is easy tosee that if f is C°O, then so is 1/f. A slightly trickier proof shows that if f isreal analytic, then so is 1/f (see Proposition 1.1.12). Recall the function classesC({Mj )) defined at the beginning of this section. When is such a function classclosed under reciprocals?

In order to answer this question, we need two new definitions:

4.2 Puiseux Series 95

Definition 4.1.21 If (M j) is a positive, monotone increasing, logarithmically con-vex sequence of numbers, then we define

. .Aj= j=1,2,...00'We will call {A j } the sequence associated with the sequence {M j ).

Definition 4.1.22 Let A1, A2.... be a sequence of real numbers. The sequenceis said to be almost increasing if there is a number K > 0 such that

As<KAj,forall1 s<j.Then we have

Theorem 4.1.23 (Rudin) Let {Mj } be an increasing, logarithmically convex se-quence of positive real numbers. If the associated sequence (Aj) is almost in-creasing, then C({M j }) is closed under the taking of reciprocals.

Theorem 4.1.24 (H6rmander) If a class C({Mj}) is closed under the taking ofreciprocals, then the associated sequence (A j ) is almost increasing.

We refer the reader to [HL 83], [RW 62], and [BJ 64], and to references thereinfor more on the lore of Gevrey and quasi-analytic classes.

4.2 Puiseux Series

A Puiseux series is a formal power series

00

ajxilk,j=N

where N is an integer and k is a fixed positive integer. For each fixed k, the set ofsuch formal power series is seen to form a field. The union of all these fields issometimes denoted by K {x), where K is the field which contains the coefficientsa j. Puiseux's theorem, in this context, is the following (see [LS 53; p. 99ff]):

Theorem 4.2.1 (Abstract Puiseux's Theorem) If K is of characteristic zero andalgebraically closed, then K{x} is algebraically closed.

Our interest is in convergent power series over the reals, so this theorem is notthe one we want to prove. We describe the situation of interest to analysts: LetA(x) and B(x) be real analytic functions of a single real variable near 0. Theirquotient A(x)/B(x) can be written as x"C(x), where C(x) is also real analyticwith C(0) 0 0, and N is an integer (possibly negative); this can be done as long


as B does not vanish identically. The family of functions of the form xNC(x),N E 1--defined near, but not necessarily at, 0-thus forms a field IC. We considera polynomial equation over that field:

xNOAo(x)Y? +xN'AI(x)Y1-I + ... +xN"-IAn-I (x)Y+xNAn(x) =0.

Here xNOAo(x), xN2AI (x), ... , xNÂ,, (x) E 1C. It is no loss of generality to as-sume that Ao 1. By replacing y with x-°y', one may assume that No < Ni,f o r i = 1, 2, ... , n, and then one may divide through by xN0; in the equationthat remains all the coefficients are real analytic. Thus it will suffice to consider apolynomial equation of the form

P(x,Y)=Yn+BI(x)Yn-I +...+Bn-I(x)Y+B.(x)=0,

where each B;(x) is real analytic near 0. We will show that there is a positiveinteger k such that, for near 0,

P(l; k, y) _ (y - RI (t)) (Y - R2O) ... (Y - Ra(4)) Q(t, y),

where each of RI, R2,..., R. is real analytic and where Q(!y, y) is a polynomialin y that has real analytic coefficients at = 0 and that has no real roots fort; nearzero, except possibly at = 0. This decomposition of P allows us to understandthe solutions of P(x, y) = 0 near x = 0.

Our approach in this section is similar to the proof of Puiseux's theorem thatappears in [BM 90]. The main tool for our investigation is an algebraic resultknown as Hensel's lemma.

First we review a few facts from algebra.

Lemma 4.2.2 (see [VDW 70; Section 3.7]) For any field K, the polynomial ringK[y] is a principal ideal domain and any two polynomials g(y) and h(y) have agreatest common divisor d (y) expressible in the form

d (y) = a(y)g(y) + b(y) h(y).

In particular, if go(y) and ho(y) are relatively prime polynomials, then there arepolynomials ao(y) and bo(y) such that

I = ao(Y)go(Y) + bo(Y)ho(Y)

Corollary 4.2.3 If go(y) and ho(y) are relatively prime polynomials of degreep and q, respectively, then for any polynomial f (y) of degree strictly less thanp + q, there exist polynomials a(y), of degree strictly less than q, and b(y), ofdegree strictly less than p, such that

f (Y) = a(y)go(Y) + b(y) ho(Y)

4.2. Puiseux Series 97

Proot By Lemma 4.2.2 there are polynomials ao(y) and bo(y) such that

1 = ao(Y)go(Y) + bo(Y)ho(Y)

Thus we have

f (Y) = f (Y) ao(Y) go(Y) + f (Y) bo(Y) ho(y)

Using division, we can write

f(y)ao(y) = gi(Y)ho(Y)+a(y),f(y)bo(y) = g2(Y)8o(Y) + b(Y),

where the degree of a(y) is strictly less than q and the degree of b(y) is strictlyless than p. Thus we have

f(Y) = [qt (y) + g2(Y)]8o(Y)ho(Y) + a(Y)8o(Y) + b(Y)ho(Y)

Since the degree of go(y) ho(y) is p + q, which is strictly greater than the degreeof f (y), it must hold that q, (y) + q2(y) = 0, proving the result. O

Lemma 4.2.4 Let x = (xt, x2, ... , xR) E R" and

P(x,Y) =y"+x,y"-t +...+xn_ly+xn.

Assume that po(y) = P(b,, b2,..., b, y) can be written as the product of rel-atively prime real factors go(y) and ho(y) of degrees p ? 1 and q > 1, respec-tively. Then there are real analytic functions

Cl (x), C2(x), ... , Cp(x) ,

and

DI (x), 1)2(x),..., Dq(x) ,

d e f i n e d near b = (bl, b2, ... , bn) such that

satisfy

and

G(x,y) = yp+CH(x, Y) = Yq + Dl (x)Yq-' + ....+ Dq(x)

P(x,y)=G(x,y)H(x,y),

G(b,,...,bn,Y)=go(Y), H(b,,...,bn,Y)=ho(Y)


Proof. Let us write

go(Y) = Yp + clyp-' + ... + cp-IY + cp,

ho(y) = Yq+dlyq-' + ...+dq_ly+dq.The plan is to show that the mapping F sending

(u1,u2,...,up;VI,V2,...,tlq)E)R"to the n-tuple consisting of the coefficients of y"-1, yn-2, ..., y, 1 in

l(yp+ulyP-1 +...+up-ly+up) (yq+UYq-1 +...+vq_1Y+vq)

has a nonsingular differential at

c = (CI,c2,...,cp;dl,d2,...,dq),

and is thus invertible in a neighborhood of (cl, c2, ... , cp; dl, d2, ... , dq). We doso by showing that this differential is surjective.

Consider a vector v = (Yl , n, ... , yp; Si, 3 2 ,--- ,S ,). We can evaluate thedifferential D.F(c) applied to v by using the formula

(Dl(c), v) = tv)r=0

We find that

(D.F(c), v)

dt(Yp + (cl + iy )Yp-1 + + (Cp-1 + tYp-1)Y + (Cp + t yp))

x dt (yq + (d1 + tSl)yq-' + + (dq_l + tSq_l)y + (d,? + tSq))]

= (Ylyp-1 +. ..+Yp_ly+Yp) (yq+d1Yq-1 +...+dq_iy+dq)

+(Yp+clyp-' +...+Cp-IY+Cp) (Slyq-1 +...+Sq-IY+Sq)

= (YIYp-1 + ... + Yp-1Y + Yp) ho(y)

+ go (Y)(SIYq-I + + 8q_1y + Sq)

By Corollary 4.2.3, every polynomial of degree less than n can be written in theform

(YIYp-I + ... + Yp-1Y + Yp) ho(y) + go(Y) (alyq-I + ... + Sq-IY + Sq)

Thus we see that DF(c) : R" -+ R" is in fact surjective. It follows that DF(c) isnonsingular and the result follows from the inverse function theorem. 0

7=0


Theorem 4.2.5 (Hensel's Lemma) Let P (x, y) be a polynomial in y of the form

P(x, y) = Yl + BI (x)Yn-1 +. _. + Bn(x),

where each Bi is real analytic. Suppose that P(0, y) factors into the product ofrelatively prime real factors go(y) and ho(y) of degrees p and q, respectively.Then P (x, y) factors into the product of G (x, y) and H (x, y) of the same degreesin y with coefficients that are real analytic in x and for which

G(0, y) = go(y), H(0, y) = ho(y).

P r o o f . W e let C1 (x), C2 W, ... , CP(x) and D1(x), D2 W, ... , Dq(x) be the func-tions defined in Lemma 4.2.4. Set

B(x) = [BI (x),..., Bn(x)).

Then we may set

G(x, y) = yP + C1 [B(x)]YP-1 + ... + CP[B(x)],

and

H(x, y) = yq + D1 [B(x)]yq-1 +... + Dq[B(x)]. O

Remark 4.2.6 The proof of Theorem 4.2.5 provides only an indirect route forcomputing the coefficients of G and H. An effective procedure for finding thecoefficients is the following: Rearrange P(x, y) by powers of x, so that

P(x,Y) = fo(Y)+xf1(Y)+...+xrfr(Y)+...,

We have that fo(y) = P(0, y) and that fo(y) is of degree n. For i = 1, 2,..., thedegree of fi (y) is strictly less than n.

We write

G(x,Y) = g0(Y)+xg1(Y)+...+xrgr(Y)+...

H(x,y) = ho(Y)+xh1(Y)+...+xrhr(Y)+...,

The polynomials 81(y), g2(y), ... are to be of degree at most p-1 in y, while thepolynomials h1(y), h2(y), ... are to be of degree at most q - 1 in y. Multiplyingtogether the above expressions for G and H and equating like powers of x, wesee that the following equations must be satisfied:

go(y)ho(y) = fo(Y) = P(0, Y) ,

go(Y)ht(Y) + gt(Y)ho(Y) = fi(y),

r-1

go(Y)hr(Y) + E gi(Y)hr-i(Y) + gr(Y)ho(Y) = fr(y),


The first equation is satisfied by hypothesis. Proceeding inductively, we suppose

that 81, 82, - - , Sr-1 and hl, h2, ... , hr_1 have been chosen so that the first requations are satisfied. The equation which must be satisfied by gr and hr is

r-1

go(Y)hr(Y) + gr(y)ho(y) = .fr(y) - gi(Y)hr-i(Y)- (4.9)

i=1

Since go and ho are relatively prime and the right-hand side of (4.9) is of degreestrictly less than n, we can use Corollary 4.2.3 to find gr and hr, of degree at mostp - 1 and q - 1, respectively, which satisfy (4.9). El

With the aid of Hensel's lemma, we can give a proof of the decompositiondescribed in the beginning of this section.

Theorem 4.2.7 (Decomposition) Let P(x, y) be a polynomial in y of the form

P(x, Y) = Y" + Bl (x)Yn-l + ... + Bn (x),

where each B, is real analytic at x = 0. Then there is a positive integer k suchthat P can be written in the form

P(tk, y) _ (y - RI (Y - R2( )) ... (Y - RaO) Y) . (4.10)

Here each of R1, R2,..., Ra is real analytic at = 0, Q(1;, y) is a polynomialin y whose coefficients are real analytic at 1 = 0, and, for t i4 0 but near zero,Q(t, y) has no real roots.

Proof. We will argue by induction on the degree n of the polynomial. Obviously,the result holds if n = 1. Let us now consider n > 1 and assume that the resultholds for each polynomial, with real analytic coefficients, that is of degree lessthan n in y.

If P(x, y) has no real roots for x 96 0 but near zero, then P(x, y) is already inthe form (4.10).

Now assume that there is a sequence x j j4 0 with xj -* 0 such that, for each j.P(xj, y) has at least one real root. Note that, as a consequence, P(0, y) will alsohave at least one real root.

There are two cases to consider: (1) P(0, y) can be written as the product ofrelatively prime real polynomials go(y) and ho(y), both of degree 1, or (2)P(0, y) _ (y - r)" for some real r.

Case 1: By Theorem 4.2.5 (Hensel's lemma), we can write P(x, y) = G(x, y) -H(x, y) where G and H are polynomials with real analytic coefficients and bothG and H have degree less than n. By the induction hypothesis, we can write eachof G and H in the form (4.10):

(Y - R1O) (Y - R2(0)... (Y - Ra(0)H( k2,Y) = (Y-SI())(Y-S2($))...(Y-Sp(y))T(t,Y).


We set k = LCM {kt, k2), and let a and b be such that LCM {kt, k2) = akj _bk2. Then we have

y) = G[(ra)'t,, y] y],

which exhibits P in the form (4.10).

Case 2: In this case, we define the new variable y' by setting

y = y' - (1/n)Bi(x)

and substitute in P(x, y) to obtain the new polynomial

P*(x, y') = (Y')n + B2* (x)(Y')n-2 + ... + Bn_, (x)Y + Bn (x)

Note that0 = B2 *(O) = B (0) _ ... = B; (0)

If B2, B3, ... , B,* all vanish identically, then P(x, y) = [y + (1/n)Bj (x)]n,and P has been put into the desired form (4.10).

So now assume that not all the B vanish identically. For each i for which B,does not vanish identically, let xP' be the smallest power of x occurring in B' , thatis, let pi > 1 be such that, at x = 0, Bi *(x) B7 (x) Ix P' is real analytic, whileB; (x)/xP'+' is not real analytic. Note that Bt *(0) 96 0. Let or be the smallestof the numbers pi / i as i varies over the integers 2, 3, ... , n for which B7 doesnot vanish identically. Let i, be such that or = pi./i, (in case i* is not uniquelydetermined, just choose one). Finally, write a = P/K, in lowest terms, with a andK positive.

Define additional new variables y" and x" by setting x = (x")K and y' _y" (x")l, where the exponents K and a are the positive integers defined in thepreceding paragraph. Substitute in P*(x, y') to obtain the new polynomial

(Y/l)n(Xll)nl + (Y")n-2 (X1l)(n-2)1 B2*[(xl/)K)

+ (Y"/)n-'(x/l)(n-l)t

B,[(x//)K)

+ Bn[(x/I)K)

which, using

we can rewrite as

Bj [(xIl)K] = (X")PIK B',[(x/I)K],

(x")"I(Y/l)n + (y")4-2 (Xl/)P2K-u B2**[(x/l)K]


+ (yi)n-' (x")P"-il B7*[(x")K]

+ (xii)PMK-2t B, *[(xri)K]J

Since we have pi Ii > e/K for all i for which Bi does not vanish identically, wesee that

ps*(x'', Y") = (Y')" + (y" )i-2 (x")P2K-21 Bet [(x/)K]

+ (X")P.K-Ml Bnt[(X//)K]

is a polynomial in y" with coefficients that are real analytic functions of x" nearzero.

Recall that there is a sequence xj 0 0 with X1 -). 0 and such that, for eachj, P(xj, y) has at least one real root. It follows that P"(0, y") has at least onereal root. Note also that, since pj,K - i.8 = 0 and Bi*`(0) 96 0, it follows thatP"(0, y") cannot have a root of multiplicity n. Thus P"(0, y") is the product ofrelatively prime real factors go(y") and ho(y"), both of degree > 1.

We again apply Hensel's lemma, this time to write

P.s(x" y") = G(xii, y") H(x" yr)

where G and H are polynomials with real analytic coefficients and where both Gand H have degree less than n. By the induction hypothesis, we can write each ofG and H in the form (4.10):

G(4' ,y") =H(tk2,y") =

We set k = LCM (k1, k2), and let a and b be such that LCM (k1, k2) = ak1 =bk2. Then we have

Y") = G{(°)k', Y,,] - y']Finally, working back through the substitutions x = (x')K = kK and

y = y' - (1/n)B1(x) = y" (x")1 - (1/n)B1((x")K) = y kl - (I/n)B1(tkK)

we see that P can be exhibited in the desired form (4.10). 0We are now in position to state a form of Puiseux's theorem. Let us denote by

P the family of functions f (4) which are defined on some open interval (0, a),a > 0, and can be written in the form

f() = tNg(r)


for some integer N, some positive integer k, and some function g which is realanalytic on an interval containing (-(a)1, (a) ). It is clear that P forms a fieldunder the usual arithmetic operations.

Theorem 4.2.8 (Puiseux's Theorem) If f (l;) is a continuous function, definedfor sufficiently small positive i , for which y = f (l;) satisfies a polynomial equa-tion

Ao(l; )Yn + A I An-I WY + An( ) = 0

with coefficients A0()..... An W in P, then the restriction of f (t;) to someinterval (0, a), a > 0, is in P.

Proof. This result follows easily from Theorem 4.2.7.

One application of Puiseux's theorem is to obtain information about the smooth-ness of solutions to polynomial equations with real analytic coefficients (or withcoefficients in P, which is no more general.) Among the possible results that onemight exhibit as typical, we have chosen the following:

Theorem 4.2.9 Let I and J be open intervals. Suppose f (x) is a continuousfunction on I such that

(1) f(x)EJforxE1,(2) P(x, f(x))=Oforx E 1,

(3) for each x E I there exists a unique y E J with P(x, y) = 0.

Here P(x, y) is a polynomial in y with coefficients which are real analytic func-tions of x. If f E Cp.1 then, for each x0 E 1, there exists a > 0 such thatf E Cp+1' in a neighborhood of x0.

Proof. Consider an arbitrary x0 E 1. By the decomposition theorem, we knowthat there are integers N and k, a positive 8, and a real analytic function g suchthat f(k) _ kNg() , (4.11)

for sufficiently small l;. Moreover the right-hand side of (4.11) satisfies the poly-nomial equation

0.

Let the powers of occurring with nonzero coefficient in the series for g be d1 <d2< .

Suppose that k is even. We claim that every di is even. If this were not thecase then, for sufficiently small l; , there would be two solutions of the polynomialequation which lie in J. Thus we can remove the common factor of 2 from k andfrom all the d1. It follows that k may be assumed to be odd.

Suppose that k is odd. For l; Ng( Ilk) to be CP'I we must have N + d1 /k > 0and either that k divides every d, or, if d;. is the least d1 not divisible by k, then


N +di./k > p + 1. In the first case, f is real analytic and, in the second case, fis Cp+1," with the number a = N +d;./k - p - 1.

Remark 4.2.10 It seems to be an open question whether a result like Theo-rem 4.2.9 is true for polynomials having coefficients which are real analytic func-tions of, or even polynomials in, more than one variable.

The next consequence of Puiseux's theorem follows readily and illustrates theprinciple that a smooth subvariety of a real analytic variety is in fact analytic.

Theorem 4.2.11 Let P (x, y) be a polynomial in y with coefficients which are realanalytic at xo.1f f E C°O is such that P(x, f (x)) = 0, then f is real analytic atxo.

There is no exact substitute for Puiseux's theorem for functions of more thantwo variables. On the other hand, [BM 90; Section 4] gives a version of Puiseux'stheorem in several variables, and, in some sense, Hironaka's resolution of sin-gularities theorem (Section 6.2) provides some of the same kind of informationin every dimension. We also point out that M. Artin's theorem on solutions ofanalytic equations [AM 68] can in some circumstances serve as a substitute forPuiseux's theorem, in particular, in generalizing the preceding theorem to the mul-tivariable setting.

4.3 Separate Real Analyticity

It is well known that a function of several real variables that is C°O in each vari-able separately does not necessarily enjoy any (joint) smoothness as a function ofseveral variables. A simple counterexample is the function

xy for (x, y) 54 (0, 0),f (x, Y) = S x2 + y2

l 0 for (x, y) = (0, 0) .

In general, a function that is separately CO0 can be expected to be no better thanmeasurable (see [KS 83]).

By sharp contrast, a function of several complex variables that is holomorphic(in the classical one-variable sense) in each variable separately is, by a deep resultof Hartogs, C°O, indeed real analytic, as a function of several variables. It alsoturns out to be holomorphic as a function of several complex variables by anyother standard definition. These matters are discussed in detail in [KS 82].

Thus it seems natural to discuss functions of several real variables that are realanalytic in each variable separately. The function f (x, y) exhibited above showsthat, in the absence of additional hypotheses, we cannot expect a separately realanalytic function to be even continuous as a function of several real variables.

4.3. Separate Real Analyticity 105

Nonetheless, it is an astonishing fact that there exist C°O functions (as a functionof two variables) on R2 that are separately analytic but not jointly analytic. Thisassertion (from [BF 61]) can be proved using category-theoretic considerations.

On the positive side, one can also use category theory to prove that a sepa-rately real analytic function is in fact real analytic (as a function of several realvariables) on a dense open set. In the past decade, Siciak [SJ 90] has completelycharacterized the singular sets that can arise for separately real analytic functions.As early as 1912, Bernstein [BL 67] showed that, in the presence of some milduniform hypothesis (such as continuity, or local boundedness), a separately realanalytic function is jointly real analytic.

Thanks to a theorem of F. Browder [BF 61] and P. Lelong [LP 61] (the resultof Lelong is more general and both results are subsumed by the later work in[SJ 69]), separate real analyticity turns out to have much in common with separatecomplex analyticity. But some ambient, or Tauberian, hypothesis is required toobtain a full positive result. It is this matter that we shall treat in the presentsection.

Siciak's proof of the theorem under discussion here uses complex methods (justas a real analytic function of one real variable is locally the restriction to the realline of a complex analytic function, so a real analytic function of several realvariables is locally the restriction to RN of a complex analytic function of sev-eral variables). Browder's earlier proof of the same result treats the real analyticfunctions directly: the proof consists in estimating the size of the coefficients ofthe Taylor expansion. This methodology is much more in the spirit of the presentbook than is Siciak's. And while Siciak's proof is softer than Browder's, it isconsiderably longer. We present the proof that appears in [BF 61].

Definition 4.3.1 Let f be a function on an open subset U of RN. We say that fis s e p a r a t e l y analytic if, f o r each j = 1, ... , N and each collection of N - 1 realvalues C = (Ch c2, ... , c j_ 1, c j+), ... , cN) such that

Uj,C = [x E R: (CI,C2,...,Cj-1, X,Cj+I,...,CN) E U)

is not empty, the function

Uj.C 3X +.f(C1,C2, ,Cj-1, X,Cj+l,...,CN)

is real analytic as a function of one real variable.

Definition 4.3.2 A function f on an open subset U c RN is called jointly realanalytic if it is real analytic as a function of several variables in the sense used inthis book, i.e., as in Definition 2.2.1.

Now we state Browder's theorem. For clarity we treat functions of two realvariables only. The proof transfers directly to the N-dimensional case.

Theorem 4.3.3 Let I be the interval (-1, 1). Let f (x, y) be a function on I x Ihaving the property that f y) E CI (I) for each fixed y E I and f (x, ) E


C°O(1) for each fixed x E I. Suppose that there is a positive constant Co > 0with the property that, for every k = 0, 1, 2, ..., we have

axk (x, y)I < Co k! (4.12)

for every (x, y) E I x I and

ayk f(x, y)I Cp k! (4.13)

for every (x, y) E I x I. Then f is a jointly real analytic function of two variablesonIxI.

Notice that the hypothesis of the theorem is not simply that f is real analyticin each variable separately but that there is some uniformity of the analyticity inthe x variable when the y variable is thought of as a parameter (and vice-versa). Itis instructive to note that similar results hold in the C°O category: Separately C°°functions need not be smooth, but if there is some uniformity of estimates on thederivatives, then joint smoothness follows. A discussion of these matters in theCO0 category appears in [KS 83].

Our proof of the theorem is broken up into several lemmas, some of which haveindependent interest.

Lemma 4.3.4 A function satisfying the hypotheses of the theorem is (jointly) CO0on I xI.

This result is of sufficient interest that we sketch two proofs.

Proof 1. By a result in [KW 33], the function f is measurable since it is separatelycontinuous. Inequalities (4.12) and (4.13) show that f and its pure derivativesare bounded. The derivatives are of course measurable since f itself is. Hencef E L°O. Thus it is easy to see that the derivatives

at

axk(x, y)

k

and ay (x,Y),

calculated as classical derivatives of a function, coincide with the derivatives wheninterpreted as distributions (this is just an exercise with integration by parts andthe definition of distribution derivative). Thus for any integer r > 0 it holds that

a ayf = X- + aye f

is bounded. Standard regularity theory for elliptic partial differential operators (ofwhich L is an example-see [BJS 63]) implies that any mixed partial derivative


of f, in the sense of distributions, satisfies

2

axm ayn E LI«

The Sobolev embedding theorem (see [SE 70] or [KP 99]) then yields that, aftercorrection on a set of measure zero, f is infinitely differentiable. But f is alreadyinfinitely differentiable in each variable separately as presented. So no correctionat any point is either necessary or possible. We conclude that f is a C°O function.

0Proof 2. As in the first proof, f is bounded and measurable. Let 4(x, y) bea C°O function of compact support in I x I that is identically equal to 1 in aneighborhood of the origin. We will prove that g - ¢ f is a C°O function.

Now the hypotheses of the theorem, together with the product rule, yield that

ak

a- (x, Y) andaks 8y(x,

Y)

are bounded functions on R2 with compact support. In particular, each of thesederivatives is an L2 function.

Let g(i,`, q) denote the usual two-variable Fourier transform of g. Then a stan-dard result of Fourier analysis yields (see [KY 76]) that

17)1 and I,7kg($, 71)I

are L2 functions for every k. But then it is elementary to see that, for any nonneg-ative integers m and n, it holds that

is L2. In turn, this implies (again see [KY 76]) that the distribution derivative

am+ng

axmayn

is an L2 function. Now the Sobolev embedding theorem yields, as before, that fis in fact C°O smooth as a function of two variables.

In what follows it will be convenient to use the notation

Dk to stand for

am+n f

ak

axk

andak

Dk to stand forkk

Y


Lemma 4.3.5 In order to prove Theorem 4.3.3 it suffices to establish the existenceof a constant CI such that, for all j, k > 0, the inequality

(DDDyf)(0,0)I <CI+k(j+k)! (4.14)

holds.

Proof. To see that (4.14) implies the theorem, we define

ft(x,Y)=E i(DxDkf)(0,0)xjyk.

jk ji.kThe inequality (4.14) implies that this series may be majorized by the series

E !k!C;+k(j+k)!Ixl' IYI".j,k

This last is the power series expansion of the function

(1-CiIxI-C1IYI) -' ,

which converges for x, y sufficiently small. Thus fi (x, y) is a real analytic func-tion in a small neighborhood of the origin. What is more, for every j, k,

(DFDkf)(0,0) = (DxDkfl)(0,0).

For x near the origin, the functions f (x, 0) and ft (x, 0) are real analytic func-tions of x with matching derivatives at the origin. Thus f (x, 0) = fl (x, 0) for xsmall. Since (4.14) implies that Dy f (x, 0) is a real analytic function of x for xsmall, a similar argument shows that Dkk, f (X' 0) = Dkk, ft (x, 0) for x sufficientlysmall. But this last enables us to argue, for small x, that f (x, ) and ft (x, ) areboth analytic near 0 and have the same derivatives of all orders at y = 0. Thusf (x, y) = fi (x, y) for x, y small. So the original function f (x, y) is real analyticin a neighborhood of the origin.

Of course we could apply a similar analysis to any other point of I x I. Thusthe new condition (4.14) implies Theorem 4.3.3. This is what we wish to prove.

0Proof of Theorem 4.3.3. We henceforth concentrate our efforts on proving (4.14).Define the function

0 if 1 < Is] ,

z(s) = 1 if Isl < 1/2,2-21s1 if 1/2 < Isl < 1 ,

and set(x, Y) = µ(x) p (Y)


Setl,r(x, Y) = (C(x, y))r+2 for r = 1, 2, .... (4.15)

Then Cr(x, y) is an (r + 1)-times continuously differentiable function with sup-port in the closed square [-1, 1] x [-1, 1]. Combining the spirit of the two proofsof the first lemma, we define a partial differential operator by

A2,,,=D2"+D2"+1.

Using a little Fourier analysis, we can construct a solution operator for Al, asfollows. For m > 1 we define

e2m(x, y) = 1,712m " + 1)-' d, d,7.IR JR

By the choice of m, the integral converges uniformly on R x R.If O(x, y) is a CZm function of compact support, then

(A2,"O) (4, ii) = (1'12"' + Igl2ni + 1) k, 7 ) .

Notice that this last expression has symbol that is the reciprocal of the symbol ofe,,,,. If 0, >li are L1 functions and their convolution is defined to be

0 **(X' Y) = IRIRO(x-s, y -t)>Ir(s,t)dsdt,

then (see [KY 76])

(0*>r)

It follows that if v(x, y) is a C21 function with support in I x 1, then

v(x, y) = e 7 , , , * (A7,,,v) f o r x E I X I . (4.16)

Now let j, k be two nonnegative integers such that j + k < 2m - 2. We maydifferentiate the expression defining e.,,, a total of j + k times under the integralsign to obtain

(Dxj Dye2",)(x, y) = jf tl+k j 7k 11112- + I)-dg dp

(4.17)By the choice of j and k this integral converges absolutely, so the Lebesgue dom-inated convergence theorem guarantees that the differentiation under the integralis justified.

It follows from (4.17) that Ds Dye,,,, is continuous and bounded for j + k <2m - 2 with a bound KO independent of j and k.


Now by differentiating the equation (4.16) under the integral sign a total ofj + k times, with j + k still being less than 2m - 2, we have

D/ Dyv(x, Y) = J (Dx Dye,.,,,)(x - s, y - t) A?,,,v(s, t) ds dt .lxt

Using our estimate on the derivatives of e2,,, we find that

I Dx Dyv(x, y)I < 4 Ko sup t)I . (4.18)(s,t)el xl

(The factor of 4 comes from the area of ! x !.) We will apply (4.18) to the function

v(x, y) = C 2 , . y)f (x, y) ,

where f is the function given in our theorem and C2n is as in (4.15). We take(x, y) = (0, 0) and v = S2m f. Recalling that ('2m is identically 1 in a neighbor-hood of the origin, we find that

I(DDDyf)(0, 0)I 15 4 Ko sup y ) )(X, y)E l x l

Now we study the term on the right-hand side of this inequality. Observe that

A2n(C2n f) = ((Dx)2m + (Dy)2n + l) (C2nf)

= C2n ((DX)2n + (Dy)2n + 1) f + R

= A2n f +7Z,

where the remainder term R involves derivatives of f that are of order strictlyless than 2m :

R = CJ,k Dl (C2n)Dxkf + C1,k Dy(C2n)Dyf

j+k=2m j+k=2mj>0 j>0

(4.19)

(This is a standard fact about commutation of differential operators, or more gen-erally of pseudodifferential operators. What we are saying here is that if P is anoperator of order 2m and Q is an operator of order 0, then P (Q f) = Q (P f) + R,where R is of order less than 2m. The verification of this assertion is a simple ex-ercise in calculus.)

We now see that, when j > 0,

y)I < M/ (2m +2)j (4.20)


and

I (Dy C2m) (x, y)I 5 Mj (2m + 2)j (4.21)

for some positive constant M. These estimates may be obtained by direct compu-tation from the explicit definition that we have given for Cam; it is convenient touse induction. Now we estimate the error term R. When the differential operatorDi' is applied to a product of functions w I w2 there result 22' terms of the formDX wt D.Pw2 with coefficient 1 (note here that it is convenient not to gather liketerms). Thus the sum of the coefficients

E ICj.kIj,k

in equation (4.19) does not exceed 22. By the hypotheses of the theorem and byestimates (4.20) and (4.21) we have (assuming, as we may, that the constant Coin the hypotheses of the theorem exceeds 1) that

IRI -<

(j+k=2msup IDkfl+ sup IDkfIICj.kI

j+k=2m

< 2.22"' sup Mf (2m + 2)1 . (Co)k k!j+k=2m

< (2. M C1 )2m sup (2m + 2)j (2m + 2)kj+k=2m

<

By similar, but simpler, reasoning one may obtain a like estimate on the termC2,,, A2m f. Combining these estimates, together with our formula for A2({2m f)and our estimate for I (Dx Dk f)(0, 0)1, we find that, for 0 < j + k < 2m - 2, wehave

I(DzDkf)(0,0)I < 4KosupIA2m(0mf)I< 4Ko(supIR.I+sup1C2m A2mfl)

)2m< Kl(2 M Cl)2m(2m)2m 1+ Zm

By Stirling's formula ([CKP 66] or [HM 68]), we know that

(2m)! (47rm)1/2(2M)2me-2m

as m -+ oo. Hence there exists an absolute constant L such that, for large m,

(2m)2m < L elm , (2m)!.


Also note that

As a result, we have

I(DzDkf)(0,0)1 e)2in(2m)!

In case j + k is odd, then we choose m so that j + k 2m - 3 and rewrite ourestimate as

I (Ds D,kf)(0, 0)I 5 C (2M C1 e2)2m , (2m)1' .(2m - 3)!y (2m - 3)!(CI)2m-3 . (2m - 3)!

(3c')j+k . (j + k)!

In the case that j + k is even, then we choose m so that j + k N 2m - 4 andimitate the last argument to obtain that

I (Dz Dk f)(0, 0)1 5 (C')i+k (j + k) !

Thus, for any choice of j, k, we have proved the estimate (4.14) (introduced inLemma 4.3.5), showing that f is real analytic in a neighborhood of the origin.This completes the proof of Theorem 4.3.3.

We remark in passing that a useful lemma of Ehrenpreis [TD 76; p. 304] givesa method for constructing cutoff functions that behave like real analytic functionsup to any prespecified finite order. Precisely, the statement is this:

Let 521 CC 122 c RN be domains. Then there is a constant Co suchthat, for any integer No > 0, one can find a function *No E C°O(122)that satisfies

(1)0<CNOS1.(2) *No-Ion 521.(3) Most importantly,

I Da I U NO 1 5 Co (CoNo)1 f o r multiindices a with la 15 No.

The function 1GNO is constructed by first selecting an intermediate domain 12 be-tween 521 and 122 satisfying 121 CC 12 CC 522. Then one chooses a fixed COOfunction's with

diam(supp rl) 5 min{ dist(121, `S2), dist(f2, `122) } .

Finally, letting X be the characteristic function of b, one defines

ENO=(A*7*...*q)*X.


Here the operation * is convolution, and the factor >f occurs No times. We leavethe details to the reader.

By using the Ehrenpreis functions, one can give a quantitative version of Proof2 of Lemma 4.3.4 and thereby present a new attack on the questions consideredhere.

We conclude with some general remarks about the material discussed in thissection. The paper [TM 72] gives a characterization of vector-valued real analyticfunctions that may be considered an obverse of our main result: directions in thetarget space are treated instead of directions in the domain. The paper [BM 90]considers functions that are real analytic along every real analytic arc. In somesense, such functions are more natural than those that are only "separately ana-lytic: ' They enjoy a number of pleasant properties. (However in [BMP 91] the au-thors exhibit such a function which is not even continuous!) The paper [BJ 67a] ofBoman proves that the analogous class of functions, with "real analytic" replacedby COO, is just the COO functions themselves.

5Results Motivated by PartialDifferential Equations

5.1 Division of Distributions I

The Cauchy-Kowalewsky theorem is perhaps the most general result in the the-ory of partial differential equations. The theory needed to state and prove a basicversion of that theorem is entirely elementary. Similarly, the specific constant co-efficient partial differential equations of mathematical physics-Poisson's equa-tion, the heat equation, and the wave equation-can be dealt with by specificelementary methods. On the other hand, the development of the general theory oflinear partial differential operators with constant coefficients is tied to the moreadvanced and abstract theory of distributions introduced by Laurent Schwartz (see[SL 50]). One historically important conjecture in the theory of distributions (see[SL 55]) concerned the problem of finding a distribution S that solves the equation

4) S=T (5.1)

for a given distribution T and a given testing function (D. (One can think of thisas dividing T by 4'.) This question arose from investigations of the solvability ofa partial differential equation

P(ax)u=f (5.2)

with constant coefficients. Here P is a polynomial in several variables and a/axrepresents the n-tuple of operators (a/axt, a/axe, ... , 8/ax,,). To see the con-nection between (5.1) and (5.2), recall that the Fourier transform of a function in

116 5. Partial Differential Equations

f E L 1(RN) is defined by

T (t) =fRN

f (t )e-lair-t dt .

It is a simple matter, using integration by parts, to verify that

rafaxj) ()=2triilf(a;)

Now if we apply the Fourier transform to equation (5.2) we obtain an equa ionof the form (5.1). Solving the equation (5.1) for S is equivalent to solving for theFourier transform a of u. (See [KS 92b] for the details.)

To prove that it is possible to divide the distribution T by the function 4) it: uf-fices to have control over the rate of vanishing of 4). Lojasiewicz [LS 59] pro tedthe requisite estimate for a real analytic 4'. In case 0 is a polynomial, whicl 1 isthe situation relevant to partial differential equations, an easier proof was fol andby H6rmander [HL 58). While Lojasiewicz's work has broad significance for thegeometry of real analytic varieties (his monograph [LS 91] has been translatedinto English), it is less accessible than Hiirmander's. In this section we will proveH6rmander's weaker version of Lojasiewicz's theorem, and in Chapter 6 we willpresent a more expository treatment of Lojasiewicz's results.

Theorem 5.1.1 Let Q(xl, x2, ... , x") be a real polynomial. Let K be a compactset. Suppose the zero set, N, of Q defined by

N = ((xl,x2,...,xn) E K : Q(x1,x2,...,xn) =0),

is nonempty, and let d(g, N) denote the Euclidean distance from

n)

to N. Then there exist positive constants c and µ such that

I Q(6, 2, ... , WI ? c - N)µ (5.3)

holdsfort = (ti, t2, ..,tn) E K.

Proving this theorem is the goal of the remainder of this section. Note that, bycompactness, to prove the theorem it will suffice to prove that there exists positiveconstants c and µ such that (5.3) holds for It I < 1.

Reformulation of the Inequality. To show that (5.3) is true for Ii;1 < 1 we willneed to restate the problem in a very precise fashion. What is needed is an un-derstanding of the set of pairs (x, y) E R2 such that x > 0 and, for that choiceof x, the value y > 0 is minimal subject to the condition that there exists a pair(i;, il) E R" x R" such that

5.1. Division of Distributions 1 117

$ is in the unit ball,

il is in the zero set of Q,

no point of the zero set of Q is closer to t than is rl,

Y = IQ()I

x is the square of the distance from i to rl.

The set of conditions listed above can be expressed symbolically as follows:

3 ($, ?7) [ Y2 = Q2($) &

II2 <

Q(n) =

Q(o 0 0)].

Now, if we let C(x, y) be the preceding set of conditions, then we see that weneed to understand the structure of

{(x,y): y>0 & C(x,y) & dw[(0<w & C(x,w))=:: y<w]}.

This is a complicated problem, but it involves only polynomial functions of thevariables, equalities and inequalities, and logical connectives. We will see that allthe variables other than x and y can be eliminated from the definition of the pre-ceding set; what remains will be only polynomial functions of x and y, equalitiesand inequalities, and logical connectives. Once this is shown, the theorem willfollow easily.

It is of historical interest to note that Hormander based his proof of the abovetheorem on Seidenberg's proof of one of A. Tarski's theorems in mathematicallogic: The decidability of the theory of real-closed fields. Since it would take usvery far afield, we shall not discuss Tarski's theorem. The interested reader shouldsee Tarski's monograph [TA 51 ], Seidenberg's paper [SA 54], or the exposition in[JN 74; Chapter 5]. Since we have before us a narrower goal, we will take a moredirect route than Seidenberg's proof of Tarski's theorem.

5.1.1 Projection of Polynomially Defined Sets

We need to understand the structure of sets of the form

S={y:3z a<z<fl & P(y,z)=0},where P is a polynomial in y and z. Notice that S is the projection onto R of thepart of the zero-set of P that lies in the strip between z = a and z = 0. Our goal


Z = I -------------

z

------------a---

y2 y3 / ya

z = 1/3 --d ---------- -b------------

Figure 5.1. Projection of a Polynomially Defined Set

will be to define a class of sets that is closed under projection operations and thatincludes sets defined using polynomial equations and inequalities.

First we illustrate what happens under projection in a case with minimal com-plications.

Example 5.1.2 Suppose that

P(y, z) = a(y)z + b(y)

Figure 5.1 illustrates this situation with a(y) 1, b(y) = -(y3/3 + y), a =-1/3, and P = 1.

First, note that if

S1 =ly

:a(y)2 + b(y)2 = 01

then S1 C_ S. In the specific case illustrated in Figure 5.1, S1 = 0.Next, let C1 be any connected component of the complement of S1. Set

S2 = {y :[a(y)a + b(y)] = 0}.

Then S2 fl Cl fl s = 0. In the specific case illustrated in Figure 5.1, S2 ={ yl, y2, y3, y4 }, with yl < Y2 < y3 < y4, where yl, y2, and y3 are the three(real) roots of y3/3 + y + 1/3 = 0 and where y4 is the unique real root ofy3/3 + y - 1 = 0. These points are indicated on the y-axis in Figure 5.1. Eachsuch point on the y-axis in Figure 5.1 is the projection of a point in {(y, z)P(y, z) = 0), but a point where either z = a or z = P.

Finally, let C2 be any connected component of the complement of Sl U S2. Thisis the same as letting C2 be a connected component of the complement of

{y:[a(y)2 + b(y)] = 01

Then either C2 c S or C2 fl S = 0. Thus S is the union of sets defined bypolynomial equations in y and connected components of complements of such


sets. In the specific case illustrated in Figure 5.1.2, the role of C2 can be played byany of the five open intervals (-oo, Y0, (Y1, y2), (y2, y3), (Y3, y4), and (Y4, oo);we see that S = (yI, Y2) U (Y3, Y4) O

Motivated by the example, we make the following definition.

Definition 5.1.3 Let C. consist of all subsets, A, of the unit n-cube II" such thatfor each xo E P there exists an open ball U centered at xo with the property thatA tl U is a finite union of sets of the form

{X : P(x)=0&x E C)

where C is in turn a finite intersection of connected components of sets of theform

U\{x: Q(x)=0}.Here P and Q are required to be real polynomials.

To generalize the reasoning illustrated in Example 5.1.2, we need to developsome algebraic tools.

Common factors of polynomials. Let

f(x) = aoxm +alxm- I + ... +am_1x +am

and

g(x)=box"be polynomials with real coefficients. We assume ao 0 0 and bo # 0. It is aclassical fact that there is a rational integral form in the coefficients of f and g,known as the resultant, which vanishes if and only if f and g have a commonfactor. (Classically, this result is proved over an arbitrary coefficient field, but wedo not need such generality: The real numbers suffice.) In this section, we shallexplore the problem of finding a common factor of f and g which is of maximaldegree. Our discussion will follow that in [VDW 70] which is used to develop theresultant.

Theorem 5.1.4 Suppose that m < n. Let

f (x) = aoxm + aIxm-t + ... + am-Ix + am

and

g(x) = box" + blx"-1 + + bm_Ix + bn

be real polynomials. There are real polynomials,

R1,R2,...,Rm,

in the coefficients ao,... , am , bo,... , bn such that f and g have a common factorof maximal degree K < m - 1 if and only if

R1 = R2 = ... = RK =01 RK+1 0 0.


Proof Suppose f and g have a common factor of degree K < m -1. Let us denotethat common factor by O, so that

f=Oh and g = Ok.

Here h is of degree m - K and k is of degree n - K. Write

h = car'"-x+clx'"-r-1+...+CM-xk = 4x"--x + dlx"-x-1 + ... + dn_K.

Since

fk = gh,we may equate coefficients on the left- and right-hand sides to find that

doao = cobo

dual + dlao = cobs + c1bo

doa2 + dial + d2ao = cob2 + clbl + c2bo

dn_K_lam +dn_Kam-1 = cm-x-lb" +c",_.bn-ld,-.,a, = c,-

,b,-This is a set of m + n - K + 1 homogeneous linear equations in the m + n -K + 1 - (K - 1) variables co, ..., Cm_K, do, ... , dn_K. It will simplify the nota-tion t o consider the variables in the above linear equations to be do, ... , do-K,-CO..... -cm_K. Then the matrix of coefficients for the system can be written as

ao 0 ... 0 0 bo 0 ... 0 0

at ao ... 0 0 bl bo ... 0 0

0 0 ... a," am-1 0 0 ... bn bn-1

0 0 ... 0 am 0 0 ... 0 bn

This matrix has n - K + 1 columns that contain a's, in - K + 1 columns that containb's, and m + n - K + 1 rows. We shall denote this matrix by MK . A necessary andsufficient condition for the linear system to have a nontrivial solution is that MKhave rank less than m+n -K+ 1. This, in turn, is equivalent to all the m +n -K + 1by m + n - K + 1 sub-matrices of MK having determinant zero.

Let us introduce the notation A (m +n +K -1, K -1) for the set of all increasingmapsof(1,2,...,K-1}into {1,2,...,m+n+K-1}.Fork E A(m+n+K-1, K - 1), let Dx denote the determinant of the square matrix obtained by deletingrows A(1), A(2), ..., A(K - 1) from M. Finally, let RK denote the sum of thesquares of the Da as A runs over A (m + n + K - 1, K - 1). We have shown that, iff and g have a common factor of degree K, then RK = 0. The converse is not true,


because while RK = 0 does imply that the above linear system has a nontrivialsolution, it might be that for all such nontrivial solutions we have co = do = 0;this implies that there is a common factor of degree larger than K.

Remark 5.1.5 We can also see how to find the common factor of maximal de-gree. Suppose the condition

RI = R2 = ... = RK = 0, RK+I 00

holds. Since MK+, is obtained from MK by eliminating the first row, the firstcolumn, and column n - K + 2, we see that the first row in MK must be dependenton the others, so in finding h and k it can be omitted. Some m + n - K columnsof the remaining matrix must be independent, and by thinking of the coefficientscorresponding to the other columns as parameters, we can apply Cramer's Ruleto solve for the coefficients corresponding to the set of independent columns. Thecommon factor is obtained by dividing the resulting h into for the resulting k intog. The coefficients of the common factor are rational functions of the coefficientsoff and g.

We shall use the above theorem to investigate the projection of sets in the familyCn.

Theorem 5.1.6 If A E Cn and fl is an orthogonal projection onto a coordinatehyperplane, then f] (A) E

Proof. We sketch the proof of the result.Let P and Q be polynomials in x = (y, z) = (y1, ..., yn-1 , z). Write

P(x) = P(Y,z) = ao(Y)zm+aI(Y)zm-1+.+am-I(Y)z+am(Y)

Q(x) = Q(y,z) = bo(Y)z" + bl (Y)z"-t + ... + bn-I (Y)z + bn (Y)

Let C be a connected component of the complement of {x : Q(x) = 0). Let aand P be real numbers. Set

S=(y:3z a<z<f3, P(x)=0, xEC).

Note that S is the orthogonal projection onto a coordinate hyperplane of the set

{(y, z) : a < z < f, P(y, z) = 0, (y, z) E C} .

Fix an arbitrary yo E R"-I. We are interested in the form of S near yo.There are now several possibilities. We shall first consider the structure of S

when we make the six simplifying assumptions which we label (A1)-(A6) below.These simplifying assumptions are in fact generic. For example, assumption (A1)is that the coefficent of the highest power of z in P(y, z) is nonvanishing at y =yo. After the discussion of the structure of Sin the presence of assumptions (A 1)-(A6), we will consider what happens when those assumptions do not obtain.


ASSUMPTION (Al). ao(yo) # 0.

ASSUMPTION (A2). bo(yo) 0 0.

Theorem 5.1.4 may be used to construct the polynomials

Ri(y), R2(y),..., R,(y),

where p. is the smaller of m and n.

ASSUMPTION (A3). RI(yo) # 0.

We also need to consider the possible multiple roots of P. Let A be the usualdiscriminant of P (see [VDW 70]).

ASSUMPTION (A4). A(Y0) # 0.

Lastly, we consider whether or not the condition a < z < p is violated.

ASSUMPTION (A5). P(yo, a) # 0.

ASSUMPTION (A6). P(Y0, 0) # 0.

By continuity, we select an open set U with yo E U such that the nonequalitiesin the above assumptions (A1)-(A6) hold for all y E U. Assumption (A6) assuresus that for each y E U there are the same number of real roots of P(y, z). Thesereal roots can be indexed zI (y), ... , zp(y) so as to be continuous in y E U.Assumption (A3) assures us that (y, zi(y)) stays in the same component of thecomplement of {x : Q(x) = 0) as y varies over U. Assumptions (A5) and (A6)assure that each root zi (y) stays always in or always out of the open interval (a, fi)as y varies over U. Thus either U g S or U n S = 0.

In the discussion above, each of the assumptions (A1)- (A6) reflects what hap-pens generically, i.e., when an appropriate polynomial in y is nonvanishing aty = yo. Below we consider what happens in the exceptional cases. Each such ex-ceptional case is dealt with by specifying which of the significant polynomials inan indexed sequence of polynomials is the first that is nonvanishing at y = yo. Werefer to these specifications of polynomials as "criteria" because each provides acriterion for choosing a particular index value in the sequence of polynomials.

CRITERION (Cl). ao(yo) = 0, , ap-I (yo) = 0, ap(yo) 0 0.

Select an integer j with 0 < j < p. We shall restrict our attention to the intersec-tion of {y : ao(y)2+aI (y)2+ +aj_I (y)2 = 0) with a connected component ofthe complement of {y : a j (y) = 0). That is, we let Ut be a connected componentofIt"-I \(y:aj (y)=0} and set

UI = UI n {y : ao(y)2 + at (y)2 + ... + a1_(y)2 = 0).


On U1 the degree of P is constantly equal torn-f, and we may as well replace theoriginal polynomial P by the following new polynomial which, to save notationalcomplexity, is also denoted by P :

P(y, z) = aj (Y)zm-j + a j+I (Y)zm-j- I + ... + am-I (Y)z + am (y),

CRITERION (C2). bo(y0) = 0, ... , bq_ 1(YO) = 0, bq (YO) 0 0.

Select an integer k with 0 < k < q. We shall restrict our attention to the intersec-tion of {y : bO(y)2+bi (y)2+ +bk_I (y)2 = 0} with a connected component ofthe complement of {y : bk(y) = 0}. That is, we let U2 be a connected componentof R"-I \ {y : bk(y) = 0} and set

U2 = U2 fl {y : bO(Y)2 + bi (Y)2 + ... + bk_ 1(Y)2 = 0} .

On U2 the degree of Q is constantly equal ton -k, and we may as well replace theoriginal polynomial Q by the following new polynomial which, to save notationalcomplexity, is also denoted by Q :

Q(Y, z) = bk(Y)Z"-k + bk+I (Y)Zn-k-I + ... + (Y)Z + bn(Y)

We apply Theorem 5.1.4 to obtain polynomials

RI(Y), R2(Y),..., Rv(Y)

satisfying the conclusion of that result, where v is the smaller of m - j and n - k.

CRITERION (C3). RI (y0) = ... = Rr-I (Y0) = 0, R,(yo) 0.

Select an integer a with 0 < e < r. We restrict our attention to the intersection ofl y : RI (y)2 + R2(y)2 + + Re_ 1(y)2 = 0) with a connected component of thecomplement of {y : Re(y) = 0}. That is, we let 03 be a connected component ofR1 -I \ (y : Re (y) = 0) and set

U3 = U3 fl (y : RI (Y)2 + R2(Y)2 + ... + Re-I (Y)2 = 01.

On U1 fl U2 fl U3 the polynomials P and Q have a common factor of constantmaximal degree e. Suppose we write P = (PH and Q = 4>K, where 4) is thecommon factor of degree a in z. Since Q does not vanish on C it is clear that thevanishing of P(y, z) is equivalent to the vanishing of H(y, z). So we may as wellreplace P by H, after multiplying by the appropriate polynomial in y to assurethat the coefficients remain polynomials in y. As before we retain the notation Pfor the new polynomial.

Because the transition from two real roots to two complex roots is marked bya double real root, assumption (A4) is the simplest way to insure that the number


of real roots remains constant. In general, to insure that the number of real rootsremains constant, one must keep the set of multiplicities of roots constant. Thisrequires considering all the derivatives of P. For each integer j less than thedegree of P, we apply Theorem 5.1.4 to P and 8i P/8zi, to produce the sequenceof polynomials

Aj.l (Y), A j.2(Y), .. .

Aj,s(YO)54 0.

Criterion (C4 j) indicates that, at y = yo, P and 8j P/azj have a common factor ofdegree precisely s -1. Because the condition A j,s (yo) 0 0 implies that A j, (y) 00 holds for y in a neighborhood of yo, we see that, for y near yo, while P andai P/8z3 might have a common factor of degree smaller than s - 1, they cannothave a common factor of degree greater than s - 1. Thus, we consider the set ofy near yo where the P and ai P/azf have a common factor of degree precisely t jwithtj <s-1.

Select an integer tj with 0 < tj < s - 1. We further restrict our attention tothe intersection of {y : Aj,1(y)2 + Aj,2(y)2 + .. + &j,r_I(y)2 = 0} with aconnected component of the complement of (y : A j,t, (y) = 0). That is, we letW j be a connected component of R"-I \ (y : Aj,tj (Y) = 0) and set

W ; = W j n {y : A j, I (Y)2 + A j,2(Y)2 +... + A.t;-1(Y)2 = 0).

On UI n U2 n U3 n Wj the polynomials P and aj P/8zi have a common factorof constant maximal degree t j. The sequence of integers ti, t2, , tm-j-I (recallthat, for y E UI, m - j is the degree of P as a polynomial in z) determines andfixes the set of multiplicities of the roots of P, so that the number of real rootsremains constant on UI n U2 n U3 n WI n W2 n W,-j-1.

Assumptions (A5) and (A6) insured that no real roots violated the constrainta < z < P. In general, one or several roots could violate the constraint.

8"-IPCRITERION (C5). P(yo, a) = . . . = (Y0, a) = 0,

azu-1

aup(Yo, a) A 0.

azuau -I P au

CRITERION (C6). P(yo, fi) _ ... = az° (Y0, ) = 0, azP (Yo, P) 96 0.

Select integers ub and vt with 0 < ub < u and 0 ut < v. We restrict ourattention to the intersection of

a -I PI y : P(y, a)2 + ... + 8z°b! (Y, a)2 + P(Yo, p)2 + ... +

8z-(Y,

#)2 = 0

with a connected component of the complement of

( a"b P a °, PY

az°b(Y, a) . azut (Y"6) = 0


That is, we let U5,6 be a connected component of

and set

a0b P 3111 P

a (Y+fi)=0I

U5,6 = U5,6 n

{Y P(Y,a)2+...+as (Y,a)2+P(Yo,p)2+...-{ awl(Y,#)2=0}

Having sufficiently restricted our attention, we see that we are now on a set whicheither lies in S or does not intersect S. That is, we have either

U,nU2nU3nw,nw2nwm_j_,nU5.69S

or

U, nU2nU3nw, nw2nwm_j_, nU5,6ns=8.

Proof of Theorem 5.1.1. We recall the discussion at the start of this section, andlet A be the set of pairs (x, y) such that x > 0 and

C(x, Y) & Y w [(0 < w & C(x, w)) = y < w] ,

where C(x, y) is the set of conditions

Y

3t,q [ y2

It I2

Q(7)

It_712V (I('_ I2<x

The description of the set can be rewritten as follows: The inequalities involving<, >, and 96 can all be replaced by disjunctions involving just =, <, and > .The logical connective can, of course, be expressed in terms of negation anddisjunction (i.e., p q is equivalent to -p V q). The quantifier V can be replacedby -, 3 Such a rewriting would make the notation very lengthy, so we shall notactually carry it out. But such a rewriting combined with repeated application ofTheorem 5.1.6 above shows that A is, in fact, in C2. We apply the definition of C2at the point (0, 0) to see that, in a neighborhood U of (0, 0), A is a finite union ofsets of the form

{(x, y) : P(x, y) = 0 & x E C)


where C is in turn a finite intersection of connected components of sets of theform

U\((x,Y): Q(x,Y)=0)From Puiseux's theorem, Theorem 4.2.8, we know that, for such a set S, eitherthere exists 8 > 0 such that

0<x<8 = 0OtClos{y>0:(x,y)ES},or there exist 8 > 0, c > 0, p > 0 such that

0<x<8 = {(x,y):0<y<cxµ}CS.The first choice cannot hold for all of the sets making up u n A. Since there areonly finitely many sets S to consider, we see there exist 8 > 0, c > 0, 1A > 0 suchthat

0<x <8 = {(x,y):0<y <cxµ}CA.It follows that if d(4, N) < 8, then IQ(!)I > c d(l;, N)µ, as desired.

In the next section, we indicate briefly how the result on division of distribu-tions follows from this estimate on the rate of vanishing.

5.2 Division of Distributions II

Following Hormander, we begin this section with the corollary of Theorem 5.1.1which is needed for the proof that a tempered distribution can be divided by apolynomial.

Recall that, for a compact set K 9 R", Theorem 5.1.1 provides a lower boundof the form c d(i;, N)', as 1; varies over K, for the absolute value, IQ(1;)I, of areal polynomial, where N is the part of the polynomial's zero set that lies in K andc and µ are positive constants; the set N is regired to be nonempty. The corollarythat we need allows l; to vary over all of R", but requires the introduction of afactor that is a negative power of (1 + It 1).

Corollary 5.2.1 Let Q(i;t,t2_., t;") be a real polynomial, and let N be its zeroset. Then either

(1) N is empty and there are positive constants c, it' such that

foralli; ER",

or

(2) N is nonempty and there are positive constants c, µ', p." such that

I Q(t)I > c (1 +Itl2)_µ

dist(l;, N)µ" , forall t E R" .

5.2. Division of Distributions II 127

Proof. We may and shall assume that Q is nonconstant. We define another poly-nomial q by

q(') = In12ni Q(77/17712),

where m is the degree of Q. We have

fore 360.

Denote the zero set of q by Z. Since Q is nonconstant, we automatically have0E Z.

Notice that if r1, 36 0, q(771) = 0, and r1i -> 0, then Q(n;1177;12) = 0 andIn;/1';121 -+ oo. Thus the origin is an isolated point of Z if and only if N iscompact.

Case 1: The origin is an isolated point of Z. Choose r > 0 so that

z n B(0, 2r) = (0).

We can apply Theorem 5.1.1 to obtain positive c and µ such that

Iq(')1 >- c dist(q, Z)`` = cl'I1, for Jill -< r.

It follows thatcl

12m-A, for ICI > l/r..

The conclusion of the corollary now follows easily by applying Theorem 5.1.1again to Q and {l; : 141 < ; ).

Case 2: The origin is not an isolated point of Z. Again we can apply Theo-rem 5.1.1 to obtain positive c and µ such that

Iq(')I > c dist(77, Z)u, for 1771 < 1.

But in this case it is nontrivial to estimate dist(77, Z), for it may well be thatdist(77, Z) < I'I.

We consider the possibility that dist(77, Z) < 1771. Let n* E Z be such thatdist(77, Z) = 177 - 77*1. Since In -'*1 < 1771, we have q* 0. Associate to 11the point 1; ='/17712 and to q* the point l;* ='*/In*12. We have 1;* E N. Thepossibility that 0 = dist('7, Z) is uninteresting, because then = * and bothsides of the inequality in (2) are 0. So we may assume that 1; i4 $*. The trianglewith vertices 0, r* is similar to the triangle with vertices 0,'*, n (the scalingfactor is 1771177*1 = 1.1-' so

It - $*1I' -'*1=ICI I VI

Thus, for 11; I > 1, we have

IQ()I >-I2m-"lr I -'Ì - 1"

> cIIti"-1LH(l;)1L,


inf {I=:'1-t 1? - r'I : r' E N} .

If H(t) < , then there must exist s;" E N with

H(s) = It-I-' It - r"I.

But then it is easy to see that It"I 5 21t 1, so

H(t) > dist(l;, N)

2It I

Since N is nonempty, there is a positive constant ct such that

dist(l; , N) 5 ct It I, for It I > I ,

so

Thus

minI dist(i;, N) dist(i;, N)

12' 2(l+ct)(1+ItI) 2(1+ct)(1+I1;I)

1001 ? C2 (1 + II)2'"-2µ dist(t, N)µ , (5.4)

fort with It I ? 1 associated with q such that 0 < dist(q, Z) < I.

For q such that dist(q, Z) = Iql, we can use the simpler estimates as before toextend the applicability of (5.4) to all 1: with ItI > 1.

Finally, the result follows easily by one further application of Theorem 5.1.1 toQ and It : ItI < 1}. 0

Now we shall apply the estimates to the division problem. We begin with somedefinitions.

Definition 5.2.2

(1) Denote by S the space of infinitely differentiable (real- or complex-valued)functions defined on all of R" which satisfy

Qa.p(f) = sup Ir D 00, (5.5)

for all multiindices a and P. Such functions are called rapidly decreasing orSchwartz functions.

(2) We topologize S by using the seminorms Qa,p for each choice of a and f. Soequipped, S is a topological vector space.

(3) A continuous linear functional on S is called a tempered distribution or Schwartzdistribution. The space of all tempered distributions is denoted by S'.

5.2. Division of Distributions II 129

To assist us in making various estimates in the remainder of this section, weintroduce some notation.

Definition 5.2.3 For an infinitely differentiable function f, non-negative integersP and in, x E R1, and a subset B C R1, set

If(x)Itm =sup {(I+IXptlDa f(x)I : Dal <ml

IfIt.m,B E B) .

In the case B = 0, we will set Ifl,m.B

equal to 0 rather than -oo.

With this notation, the seminorms

I I t,m,R^

give the same topology on S as that given by the seminorms qa, p.In general, the multiplication of distributions is ill-defined (nonetheless see re-

cent developments due to Colombeau [CJ 90]), but it does make sense to multiplya tempered distribution by a slowly increasing smooth function, in particular, bya polynomial: If T is a tempered distribution and P is a smooth function which,for each multiindex a, satisfies

IDaP($)I <Ca

for some Ca and some ka, then we set

(PT)(4>) = T(P¢),

(5.6)

(5.7)

for each ¢ E S. Certainly, P¢ is a rapidly decreasing function, so the right-handside of (5.7) is defined, and one checks easily that it is, in addition, a continuousfunctional in the topology on S.

Another way of looking at the multiplication of tempered distributions by smoothfunctions with polynomial growth is to consider first the operation of multiplyingthe rapidly decreasing functions by such a function:

Lemma 5.2.4 Let P be a slowly increasing infinitely differentiable function [i.e.,a function satisfying (5.6)]. The map Mp : S -i S, defined by

MP(4>) = PO'

is continuous.

Proof. This result follows immediately from the definition of the topology on S.0


Remark 5.2.5 In light of Lemma 5.2.4 and (5.7), the multiplication of a tempereddistribution by the function P is simply a composition of continuous functions.

The following theorem is the main result of this section.

Theorem 5.2.6 Suppose P 0- 0 is a polynomial.

(1) The mapMp:S -i S, defined by

Mp(cb) = P4',

has a continuous inverse (defined only on its image, of course).

(2) If T is a tempered distribution, then there exists a tempered distribution Ssuch that

PS=T.

(3) If T is a tempered distribution, then there exists a tempered distribution Swhich solves the partial differential equation

P(. ) S=T.

The heart of the matter is (1). Since the complement of the zero set of thepolynomial P is dense, it follows that the map Mp is one-to-one. Thus there is aninverse map from the image of Mp to S. The proof that the inverse is continuouswill clearly depend on establishing estimates on the seminorms on S.

Before we sketch the proof in the general case, we will illustrate in a simplesetting why one might expect the size of Pf to control the size of f.

Example 5.2.7 Let f : R -+ R be infinitely differentiable. Let I denote theinterval [-1, 11. Let P be the polynomial P(t) = t. It is obvious that

if (-01 = I(Pf)(-l)I. if MI = I(Pf)(1)I I

and that if the maximum off on ! occurs at some to with - I < to < 1, then

I(PfY(to)I = IP'(to)f(to)+ POW '(101 = If (101 -

Since the maximum on I either occurs at -1. 1, or at a to with -I < to < 1. itfollows that

1110.0.1 <-I P ! ,0 1 1 .

It is not to hard to show using inductive arguments that, for nonnegative integersk and m, if P(t) = rt. then there is another nonnegative integer m' and a realconstant C such that

f<C Pn....1 I f la^'.1

5.2. Division of Distributions lI 131

Indeed, by these means we could obtain similar estimates for any polynomial inone variable which does not vanish identically. The argument for a polynomial inseveral variables is significantly more difficult precisely because the zero set canbe much more complicated. Also, the seminorms on S require the inclusion of apolynomial factor, which will interfere with the easy argument we have just used.To deal with the general case, we need to define some more technical norms:

Definition 5.2.8 For an infinitely differentiable function f, nonnegative integerse and in, and a subset B c R', set

max fF

ft.m.B

=.m.B

sup {(1 + n)I/l - nlm-IaI : Jul < tit, , 0 E B,t11 1

.

Here

Rm( ,17)=f( )- I Daf(q)( -tI)alal<m

Jul!

is the remainder in Taylor's formula. As before, in case B = 0, we set I f

0.t^ B

Let i;") be a polynomial which does not vanish identically. Denoteby Bk the set of points at which P has a zero of order k or greater. A particularis in Bk if and only if

Da P(r;) = 0 for every multiindex a with IaI < k.

Define Bo = R" and Bd+i = 0, where d is the total degree of P. We have

C

Using our notation, to prove (1) of the theorem we need to show that, for eachpair of nonnegative integers 2 and in, there exist nonnegative integers e' and in'and a positive constant K such that

If( )Ir.",.Bo ` K - I V^', BO '

This is proved by an inductive argument beginning with the trivial fact that

0=1 POI £.m.BJ+I <I P)OIC,m.B0

For technical reasons, it is convenient to use the more complicated norms I' 1 f


Lemma 5.2.9 For each triple of nonnegative integers k, 1, and m, there existnonnegative integers E' and m' and a positive constant K such that

mIf1tm.B1"M"BO

Sketch of Proof. Fix k, e, and in. We argue inductively, so we may assume thatthe statement of the lemma holds fork and prove the result for k+ 1. The first stepis to prove that there exist e', m', K so that the conclusion holds for any functionf which satisfies the additional hypothesis that it vanish to order m' on Bk+t (thisextra assumption will be eliminated afterward). Once that m' is determined, wewill introduce an approximation to an arbitrary f, based on the Whitney extensiontheorem.

We shall assume that Bk+1 0; the other case is easier. We apply Corol-lary 5.2.1 to the polynomial

Q1() _ I )I2,IPI<ik

for j = 1, ... , m + 1. Notice that each Qi vanishes exactly on Bk+t and that allderivatives of Pi of order strictly less than jk vanish on Bk, so

Qi() _ E I DI (P()I )l2, fore E Bk.IPI=jk

It follows that

E l )l2 > c Bk+1)21(1 + IS I2)_µ" . (5.8)IPI=ik

where we choose the constants µ', A" large enough and c small enough that (5.8)holds for j = 1.... , m + 1. Now m' is chosen to be an integer such that

m'? µ'+km+k+m.Suppose f vanishes to order at least m' on Bk+t. For lal < in, let GQ be the

differential operator of order lal with polynomial coefficients defined by

CoF = PIQI+tDa (.).

R ,w a multiindex /3, let C.,# be the differential operator of order IaI + l/3l definedby

C.,RF = De(C.F).

Set 14 =m'-k(Ial+I)-IaI.Assuming )$1=k(lal+ 1) anddist(i,Bk+l)? 1,we have easily

)I, : ci(l+I.I)"sup(ID''(Pf)(t;)I:lyl<k(Ial+1)+lullCt IPfIIi.m'.Rt.l)' (5.9)


where we simply need to choose eI and cI large enough to dominate all the poly-nomial coefficients in £a,p. Next, we observe that if dist(t, Bk+l) < 1 while f isas before, we can find a point i;* E Bk+1 with I - t*I = dist(l;, Bk+1) Let v bethe unit vector (a; - l; *) / jt - l; * I . At r* the function Ga, p (P f) vanishes to orderat least A because Pf vanishes to order m'. Now we apply Taylor's theorem tothe function

h(r) _ (Ga,p(Pf))(4* + rv),at r = 0, to find 0 < ro < Ia; - ! *j such that

h(I

5 c2 [ dist(1; , Bk+l )111 I P f I (5.10)lim'B(Q1)

The two estimates (namely, (5.9) which applies when dist(i;, Bk+l) > 1, and(5.10), which applies when dist(i;, Bk+l) < 1) can be combined to give

I (Ga.P(Pf))OI 5 C3 [ dist(a; , Bk+l )]µ I Pf Itj.m',B(t.l)'

for all E R". Noticing that Plal+1 Da f = ,C(Pf ), we have

1i

C

IDI(PwIO'I+I)I2! IDafOIIP=(IaI I)k

/ lI= t [L I(Ga.p(Pf))($)I2\IP=(Ial+1)k

5 C4 [ dist(t , Bk+1)lµ I Pf I lg m' B( 1) 'for all 1: E R"

Then we apply (5.8) with j = jai + I to conclude that, for all!; E Bk,

I Daf 5 C5 [ dist(4, Bk+1)lµ-µ' (I + I Pf l

5 C5 [Bk+1)lµ-µ

I Pf I13.m'.B(4.l )

Since jz - µ' > 0, the distance from l: to Bk+l can be bounded by a constantmultiple of 1 + I I, so we have

SC6IPfIt',m'.B(t,J)' for all E Bk.

The constants e' and K are independent of i and f, but f must vanish to at leastorder m' on Bk+l


Similar arguments are used to obtain estimates on

ID Rm(l;. q)I/I - nlm-Dal

We refer the reader to [HL 581 for the details. The result is an estimate of the form

111* Cl I Pf l[.m.B! e,m',R"

whenever f vanishes to order m' on Bk+IWe must deal with the assumption that f must vanish to order m' on Bk+l

since, obviously, this does not generally hold true. The appropriate m' has by nowbeen fixed. Consider an arbitrary rapidly vanishing f. Using f as the source of thedata and Bk+1 as the closed set, we apply the construction from the proof of theWhitney extension theorem to produce a function g which agrees with f on Bk+1up to order m'. Now we come to the point where the more complicated normsare used. By careful consideration of the construction in the proof of Whitney'sextension theorem, one obtains the estimate

IgOlom'

< KIIflom'.B&+InB(t.p)

(5.11)

where KI and p are independent off and Bk+1 Thus to successfully estimatethe simpler norm of g, we need information about the more complicated normof f. Also, the smoothness claimed for g is only that it possess m' continuousderivatives, but that is sufficient.

We have the easy estimate

I (Pg)(Ol t, m, :5 C8 IgI[,+d.m'.B(f.P)nBt+1

based simply on the fact that P is a polynomial of degree d.

(5.12)

Combining the estimates (5.11) and (5.12) and taking the supremum over R",

we obtain

I Pg I ['.M"R" C9IfI+d.m'. BR+1

-< CIO I Pflr" ,m".R"

where the last inequality follows by the induction hypothesis. It follows, of course,that

I P(f - g)I1',m'.R" CII IPf l.m".R"

.

But f - g vanishes to order m' on Bk+1, so we can apply our earlier estimates toobtain

If-gl[.m.s `C'IPfI1,.m'.R".

5.3. The FBI Transform 135

A final application of (5.11) and the induction hypothesis gives us

K PI f t.m.Bt R by

S(Pf) = T(f), for f E S.

By (1) of Theorem 5.2.6, S is well defined and continuous. By the Hahn-Banachtheorem, there is a continuous linear S : S -+ R, that is S E S', such that

S(Pf) = T(f), for f ES,

which is the same as PS = T.

Finally, recall that the proof of (3) of Theorem 5.2.6 was sketched at the begin-ning of the previous section.

The best local properties are not always obtainable with the tempered funda-mental solutions that we have been discussing. The reader more directly interestedin partial differential equations should consult [HL 83].

5.3 The FBI Transform

The rate of decay of the Fourier transform of a function f cannot be used to givesharp information about the smoothness of f. Similarly, the decay of the Fouriertransform will not detect whether or not f is real analytic. The 0-transform (see[FJ 85]) is a serviceable variant of the Fourier transform that will give sharp resultsabout the smoothness of a function. For real analyticity the correct tool is the so-called FBI transform. The acronym FBI stands for the names of the mathematicalphysicists Fourier, Bros, and Iagnolitzer.

It is noteworthy that the FBI transform is a special instance of the theory ofwave packets as developed by Cordoba and Fefferman (see [FG 89)). Wave packettheory is an alternative method for studying propagation of singularities, a phe-nomenon that is most often understood by using Fourier integral operators (asdefined and developed by HSrmander [HL 71] and Duistermaat and H6rmander[DH 72]).

Define the Fourier transform of af Lebesgue integrable function f on R to be

T a) = f (x)e-27r' dx.

The fundamental facts about the Fourier transform are:


(i) If both f and fare integrable, then

f (X) = f f d

(ii) If f is square integrable on R, then f exists as the limit, in theL2 topology, of the functions

N

lim f (x)e-2ar"t dx.N-boo J-N

The function f satisfies

IIf IIL2 = IIf IIL2 .

(iii) If f is integrable, then f is a bounded, continuous function andIf()I < IIfIIL1 for all E R.

(iv) Recall from the previous section that the Schwartz space of rapidlydecreasing functions consists of those C°O functions on R whichhave the property that the function and each of its derivatives van-ishes at infinity at a rate faster than Ixl-N for any N. The functionh(x) = e-1X11 is an example of a rapidly decreasing or Schwartzfunction, as is any C°O function of compact support.

The space S of Schwartz functions, equipped with the semi-norms

fQa,6(f) = sup I x°

a1

aXx 6 I ,

is a FrEchet space. Its dual S' is called the space of tempered distri-butions or Schwartz distributions.

The Fourier transform takes the space S in a univalent, surjective,continuous fashion to itself. Note in particular that the Fourier trans-form maps the space CO0(R) of C°0 functions with compact supportinto S, but it does not map C0° into itself (In fact the "Heisenberguncertainty principle" asserts that the Fourier transform of a nontriv-ial (i.e., not identically zero) compactly supported function is nevercompactly supported; there are quantitative versions of this assertionas well. See (FC 831.)

(v) We have (e-'hx2) () = b-1 /2. e-"2t2/6.

(vi) If f E S then (f') (27riU) f (i; ).

(vii) If f E S then (-27rix f W) ( ) = a f W.


(viii) If f and g are integrable, then so is their convolution

and

f *g(x) / f(x -t)g(t)dtJJR

(f * g) ($) = f( )8( )By applyinnthe inverse Fourier transform to this last identity we ob-

tain (f g) f *

Further details on the elementary properties of the Fourier transform may be foundin [Kr 99] or [SW 71].

Now we define the FBI transform Trf (x, t;) of an f E L 1 (R) by the formula

Trf (x, ) = J f ds.R

Since the Gaussian expression is bounded above by I, it is plain that Trf (x, t:) iswell defined for any integrable f. In fact, we have

ITrf(x,$)I -< ]IfIIL' , (5.13)

for all t, x, and .Now we define an exponential decay condition on the FBI transform at infinity.

Definition 53.1 Fix xo E R. We say that an integrable function f satisfies thecondition RA(xo) if there are positive constants C, or, M and a neighborhood Uof xo such that, for all li; I > M and all x E U, it holds that

RA(xo) I Trf (x, t$)I < C e-or

Theorem 5.3.2 Fix xo E R. An integrable function f is real analytic at xo if andonly if f satisfies condition RA(xo).

Note that we work in R' for simplicity of notation, but the results of this sec-tion hold in any R" (see [SJ 82]). Also the theorem may be proved when f is adistribution if a certain amount of extra care is taken. However to avoid a numberof technicalities we shall assume that our function f is in C°O(R). In this way wecan concentrate on the main point: as we know from Chapter 1, real analyticityis in fact a condition on the growth of derivatives. So our job is to focus on thatcondition.

The remainder of this section will be devoted to proving the theorem with theextra hypothesis that f is C°O. The exposition here is derived from that in [FG 89].

We will divide the argument into several lemmas and propositions. We beginby showing that we can localize.

Lemma 5.3.3 Fix xo E R. Let f be an integrable function that vanishes in aneighborhood of xo. Then f satisfies condition RA(xo).


ProoL Choose S > 0 such that if Is -x0l <23 then f (s) =O. Then for Ix -xol <S it holds that

IT,f(x,tl;)1 < fR

with a = 7r32. This establishes the result. 0

The lemma has the effect of making our work local: If f satisfies RA(xo) andif f = g in a neighborhood of xo, then g satisfies condition RA(xo). In particular,if f satisfies condition RA(xo) on a neighborhood U of xo, then let * be a C00function of compact support in U which is identically 1 in a smaller neighborhoodof x0. Write f = >y f + (1 - >/r) f. The second term satisfies RA(xo) by thelemma, hence so does the first. As a result of these observations we may assumein the sequel that f is a C°O function of compact support.

We now prove the easy half of Theorem 5.3.2.

Proposition 5.3.4 If f is real analytic in a neighborhood of xo, then f satisfiescondition RA(xo).

ProoL For simplicity take xo to be 0. As indicated above, we may assume f ECr. Of course we shall only verify that f satisfies RA(0) in a small neighborhoodof 0.

By substituting z's for x's in the power series expansion of f about 0, we findthat f is complex analytic (or holomorphic) in a neighborhood of 0 = 0 + i 0 inthe complex plane. Choose 0 < S < 1 such that

{r + iv : Irl < 2S, Ivl < 8}

lies in this neighborhood.Now let 4r(r) be a C°O function with support in jr E R : lrl < 2S) such that

0 < >/r (r) < 1 for all r and *(r) = 1 when Irl < S. Then for any !; 96 0 we mayuse the Cauchy integral theorem to move the axis of integration in the definitionof T, f (x, ta;) to the contour

where S = S sgn i; = S l;/I$I Notice that the region in which !s(s) differs fromy(s) = s lies in the region where f is holomorphic; hence the Cauchy integraltheorem applies.

We see, using the new contour, that when IxI < S and It;1 0 we have

Tt f (x, it) =JR

x f(s - iaif (s)) (I - i8 f'(s)) ds .


We use the definition of S, the fact that f is bounded with bounded support, andsome obvious majorizations to see that

I Tt f (x, tt;) I < C sup e- (5.14)S

We fix ItI > S and IxI < 6/2. There are now two possibilities:

(i) If IsI < S < 1, then

28'G(s) - 82 ,(s)2 + (s - x)2 > 262 _ 82 = S2 ;

(ii) Likewise, if IsI > 6, then

28r/r(s) - 62*(s)2 + (s - x)2 > 8>Jr(s) [2 - S>[i(s)] + (8/2)2

> Sr/r(s)[26 - S] + (6/2)2

> S2/4.

In any event, the quantity in (5.14) is bounded by C e-n'(82/4)'t. Thus we have

ITtf(x,tl;)In.(a2/4).t.

Lemma 5.3.5 Let a > 0. The formula

A(f) = oa(f) = I I e27rix4-27rax2Wl(1 +iax sgnt:) f(x)dxdt;JR JR

O

defines an element of S'.

Proof. Our first job is to see that the integral converges. Let g E S. Exploitingproperty (vii) of the Fourier transform, we write

fe24erx2g(x)dx= (e >rbx2) _ ) *8(-)

\ J

for any b > 0. Now property (v) of the Fourier transform enables us to write theright-hand side more explicitly as

(b-1/2e 2t2/b)* 8(- )

Therefore, setting b = 2ai I, we have

fRe2 it e-2nal(;Ix2g(x)dx=JR(2aItl)-1/2e-,2(E-r)2/(2aIt 1) (-r)dr,


The right-hand side of the last equation is a function of t and, by inspection,vanishes at infinity more rapidly than It I-N for any positive integer N. In partic-ular, it is an integrable function. Therefore, for a > 0 and f E S, we may setg(x) = (1 +iax sgnl:) f(x) to obtain that

fR fR(1 + iax sgni;) f(x)dxdt

is a convergent integral. Our discussion of this integral shows that its convergenceonly depends on finitely many of the Qa . Therefore A is an element of S'.

Lemma 5.3.6 The functional A defined in the preceding lemma is equal to theDirac delta mass 6.

Proof For any x 96 0 we have, by the definition of the signum function,

2aixQ -2'T-'1tl - rJRe

a (1+iaxsgnl)da; = (1-iax) J e2ni(x-iax2)l d!

r

0

oo

+ (1 + iax)J

e2ni(x+iax2)l: dt

_ 1 - iax l +iax2 ri(x - iax2) 2ari(x + iax2)0. (5.15)

This shows that the distribution X is supported at the origin. Such a distributionis a sum of derivatives of the Dirac mass. We eliminate all the derivatives but thezeroth by an iterative procedure.

If f is a Schwartz function that vanishes to second order at 0 (i.e., if 0 =f (0) = f'(0)), then we notice that

I.1. e-arax2lfl (1 +a2x2)2)112

If (x)] dx dl;

fR (1 +a2x2)1121f(x)I.27rax2 dx

C f If(x)I dx+C I If(x)I dxIxl_ x2 Ixl>I Ixl

< 00.

This shows that the integral defining x converges absolutely. Thus we may applyFubini's theorem and reverse the order of integration in the integral defining X.Because of (5.15), we conclude that M(f) = 0.

Now suppose that f is a Schwartz function that vanishes to first order at 0.Write

f (x) = 0 (x) f'(0) x + (f (x) - 0 (x) [(0) . x) = ft (x) + f2 (x) ,


where 0 is an even cutoff function that is identically 1 near the origin. Then fi isodd and f2 vanishes to second order. It follows immediately that A(f2) = 0. Butif we apply A to f1, and perform the change of variable x r> -x, i; H -i; in theintegral, the result is that nothing changes except that a minus sign is introduced.It follows that A(f1) = 0.

The result of our calculations is that k = c 8. It remains to determine c. (Eventhough the exact value of c is not important for the result we seek, it is a nicecomputation and we include it for completeness.) Fix g a C°O function that isidentically 1 near x = 0. Let gk(x) = g(x/k), k = 1, 2, .... Then c = X(90 forany k. Let k -). +oo to yield

c=L1.

e2trtxte-7arax21g1(1 + iax sgn t) dx dt .

We use properties (v) and (vii) of the Fourier transform to conclude that

(e-nbx2) W = b-1/2e-n(:2/b

Therefore

and (xe-rrbx2/ (t)

C J(e-(2najEDx2) (-t,)dt + f is sgn t (xe-(2"°It)x21 (-t)dt

JR(2aItI)-1/2e-"111/(2a)dt + J(ia sgnl:)(2aItI)-312ite-7r1;1/(2a)dl; .

Notice that the second integrand simplifies to

-(1/2)(2a It I)-1/2e-n11;1/(2a)

Thus the integrals can be combined to yield

00

(8a)-1/2Loo

ItI-1/2e-n1t1/(2a) dt = 2(8a)-1/2 f t-1/2e-" 8/(2a) do

Perform the change of variable µ = 7r1; 1(2a) to obtain

7r-1/2

TOA-1/2e-µdµ27r-1/2J°° e 2ds = 1.

Thus c = 1. This completes the proof. 0

Lemma 5.3.7 If f and g are C°O functions that both satisfy condition RA (x0),then so does f g.

Proof. As usual we assume that xo = 0. By hypothesis there is a neighborhood Uof 0, and positive constants M, C, a such that, when x E U and It I > M, then

ITtf(x,tt)I - and ITtg(x,tt)I <<Ce-ar

142 5. Phrasal Dtfferentia1 Eauanons

Yut

rl(s) f(s) and r.{S1 =e-zrts-z1==3(S)

Then, by definition of the transform Tr, we have

711()=TRf(x,0 and

Also, it holds that

As a result, we have

rl (s)r2(s) = e-sr(r-`)2f(s)g(s) -

Tr (f - g) (x, tl; ) e (rl ' r2) :) ='71 s

Trnf(x, t - C) Trrig(x, C) dC

fiv:51140 f'C1-!W2A+B.

On the domain of integration in Awe have that ItI > 2t-l IC1 and hence I24 -2t C I ? I 1. Therefore

ITr/2f(x, t - C)1 = I Tr/2f(x, (t/2)(2$ - 2t-1C))I -< C e-at/2 (5.16)

holds. Next we have that

fR I Tr/2g(x, C)I dC = IIr2IIL- .

Now repeated application of properties (iii) and (vi) of the Fourier transformshows that

Ir2(U)IC 1 +11t'1 ax

2

ir- 11 Ll

This last, by inspection of the definition of r2, does not exceed C" (1 + t)2.Putting together our estimates for Tr f and Trg yields that

AI C"(t + 1)2e-ot/2 < Cme-ot/3I <

To estimate 8 we notice that

ITr/2g(x, C)I = ITr/2g(x, (t/2)(2r-1C))I <_ C . e-ar/2.

But 21 11{ I ` IC I holds. Therefore this last estimate, combined with (5.16), al-lows us to see that

1B1 < C,ne-ar/3

Combining our estimates for A and 8 yields the desired conclusion. 0

5.4. The Paley-Wiener Theorem 143

Proposition 5.3.8 Let f be a CO0 function. If condition RA(xp) is satisfied, thenf is real analytic in a neighborhood of xo.

Proof. We may assume that xo = 0. Since the distribution A equals 8, we maywrite

I.1.e2>ri(x-s)4-2rra(x-3)2141(1 + ia(x - s) sgn t)f (y) ds da;.f(x) =

(Note here that we have used a translation operator to pass from a result about theDirac mass at 0 to a Dirac mass at x.) Set r(s) = s f (s). Then r is the product ofthe real analytic function s, which satisfies condition RA(0) by Propostion 5.3.4,and f (s), which satisfies RA(O) by hypothesis. By Lemma 5.3.7, r satisfies con-dition RA(0). Therefore there are positive constants C, M, a, and 8 such that,when Ix I < 8 and I I > M, we have

IT,f(x,ti:)I and ITrr(x,tt)I (5.17)

We now introduce the notation z = x + i v with x, v real. Then

fR fRe2rri(z-s)4-2aa(z-s)2ItI(I + ia(z - s) sgn i) f (s) ds dt

= e2>rizt-41riaxvItl+brav2ItI x

[(1 + iaz sgn t)T2,itl f (x, t - 2avI I)

-ia(sgni;)T2aItlr(x, l - 2avI D] . (5.18)

We choose a = (4M + 4)-t and require that

aIxI < 8 and IuI < min 1 a

{ 16aM' 2 n' M}

The result is thate2rriz4-4siaxv141+2rrav2ItII < eaa141 and It - 2avItII 2aItIM,

hence, using (5.17) and (5.18),

f Ie2ai(z-s)4-2aa(z-s)2141(I + ia(z - s) sgn t) f (s) I ds C'e-aoltl

This absolute convergence and size estimate means that the integral

JJRe2rri(z-s)4-2rra(z-3)2141(1 + ia(z - s) sgn t) f (s) ds dt

defines a holomorphic function of z on the region in x and v specified above. Ob-viously this holomorphic function agrees with f on the real axis. Therefore f isreal analytic in a neighborhood of the origin. 0

The FBI transform is not well known in the mathematical community. It is apowerful tool that should prove useful in many contexts.


5.4 The Paley-Wiener Theorem

The FBI transform has shown us that Fourier integral operators can be used ef-fectively to detect real analyticity. This connection is, in retrospect, not surprisingbecause the exponential expression ex is real analytic. In fact the connectionswere noticed rather early in the history of twentieth century analysis by Paley andWiener [PW 34].

The gist of the Paley-Wiener theorem is that the Fourier transform of a com-pactly supported function (or, more generally, a compactly supported distribution)is an analytic function of exponential growth. The converse is true as well: Ev-ery analytic function of exponential growth arises as the Fourier transform of acompactly supported function or distribution. It is also the case that the size of thesupport is intimately connected with the rate of growth of the function.

The Paley-Wiener theorem has been influential in twentieth century analysis. Ithas made its mark particularly in the area of partial differential equations, where itsays a great deal about the existence of solutions to linear equations and to linearsystems. The related work of Malgrange and Ehrenpreis on systems with constantcoefficients is treated in some detail in [HL 63] and [EL 70].

In fact the Paley-Wiener theory of several dimensions has an interesting geo-metric flavor. It is related in spirit to the Fourier analysis of tubes over cones (see[SW 71 ]). These ideas in turn can be used to study the edge-of-the-wedge theorem(see [RW 70]).

Our purpose here is to present the central idea of the Paley-Wiener theoremwithout getting distracted by ancillary technical issues. Therefore we will presentthe result in the context in which it was first discovered: the analysis of the realline. By making this choice we can restrict any complex analysis that needs tobe done to the familiar context of the plane. We shall make a few remarks aboutmore general versions of the theorem at the end of our discussion.

As motivation for the Paley-Wiener theorem we first present an analogous the-orem in the realm of the Fourier analysis of the unit circle T =_ R/2,rZ. Of coursein practice we identify T with the interval [0, 27r] with the obvious identificationof the endpoints. Measure theory on T is defined by pulling back Lebesgue mea-sure from [0, 2$ ] under this identification. If f E L I (T) and n E Z, then weset

f (n) =2a J

f (t )e-int dt.0

Our Fourier series theorem is as follows:

Proposition 5.4.1 Let f E COO(T). Then f is real analytic on T if and only ifthere are constants c, C > 0 such that

If (n)1 < C e-clnl

Proof. By integration by parts we see that

f(n) = (in)-i . f(i)(n).


[Here the exponent (j) denotes the ju' derivative.) It is also obvious from thedefinition of the Fourier coefficients that

Ig(n)I < II8IIsup

Combining these two facts with the characterization of real analytic functionsgiven in Proposition 1.2.12 gives the result.

Matters in the noncompact setting are a bit more subtle, but exhibit the sameflavor. Recall that if f E Lt (R), then its Fourier transform is defined to be

Ta) = f f (t)e 2airg dt .

R

Notice that f E L°O(R) and II f II LO 11f II L I The Fourier inversion theorem(see, for instance, [KY 76], [Kr 99], or [SW 71]) says that in case f E Ll, then

f (x) = f f ds; .

In case f E L2, the Fourier integral must be interpreted as

_ Nf(t)e-27rut dtf( Nim

f1v

(because L2(R) ¢ Lt (R)). In this circumstance, it holds that f E L2 and

f If(X)12 dx = ft l

The Fourier inversion formula must be interpreted in a similar fashion.Notice that Fourier inversion implies Fourier uniqueness: if f (l;) = g (t;) al-

most everywhere, then f = g.Finally recall that if f, g are L 1 functions on R, then their convolution is the

function f * g(x) = fR f (x - t)g(t) dt. An elementary change of variables and

application of Fubini's theorem reveals that (f * g) (l;) = f (l;) g(1; ).

Now we have

Theorem 5.4.2 (Paley-Wiener) Let f E L2 (R). Then the following two state-ments are equivalent:

(1) There is a function F and constants a, C > 0 such that F is holomorphic inthe strip (z E C : 11m zl < a}, F(x + i0) = f (x) for all real x, and

fRIF(x+iy)12dx<C Vlyl<a.


(2) The function ealfl f lies in L2(R.)

Proof. To prove (2) implies (1), we define

F(z) = J .f e2nf z dR

Our hypothesis guarantees that f ( )e-2iyt E L2 (as a function of the x variable)as long as y = Imz satisfies I I

< a/(2n). By Fourier inversion, FIR = fFurthermore, by Plancherel's theorem,

JRIF(x+iy)12dx=JRl1( )12e 4ydt<1lfe0It111i2(R)

Thus (1) is proved with C = life°It1112L2

(R)

To prove (1) implies (2), define fy(x) = F(x + iy) for lyl < a. Observe inparticular that fo = f. We shall now prove that fy (1;) = f (t; )e-2n4y.

Let us assume for the moment that each fy is known to lie in L 1. On the onehand,

F(x + iy) = fy(x) = f ?y(t)e2-'-tR

On the other hand, we may define

H(x + iy) = f fo(t,)e2a'(x+'y)t dl; .R

Both F and H are holomorphic functions on the strip (x + iy : I yl < a). Alsothey agree on the real line, hence they must agree on the entire strip. It followsfrom Fourier uniqueness that fy(t:) = f we-2nty.

Now our hypothesis, together with Plancherel's theorem, says that

f If(t)12e-4nIyd$ <C,

R

where C is independent of lyl < a. But then the continuity of the integral (moreformally, the Lebesgue dominated convergence theorem) implies that ealtI f EL2. This completes the proof of the theorem in the presence of the extra hypothe-sis.

For the general situation, we must use the standard Fourier theory device of thesummability kernel. Fix a CO0 function }fi with the property that *(x) = 1 whenIxl < 1 and 4r(x) = 0 when Ix1 > 2. Let 0 be the inverse Fourier transform of*; so ¢(i;) = For IXI > 0 we set ¢x(t) = 1-1 Now define

00GA(z) = 01 * F = f F(z - s)4 ,(s)ds .

Then Gx is clearly holomorphic in the strip {z E C : Ilm zI < a}. Set gx,y(x) _Ga(x + iy). Then ¢a(i;) fy(). Now the uniqueness argument that we


presented in the first part of the proof shows that gay (a;) = ga0(a; )e-2irfy. Noticethat ¢( ) (just use a change of variables). Hence, when It I < 1 /A, wehave fy(i;1 = f (4)e-2'0. Since I > 0 was arbitrary, we have established thatfy(l:') = f (4)e-2irly for all 4. Now the proof is finished as before.

Corollary 5.4.3 Let g be an L t function with compact support in R. If g also hascompact support, then g - 0.

Remark 5.4.4 The corollary says that a function and its Fourier transform cannotboth have small support. There exist a variety of quantitative forms of this asser-tion as well. This circle of ideas is often referred to as the "Heisenberg uncer-tainty principle" and in fact is a mathematical model for the uncertainty principleof quantum mechanics. For more on this matter see [FC 83].

Proof of Corollary 5.4.3. Let f be the inverse Fourier transform of g. Then fsatisfies condition (2) of the Paley-Wiener theorem for any laI > 0. Take a = 1.Then, by the theorem, f is the restriction to the real line of a function F holo-morphic on {z : IImzI < 1). Since f is compactly supported, the holomorphicfunction F vanishes on an entire half-line. Hence F - 0 and f = 0.

We shall now formulate two standard variants of the Paley-Wiener theorem.The proofs involve just the same ideas, so we shall not supply those. Details maybe found in [KY 76].

Theorem 5.4.5 (Paley-Wiener, First Variant) Let f E L2(R). Then the followingtwo conditions are equivalent:

(1) There is a function F, holomorphic in the upper half plane (x E C : Im z >0), and a constant C > 0 such that

J.and

IF(x+iy)I2dx<C, vy>0

limJ I F(x + iy) - f (x)I2dx = 0.

Y10 R

(2) ?(t) = O for all i; < 0.

This version of the Paley-Wiener theorem can be considered to be a desym-metrized statement of the result: The function F is defined only on one side of thereal line (where f is supported). This explains the necessity of the convergencestatement in part (1) of the theorem. Part (2) of the theorem is in the spirit of theR and M. Riesz theorem on the circle (or the line): A measure on the circle is theradial boundary limit of a holomorphic function on the disc if and only all of itsnegative Fourier-Stieltjes coefficients are zero; in this circumstance, the measuremust be absolutely continuous with respect to Lebesgue measure.

To state our final version of the Paley-Wiener theorem in dimension one, weneed to introduce some notation (due to Landau):


Notation 5.4.6 Fix a in the extended reals, that is, a E R U {±oo}. Suppose thatg is a real-valued function that does not vanish in a punctured neighborhood of a.For a real-valued function f defined in a punctured neighborhood of a, we say fis big "0" of g as x -* a and write

f(x)=0(g(x)) asx -->a

in case

limsup I f (x)I < 00.x-a Ig(x)I

We say f is little "o" of g as x -* a and write

in case

f (x) = o(g(x)) as x -a

xlim ) =0.x-'a g(x)

Theorem 5.4.7 (Paley-Wiener, Second Variant) Let F be an entire function anda > 0. 77re following two conditions are equivalent:

(1) FIRE L2(R) and

IF(z)I = o (ealzll as Izl -> oo.

(2) There exists a function ?E L2(R) such that f () = O for ItI > a and

?(t),* dF(z) = Zn LI

This third form of the Paley-Wiener theorem is the adaptation of Paley-Wienertheory to entire functions. It has perhaps the most elegant formulation of the three.The theorem is false if the function f is replaced by a measure (that is, the little"o" in part (1) must be replaced by a big "O"). For instance, cosaz is the complexFourier transform of a compactly supported measure.

As an exercise, the reader may use Paley-Wiener theory to obtain a proof ofTitchmarsh's convolution theorem.

Theorem 5.4.8 (Titchmarsh) Let f, g be L2 functions both supported in the in-terval [ -1, 0]. If f * g vanishes in a neighborhood of the origin, then at least oneoff or g vanishes in a neighborhood of the origin.

In particular, if f * g =- 0, then either f =- 0 or g =- 0.

Both the Titchmarsh theorem and the Heisenberg uncertainty principle may beproved by real variable techniques, but the proofs are much more difficult.

Now we turn to N dimensions. What is the analogue of the interval [-a, a] ina multidimensional Euclidean space? One answer is the unit ball, but another is


the unit cube. It turns out to be most natural not to limit ourselves to these twocanonical (from the point of view of Euclidean geometry) examples, but ratherconsider any set that could be the unit ball of some norm on RN. Thus we restrictattention to sets K that are convex, compact, and satisfy -x E K whenever x EK. Such a set will be called a symmetric body.

If K is a symmetric body, then we define K* = (y E RN : X. y < 1 for all x EK). [Here "" is the standard Euclidean inner product.] The set K' is termed thepolar set of K. It too is a symmetric body. The set K * is a natural construct whenone views K as the unit ball of some norm. Clearly the Euclidean unit ball iscanonical in this context in that it is the only symmetric body that equals its polarset. In general it holds that K'* = (K')* = K.

Now if f E Lr (RN), then we define its complex Fourier transform to be

t)e ttzdt.F(z) =JRN

f(

Here z=(zi.....Zn)ECNandRecall that a function of several complex variables is said to be holomorphic if

it is holomorphic, in the classical one variable sense, in each variable separately.A holomorphic function defined on all of CN is called entire. See [KS 82] formore on these matters.

Definition 5.4.9 Fix a symmetric body L. If Z E CN, then we define

IIZIIL = sup Iz yI

We say that an entire function F is of exponential type L if for each e > 0 thereexists a constant CE > 0 such that

IF(z)I < Cfe0-W1IZIIL

all e > 0. Denote the class of all such functions by E(L).

Using this terminology, we can state the following theorem:

Theorem 5.4.10 Let f E LZ(RN) and K a symmetric body. Then following areequivalent:

(1) The function f is the restriction to RN of a function in E(K').

(2) The function f is the Fourier transform of a function supported in the sym-metric body K.

The reader is referred to [SW 71] for a proof of the theorem and for its history.

6Topics in Geometry

6.1 The Weierstrass Preparation Theorem

Suppose F(x, y), (x, y) E R" x R, is real analytic in a neighborhood of theorigin, is not identically zero, and satisfies F(0, 0) = 0. To study the locus ofthe equation F(x, y) = 0 near the origin, we would apply the implicit functiontheorem if possible, but when the linear term in the Taylor series for F vanishes,then the use of the implicit function theorem is not possible. Instead, the tool thatcan be used is the Weierstrass preparation theorem.

Example 6.1.1 Consider the locus of points satisfying

Y2

1+y2

near the origin. The function

-{-x=0

1+y2

(6.1)

l+xis real analytic in a neighborhood of the origin in R2 and U(0, 0) = 10 0. Thus,near the origin, the locus of points satisfying (6.1) is the same as the locus ofpoints satisfying

y20 = U(x,Y) 1+ 2+xY

152 6. Topics in Geometry

2 (p2)=1+x 1+y2+x =y2+1+x. (6.2)

The important feature of the polynomial on the right-hand side of (6.2) is that it isa monic polynomial in y with real analytic coefficients that vanish at x = 0. Thisclass of polynomials is named in the next definition.

Definition 6.1.2 A function W (X, y), (x, y) E R" x R, real analytic in a neigh-borhood of (0, 0) E R" x R, is called a Weierstrass polynomial of degree m,if

W(x, y) =Yn' +am-I(x)Ym-I +...+al(x)Y+ao(x), (6.3)

where each ai (x) is a real analytic function in a neighborhood of 0 E R" thatvanishes at x = 0 E R".

The Weierstrass preparation theorem guarantees that the behavior illustrated inExample 6.1.1 always occurs. For use in the statement of the theorem, recall fromDefinition 2.1.1 that A(n) denotes the set of multiindices with n entries.

Theorem 6.1.3 (Weierstrass Preparation Theorem) Let

00

(D (X, Y) E4""jxayac-A(n) j=0

x = (xl , x2, ... , xn) E R', Y E R, be real analytic in a neighborhood of (0, 0) ER" x R and suppose there is a positive integer k such that

(D0,0=4>0,1 =...=O0.k-1 = 0

and

4)0.k = I.

(1) If'P(x, y) is real analytic in a neighborhood of (0, 0) E R" x R. then thereexist unique real analytic functions Q and R,

00

Q(x, Y) = E E Qa, j xo yj ,aEA(n) j=0

00

R(x,Y)= I: Raj xay',aEA(n)j=0

with

Ra,j = 0 for j = k, k + 1, ... and for all multiindices o t, (6.4)

and satisfyingt'=Q(b +R. (6.5)

6.1. The Weierstrass Preparation Theorem 153

(2) There exist a Weierstrass polynomial W (x, y) of degree k and a functionU(x, y) real analytic and nonvanishing in a neighborhood N of (0, 0) ERn x R such that

Uc = W (6.6)

holds in N.

The proof of the Weierstrass preparation theorem requires the following lemma.

Lemma 6.1.4 Fix 0 < y < x < oo and fix a multiindex a with 0 < ka1.

(1) If ai > 0, it holds that

)I,I

yxn

(x

)(x - Y)n;Pt <ai

(2) It holds that

\Y

aln yxn

(x

x

(x - Y)n( )

Proof of Lemma 6.1.4.(1) Fix i and suppose ai > 0. We have

y)

Idl

(XY) _Pt <a,

f n

]x %_i #i=p(Y1#I

(x/Y)a' - 1 n (x/Y)aj+l - 1(x/y) - 1 ja' (x/y) - 1

laIY x - Y ja1 x-y

j#iylal+n (x/Y)°ft - 1 (x/Y)°`j+l - 1ylal+n (x/Y) - 1 j (x/Y) - 1

j#iy xaj - ya. n xaj+l - yaj+I

j#iy xaj n xaj+1

ylal x - yj=1 x - y/#+

y xlal+n-1ylal (x - y)n

(2) Conclusion (2) follows immediately from (1). 0


Proof of Theorem 6.1.3.(1) Write

'V(x,y)= E E*..jxayjaEA(n) j=o

For (6.5) to hold we must have

j'1'.,j = Raj + E Qp,v (6.7)

0<a v=0

for every multiindex a and for every nonnegative integer j.Fix a multiindex a. For j = 0, 1, ... , k - 1, we can solve (6.7) for Ra, j to find

Ra.j = Pa.j - E'> Qp.v da-A.j-vB<a v=0

jL Qp.v

osa v=o101<w

(6.8)

where we have used the fact that (D (0, j) = holds for j = 0, 1, ... , k - 1. Thuswe see that Ra,o, Ra,k-1 can be defined in terms of Qp,v with P < a,IfiI <lal,andvE (0,1,...,k-1).

With a still fixed, taking j = k in (6.7) and keeping in mind that Ra,k = 0holds for all multiindices a, we find that

k

Qa,o ='pa.k (6.9)asa v=o

I01<Ial

so Qa,o can be defined in terms of Qp,v with rg < a, Ifil < laI, and v E(0, 1, ... , k}. Finally, taking j = k + 1, f = 1, 2, . in (6.7) and noting thatRa,k+t = 0 holds for all multiindices a, we find that

k+t t-1

Qa,t ='I'a,k+t - QO,v a-O,k+l-v - Qa,v O,k+t-v , (6.10)0sa v=0 v=0

101<Ia1

so Qa,t can be defined in terms of Qa.o, Qa.l , ... , Qa.t-1 and Qp, with P < a,IfI < Ial, and V E (0, 1, ... , k). Thus we see that there are unique formal powerseries for Q and R satisfying (6.4) and (6.5). In particular, we note that

Qo,o ='1'o.k (6.11)

It remains to show that the series for Q and R converge. Since c and 41 arereal analytic, there exist positive real numbers b and c so that

max(l4a,jl, I'I'a.jl) <b c1al+j

6.1. The Weierstrass Preparation Theorem 155

holds for every multiindex a and every nonnegative integer j. Choose a positivereal number A so that

bck < 1/3.A

Then choose a positive real number B > c so that

bck+l< 1/3.B-c

Finally, choose a positive real number C > c so that

n b Bk+t cn-I< 1/3 .

ck-1 (B - c) (C - C)n -

We will prove by induction that

IQa,j I < A B3 CIaI

(6.12)

(6.13)

(6.14)

(6.15)

holds for every multiindex a and every nonnegative integer j. Note that (6.11)begins the induction. Using (6.10) (note that (6.9) is to be considered a specialcase of (6.10) since the empty sum is zero), we estimate

Qa,el

k+eb clal+k+e +

E> A By Clfil bclal-Ir9l+k+e-v

/1<a vv=OBI<InI

e-t+E A B v Cla l b ck+e-v

v=0

ABeClal [-A (B)e( c rce clal I+Be Clal b L(B/c)v (C)Ii5l

CV- -IBI<Iai

(ca

e-l

+\B/bc1' (B/c)v

v=0

We havebAk

fie)e (C )a < 1/3

by (6.12) and because B > c and C > c. Using Lemma 6.1.4, we estimate

ce Clod k+e C1181

Be Clalb (B/c)v (c /v-0

181<ial


cC clal (B/c)k+t+l 101

BL Cladb (B/c) -1 c

fis.Ipl<I0I

ct clal (B/C)k+C- - 1 n C Cn-1 C IalBC Clal b (B/c) - 1 (C - c)n (C )

Ib

Bk+t+l - ck+t+1 n c Crs-1

Btck B-c (C-c)"Bk+1

b

n c Cn-1

ck(B-c)(C-C)nwhere we have used (6.14). Finally, we estimate

It C-1

B) bck>(B/c)vv=0

(C t k I - (B/C)t\) bB e1 - (B/c)

bck+1l )t Bt - ctB B-c

bck+1(1)C Bt

B B - cbck+1< < 1/3,B-c

where we have used (6.13). Thus we have verified (6.15) and the convergence ofthe series for Q follows. It is then immediate from the equation R ='P - Q(Pthat R is also real analytic.

(2) Conclusion (2) follows from (1) by taking 41 = yk, setting W = yk - R,and setting U = Q. We see that U so constructed is nonvanishing at the originby (6.11). That W is in fact a Weierstrass polynomial follows from (6.8) appliedwith a=0.

6.2 Resolution of Singularities

Hironaka's great paper [HM 76] carries out a program of Oscar Zariski initiatedin [ZO 40) to resolve the singularities of an algebraic variety. The idea is bestcaptured with the following simple example.

Consider the variety V in R2 given by

x3+2y2-3x-2=0. (6.16)

The sketch in Figure 6.1 shows that this variety has a double point at (-1, 0).The philosophy of resolution of singularities is to exhibit the variety as the (lo-

cally) univalent, proper image under an algebraic mapping of an algebraic mani-fold without singularities. In this example, the mapping

s i-+ (-2s2+2,-2s3+3s)

6.2. Resolution of Singularities 157

FSgure6.1.TheVariety x3+2y2-3x-2=0

sends the real line algebraically and properly onto the variety. This is a particular(but certainly not the only) resolution of the singularity of the variety V.

Hironaka shows in [HM 76] that any algebraic variety over the reals, the com-plex numbers, or any field of characteristic zero may be resolved in this fashion.He shows that both complex analytic and real analytic varieties may be resolvedas well. Unfortunately for analysts, Hironaka's proof is presented in the languageof schemes and is for all practical purposes impenetrable to us.

Fortunately Bierstone and Milman [BM 891 have constructed a proof of theresolution of singularities theorem that applies to real and complex analytic va-rieties and to algebraic varieties over any field of characteristic zero. The key toresolving singularities is the beautiful classical idea of "blowing up" a point, aprocess that separates all lines passing through the point. However there is a basiccomplication that is in the nature of things and will never be removed. Namely,generic analytic varieties do not have singularities that are as simple as the sin-gularity in the variety V exhibited above. A variety is, on an open dense set, ananalytic manifold of some top dimension k, with a singular locus S of dimen-sion not exceeding k - 1. But then S is, on a relative open dense set, an analyticmanifold of dimension k - 1 with a singular locus S' of dimension not exceedingk - 2. Continuing inductively we find a stratification of the singular locus of ouranalytic variety all the way down to a discrete set of singular points. Any blowingup procedure must proceed inductively, starting at the dimension zero singularlocus and working up to the top dimension.

A second complication is that the singular locus of an analytic variety may nothave normal (i.e., transverse) crossings as in the variety V above. For instance,


Figure 6.2. The Variety y2 = (x + 1)4 (2 - x)

the variety W C R2 given by

has the property (see Figure 6.2) that the point (-1, 0) is an element of the sin-gular locus and, at that point, two branches of the curve make contact at a pointof tangency. This type of phenomenon introduces additional complexity into theblowing up procedure.

A third complication that may arise is that a singular point may be a "pinchpoint". That is, the curve does not cross itself at the point, but instead pinches inthe sense that it bends so that it is tangent to itself. For example, the curve

y2 = x3

has the point (0, 0) as a pinch point (see Figure 6.3).Because of the considerations described in the preceding paragraphs, we have

to content ourselves in this monograph with a treatment of resolution of singulari-ties in a very special situation. We shall introduce enough terminology so that thetheorem may be formulated and discussed for real analytic varieties in full gener-ality; however the proof will only consider algebraic varieties in three dimensionswith singularities that are all double and triple points with normal crossings. Abrief discussion later will explain just how special this situation really is.

As we stated above, the key to resolving singularities is blowing up a point.While formerly the sole province of algebraic geometers, this technique is nowbecoming a tool for analysts as well (see, for instance, [BF 78]). The process ofblowing up separates all the lines passing through a point P in space. A moment'sthought shows that this is a prototype for what we wish to do when resolving a sin-gularity: namely we wish to separate the tangent spaces of the different branches


Figure 6.3. The Variety y2 = x3

of our variety that pass through a multiple point P. Now we begin our formaltreatment, starting with a consideration of projective space.

Definition 6.2.1 The projective space RPN-1 is defined to be the set of (one di-mensional) lines through 0 in RN. A natural way to think about R.PN-1 is as thequotient of RN \ {0} by the equivalence relation

(Si,...,SN) (tl,...,tN)

if and only if there is a nonzero real number ,. such that

(sl,...,SO =(A.t1,...,)L tN)

The equivalence class of (s1, ... , sN) is denoted by [si, ..., sN].

In order to see that RPN-1 is a manifold, we define coordinate patches

W[io]={[s1,...,sN]:sip#0},

for io = 1, ... , N. Then local coordinates on W[i0] are given by

1sl . ... sio-1 sip+l SN

sip sip sip sip

It is a simple matter to see that the coordinate change functions are C°O, indeedreal analytic. Thus R]PN-I is a compact, real analytic manifold of real dimensionN - 1. It is sometimes geometrically convenient to think of RIP1V-1 as the unit(N - 1)-sphere with antipodal points identified. We will usually think of RIPN-1

as a collection of lines.


Definition 6.2.2 Let U be a neighborhood of the origin in RN. The blowup of theorigin is the set

U=_{(x,t)EUxRPN-':xee).The manifold U covers U in a natural way by the map n : U -- U that sends(x, C) to x, that is,

tr((x, l)) = x .

Clearly r is univalent from (U \ {0}) x RPN-1 onto U \ {0}. For if tr((x, e))7r((x', C')) and both x 0 0 and x' 0 0, then x = x', but a is the line through xand 0 and l' is the line through x' and 0 hence P = B'.

However, U separates the lines through the origin. For if Z and t' are distinctlines through 0, then ((x, f) : x E l} and {(x', t') : x' E P'} are disjoint subsetsof U.

Definition 6.2.3 The set it (0) e U is called the exceptional divisor and isusually denoted by E.

Definition 6.2.4 Let M be a manifold of dimension N and x a point of M. Let Wbe a coordinate patch on M that contains x and W U C_ RN a coordinatemap sending x to 0.

(1) Denote by Mx the "pullback" of the covering space U consisting of the set ofall ordered pairs (w, g) such that w E W, l; E U, and .(w) = n(1; ). We callMx the (local) blowup of the point x in the manifold M.

(2) The local blowup Mx is equipped with a natural projection (still called n)down to M defined by n((w, )) = w. The projection from Ms g W X 0onto the second factor will be denoted by .

(3) The set Ex = it-1 (x) = Ex is called the exceptional divisor of the blowup.

We have the commutative diagram:

Mx.

n

- U

n

MDW U

Here is projection onto the second factor (recall Mx 9 W x U).


Definition 6.2.5 If V is a subvariety of M then the proper transform of V underthe blowup procedure is defined to be

7r-1 (V \{x})=rr-1(V)\Ex.

It is the blowing up procedure that we will use to separate branches of an an-alytic variety when performing the resolution process. In order to facilitate ourunderstanding of these matters, we now consider local coordinates in M. Let(t1, ... , tN) be local coordinates on W C M. We shall focus attention on localgeometry, hence we will deal with the manifold W rather than with all of M.Therefore we shall speak only of W.. Then

W. = ((Y, f) E W x pN-1 : yj ei = Yi 1A,

where

We let

W U01 = ((y, e) : eio 54 o} .

Then on Wx[io] we can use the following N functions as local coordinates:

ej

Lio ,

Yio

forj0io,

We introduce the notation Y[io] j for these functions by defining

Y[io] j (Y, e) = L j for j 0 io ,Pio '

Y[io]io (Y, e) = Yio

by

We see that the projection n : Wx --). W is given in local coordinates on Wx [io]

(Y[i0]1, ... , Y[io]io-l, Y[i0]to, Y[io]io+1, ... , Y[io]N) ! )

(Y[io]io . Y[i0]1, , Y[io]io Y[io]io-1,

Y[io]io, Y[io]io Y[iolio+l, , Y[io]io Y[io]N)

Also the exceptional divisor Ex is given in local coordinates by

{(y, 0 : Y[io]io = 0).


Next we look at the transition functions in local coordinates. In Wx [ip] n W [i 1,io # it, we have

Y[ip]j

Y[i 1 lio

Y[ii] j

It follows that

ej

£io

£10

ej

Y[itl j = Y[i11io Y[iolj,

provided j is equal to neither ip nor i1. We have also

Y[i0lio = Yio ,

Y[itli, = Yi, .

But, on Wx,

so that

YNO'ei, =Yi,'eio,

Y[iolio = Y[itli, Y[ihlio

Y[il li, = Y[i0lio Y[ioli, .

We see that Wx is a real analytic manifold.Now it is time to study the operation of resolution of singularities. We will

study an analytic variety V C_ R3 that has only ordinary singularities of orders 2and 3. We need to define the phrase "ordinary singularity:' If P is a point of a realanalytic variety V, then define the tangent cone to V at P to be the union of alltangent lines to all analytic arcs lying in V and passing through P. In the example(6.16) with which we began this section, every point but one in the variety hastangent cone that is just a line-because every point but one is a regular (or man-ifold) point of the variety. The exception is the point (-1, 0), where the tangentcone consists of the union of the lines

Y=x+ and y=-fix-72.

Now a multiple point P of order m of an analytic variety of dimension N is called"ordinary" if the tangent cone at P consists of m distinct affine spaces each ofdimension N. Thus an ordinary double (that is, order 2) point on a curve in R2will look like X in Figure 6.4.

Generically, triple points do not occur on curves in R2. This is why we con-sider an algebraic variety V in R3. In this situation, elementary dimension theory


Figure 6A. An Ordinary Double Point

arguments show that generic triple points are isolated in the variety and the set ofdouble points form a one dimensional subvariety called the double curve. We letpt, ... , p, be the triple points and let C1, ..., CN, be the irreducible componentsof the double curve.

Now let

tri

be the blowup at the points p l , ... , p,. That is, we perform the blowup proceduresuccessively at each of the points pi through p,. Let E; be the exceptional divisorover pi. In a neighborhood of E; the proper transform Vi of V will consist ofthree smooth sheets which intersect pairwise in smooth arcs.

Of course E; is a copy of RP2, and VI intersects E; in three lines. The doublecurve of Vi is the proper transform of C, and consists of three arcs arising fromthe pairwise intersections of the three components of V1.

Let us explicitly verify the statements in the last two paragraphs using local co-ordinates. Let the coordinates about a triple point p be t1 , t2, t3; we may assume,after a change of coordinates, that V is given in a neighborhood U c R3 of p asthe zero set of the polynomial ti t2 t3. We then see that ir, t (U) is covered in anatural way by three open sets U1, U2, U3, where

U; 7r_ t (U) \ {the proper transform of the hyperplane (t : t; = 0)} .

In terms of the coordinates on U; given by

Y[i]i = ti , Y[i]j = tL , Y[l],t =tk

ti ti

we find that


1>rj 1(V) _ {Mill, Y[i12, Y[il3) t1 t2 t3 = 0}

_ {Mill, Y[i]2, Y[i]3)l

(Y[ili) (Y[i]i Y[i]3) (Y[ili Y[i]k) = 0}

{Mill, Y[i12, Y[il3)

(Y[i]i)3 Y[i]i Y[ilk = 01

{Mill, Y[i12, Y[03) Y[ili = 0}

U {(Y[ill, Y[i12, Y[il3) Y[i12 = 01

U {(Y[ill, Y102, Y[i]3) Y[i]3 = 01

Thus we see that the intersection of the proper transform VI of V with U1 equalsprecisely the proper transforms of the two coordinate hyperplanes tj = 0 andtk = 0.

Note also that the double curve C of VI is the union of the arcs Y[i ]j = Y[i ]k =0 in Ui (because we chose coordinates so that V = (ti 12 t3 = 0)). In particular,the double curve is smooth, so that the irreducible components Ci of C are disjointmanifolds of dimension one.

Now letrr2:X-+Y

be the blowup of Y along the double curve C-that is, we blow up at each pointof C. Our full resolution of the variety V will be given by n2 o r1.

Let Fi be the exceptional divisor over the irreducible component Ci, V theproper transform of VI, and E, the inverse image of Ei under 7r2. First we checkthat V is smooth. There is nothing to check except at the points of 7r2 1(C). Letc e C. We may choose coordinates so that, in a neighborhood U of c, we have

V1 =(((1,t2,t3):12.13=0) and C={(t1,t2,t3):t2=t3=0)

Now the inverse image of U under n2 is then covered by open sets U2 and U3consisting respectively of the complements of the hyperplanes (t2 = 0) and{t3 = 0).

In U2 we have coordinates

Y[211 = ti , Y[2]2 = t2 , Y[2]3 = t2

In these new coordinates we see that F = UFi = {Y[2]2 = 0)) and

n2 I(V1) _ 1(Y[2]1, Y[2]2, Y[2]3) : t2 . t3 = 01


= I(Y[2]t, Y[2]2, Y(2)3) : t2 t2 Y[2)3 = 01

{(Y[211, Y[2]2, Y[2l3) : t2 = 0}

U {(Y[2]1, Y[2]2, Y[2]3) : Y[213 = 0}

Thus we see that V is the disjoint union of smooth manifolds, hence is smooth, inU2.

Similarly on U3 we have coordinates

Y[3]1 = It , Y[3]2 = t3 Y[313 = t3

In these new coordinates we see that F = UF; _ [Y[3]3 = 0} and

n2 t(Vi) _ ((Y(313, Y[3]2, Y(313) : t2 t3 = 0}

{(Y(311, Y[312, Y[3)3) : t3 t3 Y[3]2 = 01

{(Y[3]1, Y(312, Y[3]3) : t3 = 0}

U {(Y[311, Y[312, Y[3]3) : Y[3)2 = 01

Thus we see that V is the disjoint union of smooth manifolds, hence is smooth, inU3-

In In summary, we have found that V is smooth in a neighborhood of 7r2 '(C).In fact we may note that, near c, the intersection V n F equals precisely the(disjoint) union of the two sections of the bundle F --> C that correspond to thenormal directions to C in the two branches of VI at c.

We have proved a very special case of the following theorem of Hironaka:

Theorem 6.2.6 (Hironaka) Let ft , ... , fk be real analytic functions on an opensubset U e RN, and let

V ={xEU: fj(x)=Oforj=l,...,k}

be the corresponding variety. Then there is a blowup

zr:X -->U

such that the proper transform of V in X is a smooth, real analytic manifold.

We close by noting that, for algebraic varieties, the restriction to varieties in R3(or, what is more convenient in algebraic geometry, the restriction to varieties inRP3) poses no loss of generality since dimension-theoretic considerations allow


one to reduce the general theorem-in the case of surfaces-to two dimensionalvarieties in dimension three (see [GH 78; pp. 612-613]). By restricting to singu-larities with normal crossings, and not considering even pinch points (much lessthe more complicated stratification of singularities that is typical), we have beenable to present an extremely simplified sketch of Hironaka's theorem.

6.3 Lojasiewicz's Structure Theorem for RealAnalytic Varieties

A complex analytic variety is defined to be the set of common zeroes, on someopen domain U, of a finite collection of holomorphic functions. Complex analyticvarieties are much like complex algebraic varieties: because of the completenessof the complex field, the structure theory contains no surprises and it is fairly wellunderstood. A good reference is [GU 70]. A complex analytic variety that is thezero set V of a single holomorphic function on an open set U C- C" is in factan (n - 1)-dimensional complex analytic manifold on a dense open subset VRof V. The exceptional set Eo is closed and has complex dimension at least one(real dimension at least two) less than the dimension of V. This last assertion isestablished by realizing E0 locally as the zero set of a certain resultant equationon a copy of C"-I lying in C". See [KS 82] for details.

In turn, the set E0 may be analyzed and a relatively dense open subset V, foundwhich is a complex analytic manifold of complex dimension at most n - 2 (realdimension at most 2n -4). The exceptional set Et a E0 is closed and has complexdimension at most n - 3 (real dimension at most 2n - 6).

This analysis may be continued to obtain a stratification of E into manifoldsof decreasing complex dimensions. Complete details of this construction may befound in [GU 70]. A briefer treatment is in (KS 82].

The situation for real analytic manifolds is somewhat more complicated, justbecause real analytic polynomials do not always have roots in the reals. To give anindication of the difference between the real situation and the complex situation,observe that generically the complex variety determined by k holomorphic func-tions (satisfying a natural independence condition that can be expressed in termsof the rank of the space spanned by their gradients) in C", 0 < k < n, is of com-plex dimension n - k. Nothing of the sort is true for real varieties: for example, thevariety in R3 determined by the real analytic function F(xl, x2, x3) = xi +z2+x3is the zero dimensional set ((0, 0, 0)).

Our purpose in this chapter is to give a brief description of Lojasiewicz's struc-ture theorem for real analytic varieties and his vanishing theorem for real analyticfunctions. We prove little; the primary intent is to introduce these results to thenon-specialist. In any event, the detailed proofs are extremely technical and farexceed the scope and purpose of this book. Lojasiewicz's comprehensive mono-

6.3. Lojasiewicz's Structure Theorem 167

graph [LS 91], which has been translated into English, gives a thorough treatmentof his theorem together with all necessary background. It should be noted thatthe paper [BM 881 gives a modem treatment of many of Lojasiewicz's results,providing much more accessible proofs of the theorems.

STEP I (The Structure Theorem): We begin by reviewing some terminology andresults f r o m Section 6.1. A function H(x1, ... , xk_1; xk) of k real variables iscalled a distinguished polynomial or Weierstrass polynomial if it has the form

H(xl,...,xk-l;xk) = xk +A1(xl,...,Xk_1)Xk-1+...

+Am-l(X1,...,Xk-1)Xk+Am(xl,...,xk-I),

where each Ai vanishes at (xl, ... , xk_ 1) = (0, ... , 0). It is an important factthat any analytic function is locally, up to an invertible factor, a distinguishedpolynomial. More precisely we have the following theorem (which is proved inSection 6.1).

Theorem 6.3.1 (The Weierstrass Preparation Theorem) Let f be a function thatvanishes at the origin in Rk and that is real analytic in a neighborhood of theorigin. Assume (as we may after a normalization) that f (0, ... , 0, xk) is not iden-tically zero. Then f may be written in the form

where H is a distinguished polynomial and U does not vanish in a neighborhoodof the origin.

The Weierstrass preparation theorem allows one to establish properties of an-alytic varieties by inducting on dimension. In particular, it is straightforward toprove that the collection of (germs of) real analytic functions in a neighborhoodof the origin form an integral domain, and more specifically a unique factoriza-tion domain. Thus any real analytic function that vanishes at the origin admits aunique (up to order) factorization into irreducible factors.

Likewise, if H is a distinguished polynomial, then H admits a (unique) de-composition into irreducible distinguished polynomials. If H is a distinguishedpolynomial, then the discriminant (see [VDW 70]) D(H)(xl, ... , xk_1) vanishesif and only if H(xl, ..., xk_1; xk) has a repeated irreducible factor. By using thefact that, for a nontrivial f, the discriminant cannot vanish identically, one mayprove the following result.

Proposition 6.3.2 Let f be a function that is real analytic in a neighborhood ofthe origin and assume that f (0, ... , 0, xk) is not identically zero. Then there is a(possibly smaller) neighborhood U of the origin and a distinguished polynomialHp on U such that Hp has nonvanishing discriminant on U and the zero set of fon U is identical to the zero set of Ho on U. The polynomial Hp is unique up toinvertible factors. It is called the distinguished polynomial associated to f.


By means of a careful analysis of the symmetric functions of the roots of adistinguished polynomial, Lojasiewicz is able to prove the following structuretheorem for varieties:

Theorem 633 (Lojasiewicz's Structure Theorem for Varieties) Let 4(xl,... , xN) be a real analytic function in a neighborhood of the origin. We may as-sume that 0(0, ... , 0, XN) 0- 0. After a rotation of the coordinates xl , . . , xN-1,one has that there exist numbers 3j > 0, j = 1, ... N, and a system of distin-guished polynomials

Ht(xt,...,xk;xt) (0<k<N-1;k+15 t:5 N)defined on Qk = {Ixj I < Sj, 1 < j < k} such that the discriminant V of Hdoes not vanish on Qk and the following properties are satisfied:

(1) Each root c of H4 (xl , ... , xk; ')On Qk satisfies I I< St.

(2) The set

Z-(x=(xl,...,xN):Ixj I <SjVjand(b(x)=0)

has a decompositionZ=VN-1 U...UV°.

The set V O is either empty or consists of the origin alone. For 1 < k < N -1,we may write Vk as a finite, disjoint union

Vk = Ux rx

oft-dimensional subvarieties which have the following explicit description:

(a) (Analytic Parametrization) Each 1'x is defined by a system of N -k equa-tions

xk+l = x qk+l (xl , ... , xk) ,

XN = xgN(xl,...,xk),

where each function x qk is each real analytic on an open subset S2x C

Qk CRk,Hk (xt, ...,xk; xqk) 0,

andDIk(x1,...,xk) # 0

for all (xi,...,xk)Enk-,t=k+1,...,N.

6.3. Lojasiewicz's Structure Theorem 169

(b) (Non-Redundancy) For any integers k, X, X', either pk = f2k, or c4 n

!ak, = 0. In the second instance one has, for any e = k + 1, ... , N,

either x r X'71 on SZX or x ?1k (xl , ... , xk) 54 x qi (xl , ... , xk) for allxESZX.

(c) (Stratification) For each k the closure of Vk contains all the subsequentVi 's: that is, Q fl Vk 2 V1 U .. U V0. (This property, while technical,is an important point. The lower dimensional varieties Vi, j < N - 1,do not occur as isolated sets; they are in fact the zero sets of certaindiscriminants and (in a sense) form the boundaries of the componentsI'x+1 of Vj+1 .. , vN'1. The example (6.16) at the beginning of thischapter illustrates this principle.)

Lojasiewicz's theorem teaches us that a real analytic variety can be stratifiedinto submanifolds of dimensions 0, 1, ... , N - 1. The statement in the theoremthat the zero dimensional manifold can be (locally) taken to be the origin is justanother way of saying that a zero dimensional manifold is a discrete set of points.

Of course Lojasiewicz's theorem is trivial when N = 1. For N = 2 it may bederived as an easy consequence of the local Puiseux series expansion. Howeverfor N > 3 it is deep and new. Now we present the first principal application (cf.Corollary 5.2.1).

Theorem 6.3.4 (The Vanishing Theorem) Let f be a nonzero real analytic func-tion on an open set U C R'''. Assume that the zero set Z off in U is nonempty.Define

dist (x, Z) = inf(Ix - zl : z E Z).

Let E be a compact subset of U. Then there are a constant C > 0 and aninteger q > 0, depending on E, such that

If (x) I > C dist (x, Z)4

for every X E E.

We have already discussed results of this type in Chapter 5, particularly inTheorem 5.11 and Corollary 5.2.1.

Notice that in one variable this result is trivial: by a compactness argument wemay take U to be so small that it contains a single, isolated zero P of f. Then fvanishes to some finite order m at P and we may take q = m.

For N > 2 matters are less obvious. However consider a special case. In caseZ has the special form

Z= {(Xl, ... , XN) : XN =0),

then we may write f in the form


P X1, ... , XN) = (XN)m . g(xl, ... , XN),

where g is real analytic and does not vanish. Since g does not vanish it followsagain that the desired inequality holds with q = m.

Now it is too much to hope in general that Z has the simple form of a hyper-plane. However one might hope that Z is (the union of sets each of which is) abi-lipschitzian manifold; more particularly, we might realize Z as (the union ofsets each of which is) the graph of a real analytic function that is in some Lip-schitz class. [The form of the Puiseux expansion suggests rather explicitly howthis might come about in two dimensions.]

Consider the examplef (X' Y) = Y2 - x3 .

For this f there is no problem verifying Lojasiewicz's inequality on a compact setE that misses the origin: just perform a real analytic change of coordinates andreduce to the hyperplane case. However the zero set of f has a cusp at the origin,and the simple device of a change of coordinates does not apply there. Instead wenotice that, near 0, Z = Z(f) can be realized as the union of the sets

rj =((x,y)ER2:y2=x3,y>0),

r2={(x,y)ER2:y2=x3,y <0),

and

r° _ {(0, o)).

Because each of r,1 and r2 is the graph of a real analytic function in the variabley that is Lipschitz 2/3, it is not difficult to see that

f(x, y) > C y2 = C' . ([y2]1/3/3

> C" disc ((x, y), Z)3.

(In this particularly straightforward example the set ro plays no explicit role inthe analysis; however see the discussion below.)

For the general case, an important part of Lojasiewicz's analysis involves show-ing that the varieties rX are the graphs of the functions X nl and that these func-tions are in fact Lipschitz of some positive order. In the two dimensional examplejust discussed, the (implicit) role played by the zero dimensional variety ro is toenable us to deduce that the worst points to consider are those on the coordinateaxes. Once we have this piece of information, the analysis becomes one dimen-sional. In higher dimensions, the exceptional set V N-1 U... U VO is more complexand one must obtain the estimate by inducting on the Vi's. We can say no moreabout the matter here.

Our last application is the following theorem:

6.4 The Embedding of Real Analytic Manifolds 171

Theorem 6.3.5 (The Lojasiewicz Division Theorem) Let 0 be a real analyticfunction on an open set U C RN that vanishes identically on no connected com-ponent of U. If T is a distribution on U, then there exists a distribution S suchthat

4) S=T.

We sketched the proof of this theorem in the previous chapter, in the case when4> is a polynomial. The vanishing theorem, Theorem 6.3.4, provides the critical es-timate so that the same proof can be used for 4> real analytic. In fact, Lojasiewiczproves that any infinitely differentiable function 0 whose zero set satisfies theconclusions of the structure theorem for analytic varieties, and with the additionalhypothesis that the functions X nt vanish only to finite order-in a rather strong,quantitative sense that is implied by the vanishing theorem-also satisfies the con-clusion of the division theorem. We refer the reader to [LS 59], [LS 91] for furtherdetails.

The thinness of the zero set of a nonconstant real analytic functions can fre-quently be a powerful analytic tool. In [DT 81] it is used to give a strikingly easyproof of the local solvability of constant coefficient partial differential operators.See Chapter 5.

6.4 The Embedding of Real Analytic Manifolds

Recall that a manifold of dimension N is a paracompact Hausdorff space M thatis equipped with a locally Euclidean structure in the following fashion: Thereis a covering U = t of M by open sets and there are homeomorphisms4i : Ul -+ B, where B C RN is the unit ball. We specify additional structureon the manifold by imposing conditions on the transition from one coordinatepatch Uj to another. That is, the manifold is Ck for some k = 1, 2, ... if all of thetransition functions

4joOkt :ckoOjt(B)-+Ojo4kt(B)

are Ck. Notice that the condition that we check here is on a function (namely4j o 4k t) from Euclidean space to Euclidean space; therefore it makes sense apriori to discuss smoothness of the function. When the condition holds for k = oothe manifold is then said to be C°O or "smooth:' When the maps are real analyticthen the manifold is termed real analytic.

In the case that N = 2n is even, then we may identify RN with C" in a nat-ural way. If the transition maps ¢I o Ok t are holomorphic, then we say that themanifold M is a complex analytic manifold or, simply, a complex manifold.

Function theory on an abstractly presented manifold (as above) can be incon-venient and tedious, for one must make constructions locally on the coordinatepatches Uj and then paste them together (usually with a partition of unity). If the


manifold can be realized in a natural fashion as a subset of Euclidean space, thenthe manifold inherits the function theory of the Euclidean space-by restriction.Thus we are led to consider embeddings.

In order to give a precise description of an embedding, we first must define thenotion of a smooth (respectively real analytic, complex analytic) function on amanifold. If M is a smooth manifold, then a function f : M -> R is called CO0 orsmooth if for each coordinate mapping O j : Uj -+ B it holds that f o 071 : B -R is COO. The definition of real analytic function and complex analytic functionon a real analytic or complex analytic manifold is of course analogous.

Now a smooth mapping of a smooth manifold M of dimension N into a Eu-clidean space Rk is a function

F=(f1, ,fk),

where each f j is a smooth function from the manifold M into R. The mappingis called an embedding if it is a homeomorphism onto its image. Of particularinterest and utility are proper embeddings: an embedding F : M -+ Rk is calledproper if, for any compact K C RN, it holds that f -t (K) is compact in M.Another, more informal, way to think about the concept of "proper" is that if (p j )are points of M that "run out to the edge" of M, then their images F(p j) "run outto infinity" in RN.

In general, a manifold of dimension N does not embed into RN. For example,a sphere is a two dimensional manifold but will not embed into R2. A Klein bottleis a two dimensional manifold that will not embed into R3.

In 1936, H. Whitney {WH 36] proved that any smooth manifold of dimensionN can be smoothly, properly embedded in R2N+r. This result is sharp. In theperiod 1930-1960 one of the major unsolved problems in manifold theory wasto properly embed a real analytic manifold into some Euclidean space. Whitney[WH 36] was able to prove that there is a COO embedding of such a manifoldwhose image in Euclidean space is a real analytic submanifold of space; but sucha result is of little use since the map does not preserve the real analytic structureof the manifold.

In order to understand why the real analytic embedding of a real analytic mani-fold is difficult, we briefly discuss the proof in the C°O case. By the very definitionof manifold, one is given a local embedding: that is, the coordinate function O jis an embedding of Uj into RN. For each j let Aj be a C°O function of com-pact support in U j such that F_j k j (x) = 1 on M (such a family of functionson a manifold is called a partition of unity and is a standard construct in mani-fold theory-see [MJ 66]). Naively, one might hope that F(x) = E j Xjoj is anembedding of M into RN. But of course this map will generally not be one-to-one. So we must pass to higher dimensions to separate the images of the differentcoordinate patches. This is the spirit of Whitney's proof.

The problem with emulating the preceding argument in the real analytic cat-egory is that partitions of unity do not exist. A real analytic function, either on

6.4. The Embedding of Real Analytic Manifolds 173

Euclidean space or on a manifold, that is compactly supported (more generallythat vanishes on an open set) must be identically zero. Thus entirely differenttechniques must be developed to treat embedding of real analytic manifolds. Theproblem comes down to constructing a large family of globally defined real ana-lytic functions on the manifold. By the way that a manifold is defined, one onlyhas the ability to construct functions locally (on the coordinate patches). Thus oneneeds a way to patch locally defined objects together in the real analytic category.Much in the spirit of the Stone-Weierstrass theorem, it suffices for our purposes tofind globally defined real analytic functions on the manifold that separate points.

There are three known ways to address the technical problem described abovein the real analytic category. Each of these methods requires deep and detailedbackground in either sheaf theory, several complex variables, differential geom-etry, or partial differential equations. Limitations of space and scope make it im-possible for us to present in detail any of these methods; however we shall brieflydescribe each of them.

The first method, for compact manifolds, proceeds as follows (for details, see[RH 60]): Suppose that one is given a compact real analytic manifold M thatcomes equipped with a real analytic Riemannian metric. Associated to this Rie-mannian metric is its Laplace-Beltrami operator C-a second order, positive,elliptic partial differential operator on M that is invariant under isometries of themanifold. The eigenfunctions of the operator C are well understood: they will bereal analytic (by the real analytic hypoellipticity of elliptic partial differential op-erators), they are countable in number, and they will separate points in a suitableway. In fact this last assertion follows from Hermann Weyl's theory of eigenval-ues of elliptic operators on a compact manifold: the geometry of the manifoldcan be reconstructed from the spectral theory of a suitable elliptic operator on themanifold (see [Cl 84] and the more general index theory of Atiyah and Singer[PR 65]). Thus, with some additional technique, the eigenfunctions of C can bepatched together to manufacture an embedding of the manifold.

The difficulty with the approach just discussed (certainly the simpler of thethree) is finding a real analytic Riemannian metric. To construct a C°O Rieman-nian metric on the manifold is an exercise with partitions of unity. But the con-struction of a real analytic metric, that is a matrix {g;,j(x))N=, of functions thatis positive definite for each x, begs the problem of constructing real analytic func-tions on a real analytic manifold. While in some contexts the necessary functions,indeed the metric itself, are given to us from the problem being studied, in generalthe problem of constructing a real analytic metric is no simpler than constructingan embedding (note here that once the manifold is embedded then a Riemannianmetric is automatically inherited from the ambient Euclidean space). Thus thisapproach, while appealing, does not completely settle the embedding problem.

The partial differential equations approach to the embedding problem, whichagain only applies in the compact case, is due to C. B. Morrey [MC 68]. It can besummarized as follows: One first constructs a positive, elliptic, second order par-tial differential operator with real analytic coefficients on the manifold M that has


characteristics similar to the Laplace-Beltrami operator described in the discus-sion of the first method. Then the eigenfunctions of this operator become the basictools for constructing the embedding. We shall say no more about this method.

The third method, due to H. Grauert [GH 58), applies to any real analytic man-ifold, compact or noncompact. It is not in the spirit of the present book because itreduces the embedding problem to an even deeper and more difficult problem inthe complex analysis of several variables; but Grauert's is the only known tech-nique for solving the general embedding problem. In order to avoid an extremelytechnical digression into the lore and machinery of several complex variables, wegive but a brief description of Grauert's ideas.

Let U be an open subset of RN and let (XI, x2, ... , xN) be the Euclidean coor-dinates on U. We may think of U as a subset of C 'V in a natural way by means ofthe mapping

(XI,X2,....XN) F-- (XI +10,X2+i0,...,XN+i0).

In this fashion we are considering the (trivial) real analytic manifold U as a sub-manifold of the complex manifold

U = ((XI +iyl,X2+iy2,...,XN+iyN): (xl,x2,...,XN) E U).

The manifold U is called a comple.xiication of U.If 4,(x) is a real analytic function on U and P E U, then 0 has a power series

expansion about the point P :

4,(x) = Eaa(x - P)a.

Of course there is an r > 0 such that the series converges absolutely and uni-formly when Jxj - P j I < r f o r j = 1, 2, ... , N. But then the function

4,(z) = a. (z - P)a

is well-defined and the series is absolutely and uniformly convergent when 1z j -Pj 1 < r, j = 1, 2.... N. The function ¢(z) is a holomorphic function of sev-eral complex variables (that is, it is holomorphic in each variable separately-see[KS 82] for a discussion of several equivalent definitions of holomorphic func-tion of several complex variables). Thus the function 0 is a complexification ofthe original real analytic function 0. We may perform this complexification pro-cedure on the power series expansion of 0 about each point P of U. Of course,by the uniqueness of analytic continuation, two different complexifications abouttwo different points of U must agree on their common domain. As a result of thisprocedure we obtain an open subset U of CN with U C U and a complex analyticfunction 4, on U such that The function 0 is the complexification ofthe original analytic function U.

6.4. The Embedding of Real Analytic Manifolds 175

Now if M is a real analytic manifold then, by a procedure analogous to that de-scribed in the preceding paragraph, each of the inverse coordinate functions -1

may be "complexified" to a function Vii. The image of the complexified functionwill lie in an N-dimensional line bundle over the coordinate patch Uj. We shallnot provide details here, but refer the interested reader to [BW 59]. That the tran-sition functions f o ;k t are holomorphic functions of several complex variablesis a formality that follows immediately from the Bruhat-Whitney construction.This procedure creates a complex manifold M that is a submanifold of an N-dimensional line bundle over the original real analytic manifold M and which hascomplex analytic coordinate functions. Thus M is realized in a natural fashion asa real analytic submanifold of the complex manifold M.

Grauert in fact proves an embedding theorem for (a small modification of) thecomplex manifold M. By restriction, this provides an embedding of the originalreal analytic manifold M. In order to give a description of the procedure, we needa new definition. Let U be an open subset of C". Let u be a continuous functionon U. We say that u is plurisubharmonic on U if for each fixed a, b E C" suchthat Ua,b e C and a + b E U) # 0 it holds that the function

Ua.b 3 r-) u( C a + b)

is subharmonicl in the classical sense of function theory of one complex vari-able. Subharmonic functions are much more flexible objects than are holomorphicfunctions. For instance, they are closed under the operation of taking a maximum.They may be constructed as potentials of positive measures. Plurisubharmonicfunctions are likewise flexible. And just as the Riesz representation (see [TM 59])can be used to manufacture harmonic functions from subharmonic functions, sothere are analogous devices in the theory of several complex variables to passfrom plurisubharmonic functions to the real parts of holomorphic functions.

Naturally a function u on a complex manifold W is termed plurisubharmonic ifeach of the compositions u o Oi 1 with inverse coordinate functions is plurisubhar-monic. By means of an extremely ingenious argument, Grauert constructs on (aslightly shrunken version of) M a plurisubharmonic function p with the propertythat for every positive real number r > 0 the set a-1({x E R : x < r)) is compactin M. Such a function p is called a plurisubharmonic exhaustion function for M.Grauert proves that any complex manifold that has a plurisubharmonic exhaustionfunction is a Stein manifold.

What is a Stein manifold? A Stein manifold W is a complex manifold that sup-ports a great many holomorphic functions. Indeed, given any two point a, b E Wthere is a holomorphic function f on W such that f (a) 96 f (b). As indicated inthe first portion of this section, such functions are the basic tools for constructing

IAn upper semicontinuous function f : U -+ R U l-ooi is subharmonic if, for every x E Uand r > 0 with B(x. r) e U and for every real-valued, continuous function h : B(x. r) -. R that isharmonic on B(x. r) and satisfies h > f on aB(x, r), it holds that h > f on B(x. r).


an embedding. It is not too difficult to imitate the Whitney construction, usingGrauert's separating functions, to construct an embedding of the Stein manifoldM. We mention, however, that a deeper theorem provides even a proper embed-ding of M. This, by restriction, properly embeds the original real analytic mani-fold M and solves the embedding problem. The proper embedding theorem wasannounced by R. Remmert in [RR 54]. Complete proofs appear in [BIS 61] and[NAR 60] (see also the discussion in [BN 90; Section 3]).

We conclude this section by recording some results which are related, at leastphilosophically, to the subject proper of the present section.

Riemann first developed the concept of an abstract manifold with a metricstructure (what we now call a Riemannian manifold) in 1854 (see [LD 99; p.219ff]). In attempting to understand this circle of ideas, it is natural to wonderwhether every such abstractly presented manifold has a realization as a metricsubmanifold of Euclidean space. It should be borne in mind here that the questionof embedding the manifold differentiably is a much simpler one and amounts,from our modem perspective, to an exercise in the concept of general position(see [HM 76]). However the problem of obtaining an isometric embedding isquite subtle. It was solved, using an ingenious argument, by John Nash in 1956(see [NJ 56]). A nice history of the problem is given in that paper.

Our interest in the present section of the book is in real analytic manifolds.Since a real analytic manifold is a fortiori COO, it follows from Nash's theoremthat a real analytic Riemannian manifold has a Coo isometric embedding. It isnatural to ask whether there is a real analytic isometric embedding. In 1971 thefollowing result was proved by Greene and Jacobowitz ([GJ 71]).

Theorem 6.4.1 Let M be a compact, real analytic Riemannian manifold of di-mension n. Then there is a real analytic, isometric embedding of M intoR(3"2+1ln)/2.

The principal analytic tool in the proof of all the Nash-type theorems is a pow-erful version of the implicit function theorem (or inverse function theorem). Theclassical inverse function theorem says, in effect, that a smooth mapping of Eu-clidean spaces is surjective in a neighborhood of any point where its derivativeis surjective. Nash [NJ 561 provides an implicit function theorem for mappingsof function spaces in which the classical notion of derivative is replaced by theFrechet derivative.

The additional complication that must be dealt with in embedding problems isthat there is a loss of derivatives that makes the most natural application of the im-plicit function theorem unworkable. Thus Nash used an iteration scheme involv-ing alternate applications of smoothing operators and implicit function theoremestimates. Discussions of Nash's theorem can be found in [KP 02; Section 6.4] orin the extensive survey of Hamilton [HA 82].

We can say no more about this rather technical material here. A nice introduc-tion to the subject appears in [GR 70]. Additional work, for non-compact real

6.5 Semianalytic and Subanalytic Sets 177

analytic manifolds, appears in [GM 70]. That paper also contains results aboutlowering the dimension of the target space in which the Riemannian manifoldis embedded. The final word about embedding of Riemannian manifolds has notbeen heard, and there is still activity in the field.

6.5 Semianalytic and Subanalytic Sets

6.5.1 Basic Definitions

The theory of semianalytic and subanalytic sets is concerned with sets of pointswhich can be described using real analytic functions. Here we will not be ableto give complete proofs or even a complete exposition. We shall try to cover thehighlights. The reader interested in a deeper treatment is referred to the book ofLojasiewicz [LS 91] and to the paper of Bierstone and Milman [BM 88] and tothe references cited there. Our presentation closely follows [BM 88].

Definition 6.5.1 An algebraic subset of R" is a set of the form

R"fl{(xt,...,x"): P(xl,...,x,,)=0), (6.17)

where P is a real polynomial.

Clearly, algebraic subsets are those which can be described by polynomialequations. If we enlarge the allowable types of descriptions to include inequal-ities, conjunctions, disjunctions, and negations, then we have the following largerclass:

Definition 6.5.2 The family of semialgebraic subsets of R" is the smallest fam-ily which contains the algebraic subsets of R" and which is closed under finiteintersection, finite union, and complement.

There is another class of logical connectives: The quantifiers. The use of the ex-istential quantifier corresponds to projection. In this way we obtain what appearsto be a larger class.

Definition 6.5.3 A subset S of R" is subalgebraic if, for some m, it is the imageof a semialgebraic subset of R"+m = R" x R'" under projection onto the firstfactor.

Actually the term "subalgebraic set" turns out to be redundant. This is becauseof the following theorem.

Theorem 6.5.4 (Tarski-Seidenberg) Every subalgebraic set is semialgebraic.

In light of this theorem and the logical equivalence of the universal quantifier,V, with a combination the existential quantifier and negations, namely, - 3 it isalso true that no new sets will be introduced by the use of the universal quantifier.


Now we consider replacing the polynomial in (6.17) above by a real analyticfunction.

Definition 6.5.5

(1) Let U be an open subset of R". An analytic subset of U is a set of the form

Ufl{(xi,.. ,xn):F(xl, ..,x")=0),

where F is a real analytic function on U.

(2) Let U be an open subset of R". The family of semianalytic subsets of U is thesmallest family which contains the analytic subsets of U and which is closedunder finite intersection, finite union, and complement.

(3) A subset S of iR" is semianalytic if each point p E S has an open neighbor-hood U such that S n U is a semianalytic subset of U.

(4) A subset S of R" is subanalytic if each point p E S has a neighborhood Usuch that, for some m, S n U is the image of a relatively compact semianalyticsubset of R"+m =1R" x Rm under projection onto the first factor.

The compactness of a topological space is an intrinsic property of the spaceand not of how it is embedded in another space. Thus it is relevant to recall thefollowing definition from general topology.

Definition 6.5.6 A subset K of a topological space X is called relatively compactif the closure of K in X is compact.

A simple generalization of these ideas is made by replacing R' by a real ana-lytic manifold:

Definition 6.5.7 Let M be a real analytic manifold.

(1) Let U be an open coordinate neighborhood in M. An analytic subset of U isa set of the form

Un((xl,...,x"): F(xI,...,xn)=0),

where F is a real analytic function on U.

(2) Let U be an open coordinate neighborhood in M. The family of semianalyticsubsets of U is the smallest family which contains the analytic subsets of Uand which is closed under finite intersection, finite union, and complement.

(3) A subset S of M is semianalytic if each point p E S has an open coordinateneighborhood U such that S n U is a semianalytic subset of U.

(4) A subset S of M is subanalytic if each point p E S has a neighborhood Usuch that, for some real analytic manifold N, SnU is the image of a relativelycompact semianalytic subset of M x N under projection onto the first factor.

6.5. Semianalytic and Subanalytic Sets 179

For the purposes of analysis, the main results (due to Hironaka) are the follow-ing:

Theorem 6.5.8 (Uniformization) Suppose that S is a closed subanalytic subsetof the real analytic manifold M. Then there exists a real analytic manifold N anda proper real analytic mapping ¢ : N -+ M such that 0(N) = S. Further N canbe assumed to be of the same dimension as S.

Theorem 6.5.9 (Rectilinearization) Suppose that S is a subanalytic subset of thereal analytic manifold M of dimension in. Let K be a compact subset of M. Thenthere exist finitely many real analytic functions Oi : R' -> M, i = 1, ... , p,such that

(1) there are compact sets LI c R', i = 1, ... , p, for which tJi Oi(Li) is aneighborhood of K in M,

(2) for each i, 01 I (S) is a union of quadrants in Rm, where a quadrant in Rm isa set of the form

((XI,...,Xm) : xI UI 0. ..., xm vm 0},

with on E "< ", "> "} for each i.

We shall also need the notions of "semianalytic function" and "subanalyticfunctions."

Definition 65.10 Let M and N be real analytic manifolds. Let S be a subset ofM, and let f : S -+ N be a function.

(1) We say that f is semianalytic if its graph is semianalytic in M x N.

(2) We say that f is subanalytic if its graph is subanalytic in M x N.

There is also a notion of "semialgebraic function" that is defined similarly.

Definition 65.11 Let S be a subset of R". We say that f : 1R" -+ R' is semial-gebraic if its graph is semialgebraic in 1R" x Rm.

6.5.2 Facts Concerning Semianalytic and Subanalytic Sets

We state without proof some of the fundamental facts about semianalytic andsubanalytic sets. The main tool used in developing these results is the Weierstrasspreparation theorem.

Theorem 6.5.12 Let S be a semianalytic subset of the real analytic manifold M.Then:


(1) Every connected component of S is semianalytic.

(2) The family of connected components of S is locally finite.

(3) The set S is locally connected

(4) The closure and interior of S are semianalytic.

(5) Let U be a semianalytic subset of M with U C S which is open relative to S.Then U is locally a finite union of sets of the form

s n {x : f1(x) > o, ... , fk (x) > o) ,

where f1, ... , fk are real analytic functions.

(6) If S is closed, then S is locally a finite union of sets of the form

{x : fi(x) > 0,..., A(x) ? U),

where f1, ... , fk are real analytic functions.

The next theorem of Lojasiewicz allows us to see that, in contrast to the alge-braic situation, not all subanalytic sets are semianalytic. We will find it convenientto use some additional notation.

Notation 6.5.13 Let M be real analytic manifold of dimension k and let S g Zbe subsets of M. We will use Clos(S) to denote the closure of S in M and we willuse Closz(S) to denote the closure of S in Z using the relative topology on Z.

Theorem 6.5.14 Let M be a real analytic manifold of dimension k. Let S be asubset of M. Necessary and sufficient for S to be semianalytic of dimension lessthan or equal to k is that there exist an analytic set Z of dimension less than orequal to k such that

(1) S C Z,

(2) Clos(S) \ S is semianalytic of dimension less than or equal to k - 1,

(3) S \ Closz(S) is also semianalytic of dimension less than or equal to k - 1.

By the theorem, if a semianalytic subset of R" is of dimension less than n,then, in a neighborhood of each point, there must be a nontrivial analytic functionwhich vanishes on the subset. We consider the following example of Osgood (see[OW 65; Part 1, Chap. 2, §22] or [OW 16]). Set

S = {(x, y, z) : 3(u, v) such that x = u, y = uv, z = uve°) .


Clearly, S is subanalytic; if S were semianalytic, then there would be some realanalytic function f (x, y, z) defined near (0, 0, 0), not identically zero, which van-ishes on S. Assuming such an f exists, we write

00

f (x, y, z) = T fj (x, y, Z),j=o

where f j (x, y, z) is homogeneous of degree j. For (u, v) near the origin in R2we must have

00

0 = f(u, uv, uve°) _ uj fj(1, v, ve°),j=O

so that, for each j,u)0 = fj(l,v,ve

Since f j is a homogeneous polynomial of degree j, we must have f j - 0, acontradiction. Thus S is subanalytic, but not semianalytic.

For the semialgebraic sets, the Tarski-Seidenberg theorem shows that projec-tion does not lead to a larger class of sets, i.e., the projection of a semialgebraic setis semialgebraic. It follows a fortiori that the subsequent use of the complementwill not lead to a larger class. For the semianalytic sets, this afortiori argumentcannot be used. In spite of this, we have the following theorem (see [BM 88;Theorem 3.10] or [DLS 79]).

Theorem 6.5.15 Let M be a real analytic manifold and let S be a subanalyticsubset of M. Then M \ S is subanalytic.

An important result on subanalytic functions is the following theorem firstproved by A. M. Gabriblov (see [BM 88; Theorem 3.14] or [DLS 82]).

Theorem 6.5.16 Let M and N be real analytic manifolds, and let S be a rela-tively compact subanalytic subset of M. For a subanalytic function f : M -> N,the number of connected components of a fiber f -1 (p) is locally bounded on N.

6.5.3 Examples and Discussion

It was asserted earlier that for an analyst the main results concerning semianalyticsets and subanalytics sets are the uniformization theorem and the rectilineariza-tion theorem. In this subsection we shall illustrate this point. We start with anelementary inequality.

Definition 6.5.17 For n a positive integer and ! E R set

R ( ) = III if t:>0,-I$I^ if l: <0.


Lemma 6.5.18 Let n be a positive integer. If ti, t2 E R, then

IRn(t2) - Rn(ti)1 < 21t2 - t1 I"

Proof. Set 1:, = Rn (ti ), for i = 1, 2. We may assume l;1 < 6. There are twocases depending on whether or not 441 and 1;2 have the same sign.

First we supposeI 0:5 6

Set M = t2}, so Mn < It2 - tt 1. Then we have

12-IIn = (2+It1I)n<

nn (n)M,.

i _-O

= 2nMn

2nIt2-t11.

Next we suppose0-< 1 S t2 -

In this case, we estimate

n-1 n - 1 n-1( 1)in-1-ii n 1-iiJ 2 I _E2 2 1

i=0 i-0

so that

n _ t)1t2-tII" = (t2-ti)(E(ni 1),12-I , I

i _-O

(n-12n (t2 - 1) t2-1-i

1

i=0= 2nIt2tI1.

Lemma 6.5.19 Let I be an open interval with 0 E I. Suppose h: I -+ R is realanalytic and vanishes only at 0. If

h(l)(0) = ... =h(n-1)(0)

= 0, and h(n)(0) > 0,

then g: I -> R, defined by setting

g(t)= Ih(t)I ift>_0,Ih(t)In ift < 0,

is continuously differentiable on I with g'(0) = [f (n) (0)]1.


Proof. The derivative of g is easily calculated away from 0, while the behavior at0 is determined by using the power series for h.

Lemma 6.5.20 If f : R - R is a continuous subanalytic function, then f is lo-cally Holder continuous.

Proof. The continuity off implies that the graph is closed, so the uniformizationtheorem is applicable. Thus there exist a one dimensional real analytic manifoldM and a proper real analytic map 0: M -> R x R such that the graph off is theimage of 0. Since we need to prove a local statement, we may assume M = R.

Fix po E R and xo E ¢-t (po). Let ilt and 112 be the projection of R x R ontothe first and second factors, respectively. We know that

h (t) = III o ep (t + xo) - Po

has an isolated zero at t = 0; suppose it is a zero of order n. Let a be the sign ofh (n) (0). Set

8(t)={ aIh(t)I^ ift>0,-a Ih(t)I^ if t <0.

By Lemma 6.5.19, the inverse function theorem applies to g, so g-t is definedand continuously differentiable in a neighborhood of 0. Now note that, with s,Othe translation operator defined by ta(x) = x + xo and with R" as in Defini-tion 6.5.17, we see that

f(P) = og-t o R.(P - Po)

holds

(i) in an open interval (po - 8, po + 8) if n is odd,

(ii) in a half-open interval [po, po + 8) if n is even and a = +1,

(iii) in a half-open interval (po - 8, pp] if n is even and a = -1.

By Lemma 6.5.18, f is Holder continuous on the interval where the above in-equality holds. Since 0 is proper, there is either an xo E 0-1 (po) for which n isodd, or there are X1, x2 E ¢-t (po) with n even and with opposite signs for a, sothat f is Holder continuous in an open interval about po.

Proposition 6.5.21 Let f : R'" -+ R" be a subanalytic function. If f is continu-ous, then f is locally Holder continuous.

Proof. Let U be a bounded open set. Consider

A = {(s, t) : 3(x, y) E U such that Ix - y12 = S2 & If (x) - f (y)I2 < t2] ,


B = {(s,t):s>0 & t>0},C = {(s,t):s50 & t>0},D =F =

(Af1B)UC,D\b,

where 15 denotes the interior of D. Then F is the graph of a continuous subana-lytic function from F : R to R, which by Lemma 6.5.20 is locally Holder contin-uous. The result follows from the Holder continuity of F at 0 since F restrictedto (0, oo) represents the modulus of continuity of f. 0

Note that the continuity hypothesis is necessary since

{(x, y) : x > 0 & y = sin(x)} U {(x, y) : 0 > x & y = 0}

is a semianalaytic function (which even has the intermediate value property), butis not continuous at x = 0.

We also have the following proposition:

Proposition 6.5.22 Let S be a subanalytic subset of R'". Then the distance func-tiond:R' -+ Rdefinedby

d(x) = dist(x, S)

is subanalytic and Holder continuous.

Proof Clearly, the distance function is continuous-indeed the Holder continuity(with exponent 1) is an exercise. Set

T ={(x,z)ERm xR:2ysuch that yE S & z> Ix - yI).

Then, letting t denote the interior of T, we see that T \ fi is the graph of thedistance function and is subanalytic. O

In the context of the preceding proposition, we mention the following result.

Theorem 6.5.23 (Pblya-Raby) Let S be a closed subset of R' and let d: R' -,R be the distance function given by d (x) = dist(x, S). The square of the distancefunction is real analytic in an open neighborhood of xp E S if and only if S is areal analytic submanifold in an open neighborhood of xp.

6.5.4 Rectilinearization

The proof of the uniformization theorem makes use of the notion of blowing upwhich was discussed in Section 6.2 in the context of resolution of singularities,so we will not discuss that here. But another useful consequence of blowing upinvolves the following definition.


Definition 6.5.24 Let M be a real analytic manifold and let O(M) denote thering of real analytic functions on M. For f E O(M) we say that f is locally nor-mal crossings if each point of M has a coordinate neighborhood with coordinatesxl,...,xm such that

f(x) = ...xm g(x),

with each ri a nonnegative integer and where g is nonvanishing in the neighbor-hood.

Using the blowing up technique, one can prove the following theorem:

Theorem 6.5.25 Let M be a real analytic manifold and let 0 0 f E O(M).Then there exist a real analytic manifold N and a proper surjective real analyticmapping ¢: N -- M such that

(1) f o 0 is locally normal crossings on N,

(2) there is an open dense subset of N on which 0 is locally a dif eotnorphism.

In this subsection, we shall show how the rectilinearization theorem followsfrom the previous theorem and the uniformization theorem. Recall the rectilin-earization theorem stated in Subsection 6.5.1:

Theorem 6.5.9 (Rectilinearization) Suppose that S is a subanalytic subset of thereal analytic manifold M of dimension in. Let K be a compact subset of M. Thenthere exist finitely many real analytic functions qi : Rm -* M, i = 1, ... , p,such that

(1) there are compact sets Li c_ Rm, i = 1, ... , p, for which Ui Oi(L1) is aneighborhood of K in M,

(2) for each i, Oi- I (S) is a union of quadrants in R', where a quadrant in Rm isa set of the form

((x1,...,xm):XI U1 0,...,xm om 0),

with ai E "< ", "> "} for each i.

Proof. The result is local, so we may assume that M = Rm. Next, we find aneighborhood U of K such that there are closed subanalytic subsets S1, j with

r

SnU=U(Si,1 \S1,2)i=1

It is known from the previous section that the distance function to a subanalyticset is subanalytic so, defining di,j: U -+ R by setting

di,j (x) = dist(x, Si,j),


we obtain a collection of continuous subanalytic functions.We shall show that there exist a real analytic manifold, N, also m-dimensional,

and a proper, surjective real analytic mapping 0: N -* U such that each d;jis real analytic on N. Define the subanalytic mapping f : U -> R2' by f =(di,1, ... , dr,2). By the uniformization theorem applied to the graph of f, thereexist a real analytic manifold N of the same dimension as the graph of f, that ism-dimensional, and a proper real analytic mapping 4): N -- U x R2' such thatthe image of 4) is the graph of f. Let 111 and 172 denote projection of RI x R?'onto the first and second factors, respectively. Setting O=n, o 4), we see that 4,is surjective and n2 o 4) = (d1,1 0 ¢, ... , dr,2 o ¢) is real analytic.

Applying the above theorem, we obtain another real analytic manifold N ofdimension m and a proper surjective real analytic mapping *: N -> N such thateach d;j o 0 o * is locally normal crossings, from which the result follows. 0

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[WH 36] Differential manifolds, Annals of Mathematics (2) 37 (1936), 645-680.

Bibliography 201

KBsaku Yosida

(YK 71] Functional Analysis, Springer-Verlag, Berlin, 1974.

Oscar Zariski

[ZO 40] Local uniformization theorems on algebraic varieties, Annals of Mathemat-ics (2) 41 (1940), 852-896.

Antoni Szczepan Zygmund

[ZA 59] Trigonometric Series, 2nd edition, Cambridge University Press, Cambridge,1959.

Index

Abel, Neils Henrik, 2 22algebraic set, 122almost increasing sequence, 25analytic continuation, 15analytic set, 178Artin, Michael, 104associated function, 88associated sequence, 25Atiyah, Michael Francis, 173

Bang, Thoger, 24base space, 56Bernstein's lemma, 85Bernstein's theorem, 80Bernstein, Serge, 79, 105Besicovitch's theorem, 22Besicovitch, Abram Samoilovitch, 22Bierstone, Edward, 113, 157, 172big "0", 148blowup, 160blowup, local, 160Boas, Ralph Philip, 68Boman, Jan, L13Borel's theorem, 22

Bore], F61ix Edouard Justin Smile,72.84

Browder, Felix Earl, 1Q5bundle, 56bundle projection, 56

Cauchy data, 42 63Cauchy problem, 42Cauchy-Kowalewsky theorem, 42Cauchy-Kowalewsky Theorem, 64complex analytic function on a man-

ifold, 122complex manifold, 171complexification, 174consistent equation, 64convolution, L45

de la Vall6e Poussin, Charles-Jean,18

Denjoy-Carleman theorem, 88derived series, 8Dini, Ulisse, 38direct limit, 51)direct limit topology, 51, 53

204 Index

direct system, 50distance function, 184distinguished polynomial, 167domain of convergence, 2$

Ehrenpreis, Leon, 112embedding, L72entire function, 142Euler, Leonhard, 67exceptional divisor, L60exponential type, 149

Fai di Bruno, Francesco, 16formula, 11

fibre, 56Fourier transform, 133

GabriBlov, Andre M.,181Gevrey classes, 94Grauert, Hans, 174Greene, Robert E., 176

Hadamard formula, 6Hartogs's theorem, 104Hensel's lemma, 29Hironaka, Heisuke, 156, 16 179holomorphic function, L49Hormander, Lars, 95 116, 126

implicit function theorem, 40 49inductive limit, 50infinite order vanishing, 82infinitely differentiable, 11initial condition, 42initial data, 42initial object, 52interval of convergence, 2inverse function theorem, 22 42inverse limit, 52inverse limit topology, 52inverse system, 51

Jacobowitz, Howard, 176joint analyticity, 105

k times continuously differentiable,Ll

Laplace-Beltrami operator, 173Lelong, Pierre, 195little "o", L48local triviality, 56logarithmic convexity, 28 85Lojasiewicz's division theorem, 171Lojasiewicz's structure theorem, 1. i$Lojasiewicz's vanishing theorem, L69Lojasiewicz, Stanislaw, 180

m-linear function, 61majorant, 32majorization, 42manifold, 171Milman, Pierre D., 113, 157, 172Morrey, Charles Bradfield, Jr., 173multiindex, 25multilinear function, 61multinomial theorem, 1&multiplication of distributions, 129

Nash embedding theorem, 176Nash, John F., 176Newton's binomial formula, 20non-characteristic equation, 64normal bundle, 58normal crossings, 185normal space, 56normal symmetric algebra bundle, 61normal symmetric form bundle, 61

ordinary singularities, L62Osgood, William Fogg, 180

Paley-Wiener theorem, 145. 1, 14$partition of unity, 76 L72plurisubharmonic exhaustion function,

115plurisubharmonic function, L75polar set, 142power series, 1power series in m variables, 27

Index 205

Pringsheim, Alfred, 68Pringsheim-Boas Theorem, 62projective limit, 52projective space, 152proper embedding, L22proper transform, 161Puiseux series, 95Puiseux's theorem, 103Puiseux's theorem, abstract, 95P61ya-Raby theorem, 184

quasi-analytic class, 82

radius of convergence, 2rapidly decreasing functions, l113.6real analytic at a point, 3real analytic function, 29 55real analytic function on a manifold,

172real analytic manifold, 171real analytic on a set, 3real analytic section of the normal

bundle, 60real analytic section of the normal

symmetric form bundle, 61real analytic section of the tangent

bundle, 60real analytic submanifold, 54real vector bundle, 56rectilinearization theorem, 1. 185relatively compact, 178Remmert, Reinhold, 176Rudin, Walter, 95

Schwartz distribution, 128 136Schwartz functions, 128. 136Schwartz space, 136section of the tangent bundle, 60Seidenberg, Abraham, 117

semialgebraic function, 172semialgebraic set, 172semianalytic, 178semianalytic function, 172semianalytic set, 178separate analyticity, L05Siciak, J6zef, 105silhouette, 27Singer, Isadore M., 173smooth function on a manifold, L72smooth mapping of a manifold, 172Stein manifold, L75subanalytic, 178subanalytic function, 172subanalytic set, 178subharmonic function, L75symmetric body, 142symmetric function, 61

tangent bundle, 57tangent cone, L62tangent space, 56Tarski, Alfred, L17Tarski-Seidenberg theorem, 122tempered distribution, 128, 136total space, 56transition functions, 171

uniformization theorem, 172.

Weierstrass polynomial, 152, 167Weierstrass preparation theorem, 152

167Whitney approximation theorem, 78Whitney decomposition, 76Whitney extension theorem, 78 84Whitney, Hassler, 122

Zariski, Oscar, 156

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S.G. Krantz, H.R. Parks, A Primer of Real Analytic Functions2002. Hardcover. Second EditionISBN 0-8176-4264-1

The subject of real analytic functions is one of the oldest in modern mathematics andis the wellspring of the theory of analysis, both real and complex. To date. there is nocomprehensive book on the subject, yet the tools of the theory are widely used bymathematicians today.

Key topics in the theory of real analytic functions that are covered in this text andare rather difficult to pry out of the literature include: the real analytic implicitfunction theorem, resolution of singularities, the FBI transform, semi-analytic sets,Faa di Bruno's formula and its applications, zero sets of real analytic functions.Lojaciewicz's theorem, Puiseaux's theorem.

New to this second edition are such topics as:

A more revised and comprehensive treatment of the Faa di Bruno formula

An alternative treatment of the implicit function theorem

Topologies on the space of real analytic functions

The Weierstrass Preparation Theorem

This well organized and clearly written advanced textbook introduces students to realanalytic functions of one or more real variables in a systematic fashion. The first partfocuses on elementary properties and classical topics and the second part is devotedto more difficult topics. Many historical remarks, examples, references and anexcellent index should encourage student and researcher alike to further study thisvaluable and exciting theory.

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