PRINCETON MATHEB6AmCA.L SERIES Edba: -TON MORSE and A. W. T ~ o ~ R 1. The Classical Omups, Their In-ts and Repre~8nfations. By ~RMANN WEYL. 2. Topological Groups. By L. PO~AQIN. Translated by Ennu LEHMER. 3. bn Introduction to Differential Geometry with Uee of the Tensor Celaulus. By Lmmm Prm EISENHART. 4. Dimension Theoq. By Wmom H ~ a z and HENBY W~ZMAN. 6. The mytioal Foundations of Oelmtid Mechanio~. By A ~ L W I ~ R . 6. The Laplaoe Tramform. 13y DAVID V ~ O N WIDDBIE. 7. Integration. By EDWARD J-s Mas-. 8. Theory of Lie Groups: I. By Cumn ~ W Y . 9. Matharmtical Methods of Statistics. By FTla*r.TI WR. 10. Several Complex Variables. By S. BOOHNER and W. T. M~RTIN. 11. Introduction to Topology. By SOLOMON ~~SOHBITZ;. 12. The Topology of Sdwes and their ~omtions. By JHOB N~LSEN aind WE- ~~. 13. Algebraio Curvea. By ROB- J. Wm. 14. The Topology of Ebre Bundl.8~1. By NORMAX STEENROD. 15. Foundations of Algebraio Topology. By S A ~ L EIL~ENBERG and No- STEEXROD. 16. hotionsls of Finite Riemwm Surfwee. By ~ ~ N A S ~ M 80- and DONU C. SPEN~R. 17. htmduction to Matheansteal Logic, Vol. I. By ALONZO C ~ O B . 18. &@bX%i0 &90m9tqm By 8. ~E'sUHETZ. 19. Homologid Algebra. By Hwrar CARTAX end Saaarwr, Em=-. 20. The Convolution Transform. By I. I. Hrsrta- and D. V. WIDDBR
Transcript
PRINCETON MATHEB6AmCA.L SERIES
E d b a : -TON MORSE and A. W. T ~ o ~ R
1. The Classical Omups, Their In-ts and Repre~8nfations. By ~ R M A N N WEYL.
2. Topological Groups. By L. P O ~ A Q I N . Translated by Ennu LEHMER. 3. bn Introduction to Differential Geometry with Uee of the Tensor Celaulus.
By Lmmm P r m EISENHART. 4. Dimension Theoq. By Wmom H ~ a z and HENBY W ~ Z M A N . 6. The mytioal Foundations of Oelmtid Mechanio~. By A ~ L W I ~ R . 6. The Laplaoe Tramform. 13y DAVID V ~ O N WIDDBIE. 7. Integration. By EDWARD J-s Mas-. 8. Theory of Lie Groups: I. By C u m n ~ W Y .
9. Matharmtical Methods of Statistics. By FTla*r.TI WR. 10. Several Complex Variables. By S. BOOHNER and W. T. M~RTIN. 11. Introduction to Topology. By SOLOMON ~ ~ S O H B I T Z ; .
12. The Topology of S d w e s and their ~ o m t i o n s . By JHOB N ~ L S E N aind WE- ~~.
13. Algebraio Curvea. By ROB- J. W m . 14. The Topology of Ebre Bundl.8~1. By NORMAX STEENROD. 15. Foundations of Algebraio Topology. By S A ~ L EIL~ENBERG and
No- STEEXROD. 16. hotionsls of Finite Riemwm Surfwee. By ~ ~ N A S ~ M 80- and
DONU C. S P E N ~ R . 17. htmduction to Matheansteal Logic, Vol. I. By ALONZO C ~ O B . 18. &@bX%i0 &90m9tqm By 8. ~ E ' s U H E T Z .
19. Homologid Algebra. By Hwrar CARTAX end Saaarwr, Em=-. 20. The Convolution Transform. By I. I. Hrsrta- and D. V. WIDDBR
THE CONVOLUTION
TRANSFORM
BY
I. I. HIRSCHMBN
AND
D. V. WIDDER
1966
PRINCETON UNIVERSITY PRESS
PRINCETON, NEW JERSEY
Pubt&lrad, 1966, by ~~ Udww Prsse Lon&n: Qmflhy Cudwbge , Oqford Uniwgidg P m
L.C. C a a ~ 54-4080
Preface
THE operation of convolution applied to aequencea or functiom is basic in analysis. It aria08 when two power series or two Laplace (or Fourier) integrals ere multiplied together. Also most of the classical integral transforms involve integrals which define convolutions. For the present authors the oonvolution transform oame 88 a natural generalization of the Laplaae transform. It wa0 early recognized that the now familiar real inversion of the latter is essentially scoomplished by a partiozJcr linear differential operadar of infinite order (in which translations are allowed). When one studies genera2 operators of the same nature one encounters immediately general convolution transforms as the objects which they invert. This relation between differential operators end integral trans- forms is the baeic theme of the present study.
The book may be read easily by anyone who ha^ s working knowledge of real and complex variable theory. For such a reader it should be oomplete in itself, except that certain fundamentals from The L o v p b T r a m f m (number 6 in this series) are assumed. However, it is by no means necessary to have read that treatise completely in order to under- atand th is one. Indeed some of those earlier resultrs can now be better understood as special cases of the newer developments.
In oonclusion we vish to thank the editors of the Prinoeton Bhthe- matical Series for including tbis book in the series.
. . . . . . . . . . 10 . Determining function in L* 238
MISCELLANEOUS TOPICS SEOmON
1. Introduction . a a a a
2. Bemstein p0lynom.ialaD . • a . a
3. Behaviour at S ~ i t y a . I a
4. The analytic character of kernels of clasae~ I and I1 , 5. Quasi-analyticity a a
BIBLIOGRAPHY . • I rn
SYMBOLS AND NOTArnONS
THE CONVOLUTION TRANSFORM
CHAPTER I
Introduction
1. INTRODUCTION
1 .I. In this preliminary chapter is. presented a heuristic introduction to the material whioh is to be given detailed treatment in later chapters. The method here is to illustrate rather than to prove. As illustrationa we use four special examples of convolution transforms which taken together will ahow clearly the variety of properties which such tramforms may have. The first of these examples involvee an exponential kernel and is, in a sense, trivial. However, from another point of view, this kernel is the atomic materid from which all others are constructed, so that its use for introductory purposes is perhapa mandatory. The last two examples reduce to the Laplace and Stieltjes transforma. Since we regard the fundamental properties of these as known, any new reaulte about the convolution transform can be checked against the corresponding known ones for theae two claarsic transforms.
2. CONVOLUTIONS
2.1. When two Lament series
are multiplied together formally a new series of the same type results 00
where the new ooefficients c, are related to the old ones as follows: a
The aequenoe {c*}" is called the cavolction of the sequences {a,}:, and -00
{b,} - 0 0 " . We arrive at the oontinuous analogue of this operation when we multiply together two bilateral Laplace integrals,
a a ~ ( 8 ) = j"'4ta(t) dt.
3
4 INTRODUCTION
The result is an integral of the same form,
where
J - w
This oombination of functions occurs so frequently that it may be regarded aa one of the fundmental operations of analysis. The function c(x) is called the convoZutkm of a(%) and b(x), and the integral (1) is commonly abbreviated aa a(x) * b(x) or M a * b.
If we take one of the functions, say a(%), as b e d equation (1) may be considered as an integral equation with c(x) the given function and b(x) the unknown. Alternatively, the equation may be thought of as, an integral transform. Our mud notation will be
and this wi l l be called the con,wZution trawf' with k e d Q(x) of the jumtion ~ ( x ) into f(x). Let u8 list the following four examples which will serve as the illwtratione mentioned in tj 1.
A. Choose the kernel as
= o O ~ Z < ~ . Then
0 g * tp = j':fP(z - 1) dt = e
B. If #(x) = (1 /2 ) r 1 *I , then
c. If a(%) = (%)-I sech (t/2), then (2) becomes
l a x - t f(x) = g j" ?h - q(t) dt. 2
However, if we make an exponential change of variable, replacing ex and et by x and t, respectively, this becomas
where f (x) = emia~(e@), ~ ( x ) = ?redg@(ea).
This ia the familiar Stieltjes transform.
I). If -P Ir = e f(5) = F(h)8, p(z) = O(ea),
then (2) becomes the Laplace transform
Thus we see that both the Stieltjea tramform and the un&teraZ Laplace transform are special cases of the convolution transform, as predicted in 3 1.
3. OPEUTIONAL CALCULUS
3.1. Very useful as a guide to the following theory is a rudimentary knowledge of operational calculus. This is a technique which treats an ~perat iond symbol suoh as "D," for differentiation, as if it were a number throughout a calculation and at the lmt step gives the symbol its original meaning. We shall not be concerned with the justifimtion of this pro- cess, whioh of ooume . lies in the fact that there is a one-to-one correspondence between the laws of combination of the operation on the one hand and the fundamental operations of arithmetic ibn the other. We take rather the point of view that a suitable meaning for a new opera- t ion may be deduoed by algebraic procedures on old ones and then adopted aa a defingtion; that new results may be oonjectured by operational calculus and then proved.
As a h t illustration let us deduce s, meaning for the operation eaD. W e expand this exponential in power wries &a if 1) were a number and then interpret LYI as a derivative of order k, giving the Meolanrin series:
eUDf(x) = = " f ( z ) = f ( ~ + a). &-0 k!
Finally, we depne eaDf (2) as f (x + a) end observe that in spite of the background of the definition saDf(x) is well defined even if f(x) is not differentiable.
As a second application of the operational caloulus let us solve the diffarential system
6 INTRODUCTION [OH. I
where q ( x ) € 0 B (is oontinuoua and bounded) for all x. We observe first that if a solution exists, it is unique. For, the corresponding homogeneous equation
has the general solution Ass, and the latter solution can satisfy (3) only if the constant A is zero. That is, the system (4) (3) has only the identically v d h i n g solution.
The symbolic solution of (2) is
so that an interpretation of the operator (1 - D)-1 is needed. This may be supplied by equation (1) and the familiar bplace integral
1 00 0
(5 ) -- - J' ;4tg(t) dt = 1' Ftet at -0o < s -=. 1, 1 - 8
where g(t) is the kernel of Example A. Replacing s by D in (5) and using (1) for the interpretation of efD, we obtain
That is, the operational solution of the system (2) (3) is
(6) = g * 92 = ex J''ae-~y(t) at. We can now 8how directly that thh is the actual solution. Since q(x) € I3 the integral (6) aonverges, so that the boundary condition (3) is clearly satisfied. Since q ( x ) E C, formal differentiation of equation (6) shows that f ( x ) satisfies (2).
In spite of the trivial nature of this result, let us record it as a theorem, for we shall need to refer to it in an inductive proof of Chapter 11.
THEOREM 3.1. Ifpl(x) EC Bin (-a, a), a d i f
f(4 = g * q = em lDe-tv(t) at,
t h n f ( x ) is the unique aoloction of the 8y8km (2) (3). In a similar way we could show that the unique solution of the system
is f (x ) = (p * a, where Q is the kernel of Example B. The role of the hplace transform in these examples should be noticed.
If r function F(a) can be expremed as a Laplace integral, then an inter- pretation of the operator F(D) is immediately available by use of (1).
4. GREEN'S FUNCTIONS
4.1. It is a, familiar fact that the Green's function of a, non- homogeneous differential system enables one to solve the system explicitly. Aa a simple illustration let us consider the dserential system, (2) (3) of 5 3. Formally, the Green's fixnetion U ( z , t ) of the system is a function of x and of a parameter t such that
Here, as previously, D stande for differentiation with respect t o a, the parameter t being held fixed. The function 4s) is Dirac's ~ymbolic function with the following propertiee:
J-n I n terms of G(x, t) the solution of the given system (2) (3) of 8 3 is
cO
(3) f (4 = 1 =a&, t)c(t) a? at least fom11y. For, differentiation under the integral sign gives
Also f ( x ) satisfies the desired boundary condition by (2). The above heuristic procedure may serve aa a guide. We take as our
mtual definition of G(x, t) a function of x whioh satisfies the equation (1 - D)Q(x, d ) = 0
in eaoh of the intervals (-00, 1) (t , co), has a unit jump at x = t ,
and satisfies (2). Let us computa U(x , t). It must evidently have the form
a(%, t) = A(t)em --a < x I t,
By (2), B(t) = 0. By (4), A(t) = 1. That is, Q(x, t ) = g(x - t ) , where g(t) is the kernel of Example A of $2 . That the function (3), deduced formally ss a solution of the given system, is indeed the solution now follows by Theorem 3.1 provided that y(x) E C B.
In a similar way we could show that the Green's function of the system (7) (8) (9) of 5 3 is the kernel of Exaaiple B. In a later chapter we shall see t h t the kernels of Examples C and D may also be regarded as Green's funotions of differential systems of inh i te order.
5. OPERATIONAL CALCULUS CONTINUED
5.1. We may apply the operational procedure of 8 3 to obtain an inversion formula, for the convolution traauform
r03 (1)
Let
be the bilateral h p h o e tr~mform of a(t). we have
= f(x) Multiplying by E(D) we obtain
our desired inversion formula. The funation i ( 8 ) defmed by (2) will be called the inveri&n function corresponding to the kernel G(t).
Let ua compute the bilateral Laplace transforms of the kernel8 considered in 8 2. In these formulea 8 = o. + k is a complex variable.
1 A. J-'p(t)e4t dt = - -a < u < 1,
1-8
Here we have used severel familiar formulas from the theory of the Gamma function. Compare, for example, E. C. Titchmmh [1939; 1061.
84 OPERATIONAL UA LO US US CONTINUED
A. If cD
f(4 = 1 /(x - t)fp(t) &.
then formula (4) implies that
(1 - Dlf (x ) = ~ ( 4 We have already verified this.
B. If
then formala (4) implies tha t
(1 - (1 + D ) f ( x ) = v ( x ) * We have
f(z) = ,- I e4'F'(t) dt + f [ PeB-tp(t) dt,
fkom whioh we obtain
Referring to Example A we see that
then the formub (4) implies that coa .rrD f(x) = q ( x ) .
This formula is inoomplete however sinoe no definite meaning attaahea to cos VD. We have, see E. C. Titohmarsh [1939; 1141,
cos T X = lim ebnx 2x ) eb~ ( 2 ~ 1 1 ir ('-2E+ 1 woo k- -n
where the {b,,} are real and lim b, = 0, and this suggeste t l r+QS
as a posaible inversion formula for (5 ) .
then the f o d a (4) implies thst
and since, see Titchrah [1939; 2571,
we may conjecture that
Other expamion8 for COB m end l/I'(l- x ) would lead to different "definitions" of COB n~ a;nd l/r(l - D). The product definitions given here are charaoterhtio of OUT theory.
If mitable choices are made for b, then the formulas (8) and (8) become, after a change of variebles, well known operational inve~ian formulaa for the Stieltjes and h p b o e transforms. Let us verify this in d e w for the Laplace tramform. We have shown in 5 2 that if in
J- co we put
(9) f(x) = @F(@), ~ ( t ) = 0 (e-$1, = log y, t = --log u,
then we obtain
F(v) = La e-uu@(u) du.
If we define
b,, = log n - 1 Z ;+Y n = ~ , 2 , = . . k-1
?
then lim b, = 0 ~ n d
(10)
where
8 61 THE QBNERATION 02 KERNELS
We have ,
Making uae of (9) we see that (10) is equivalent to the familiar inversion formula,
see D. V. Widder [1946; 2881. For a similar diecus&ion of the Stieltjes transform see 8 9 of Chapter 111.
6. THE GENERATION OF KERNELS
6.1. Let b, {ak}* be real numbers such that Em1
The condition (1) insures that the idinil;le product (2) is convergent. See E. C. Titohmamh 11939; 2501. If there exists a function U(t) such thalt
(3) 1
e41U(t) olt = - B(8)
then the considerations of the preoeding section suggeat strongly that the oonvolution transform
ie inverted by the formula
(5)
where
Hem the b, ase real numbers such that lim b, = 0. -00
The complex inversion fornula for the bilateral Laplace transform asserts that if the tramform
converges absolutely in ths strip cr, < e < 0, then (under certain restrictions)
a(t) = - e8?d (a) ds a l < d < a , , - c ~ < t < ~ . 2& a-ia,
We therefore r~et
We shall ultimately prove that U(t) defined by (6) satisfies (3) and that the convolution transform (4) is indeed inverted by the operational inversion formula (6).
In point of fact we shall treat a slightly more generd class of kernels. Let a,, k = 1,2, , b, c be real and such that
and let
It is to the lstudy of the kernels (8) and their essociated convolution tromforms that the present book is devoted.
7. VARIATION DIMINISHING CONVOLUTIONS
7.1. It is n a t d to ask why when our operational procedures apply, at least formally, to every convolution transform we have limited om- selves to the kernels 6.1(8). The reasons for this lie somewhat deeper then the opemtioml oalculus, and depend upon the following result.
o 71 VARIATION DIMINISHING CON VOLUTIOT
Lr)t a(t1E L(--00, a),
J-: Q(t) dt = 1,
and let
f (4 = J-'a(x - t ) v ( t ) at,
where g(t) is bounded and continuous. The kernel G(t) will be said to be variation diminishing if the number of changes of sign of f (x) for -GO < x < co never exceeds the number of changes of aign of ~ ( t ) for --co < t < a. It haa been shown by Schoenbarg [1Q47, 1948bl that G(t) is variation d i d h i n g if and only if it is of the form 6.1(8). See Chapter IV for a proof in a form applicable to our theory.
As an example we may veaify that
is variation diminishing. We have from 5 3 that
Iff ( x ) haa n changes of sign, then there exist points
- - ~ < X , < X ~ < ' ' ' < x , < o o suoh that
f(*J = 0 i = ~ , ~ = = ~ n ,
and such that f(x) is not identically zero in any of the intervals
Applying Rolle's theorem we see that p(x) has a t least one change of sign in -oh of the intervals
Thus ~ ( x ) hae et least n changes of aign. Note that in the above azgument use was made of the faot that l irn,f (~)e-~ = 0.
-3.00
Thus the kernels 6.1(8) are characterized by an important intrinsio property. Since this property of being variation danumdu . . .
' 43 plays essential role in our theory our ohoice of kernels is seen to be diatated not by convenience but by intrinsio ma;thema;ticd structure.
8. OUTLINE OF PROGRAM
8 In this seotion we e h d very briefly describe and illustrate the program of the following ohepters. Chapters I1 through V are devoted to the study of the kernels G(t) defined by 6.1(8). We list a few typical properties. It is shown thaf G(t)'is a frequency function with mean b and v d n c e 2a + 2az2; that is
k
@(t) 2 4
I-: Q(t) cEt = 1,
(1) U(t)(t - b) dt = 0,
We also prove, as we indicated before, that U(t) ia variation diminbhing, a fact whbh has many interesting consequenoes, among them that Gcn)(t) has, for each value of e for which it ia defined, exactly n changes of sign, and tbt -log G(t) is convex.
In chapters M and V I I we study the convolution transform with kernel G(t) d&ed by the formula 6.1(8) with o = 0. These kerneh mwt be subdivided into h e classes:
#(t) E Jass I if there are positive and negative root8 ak; - 00' U(t) E c l a a I1 if thre are only positi'ue roo& a, and if zag1 -
E G(t) € clQea III there we only positive roo& a, and if za;a,l< 03.
k;
If G(t) s&tisfies the restriction that c = 0 then Q(t) or G(-t) belongs to . .
one of theae classes. The following results may serve as examples. The kernel of Example C belongs to class I. It is well known that if
the Stieltjes transform
O(U) au
converge8 (conditionally) for any value of y > 0 then it converges for all y > 0, see D. V. Widder [1946; 3261. After an exponential change variable we fiad tht if
seoh [f (x - t ) lv( t ) dt
converges (aonditiondy) for m y value of x, --a < x < ao, then it converges for all z. This convergence behaviour is chara,cterisfio of class I kernels.
8 81 OUSPLINB 03 PROGRAM 15
The kernel of Example D belongs t o class 11. If the Laplace transform
P(g) = l y e - v ~ (u) du
aonvergea (conditionally) for any value of y > 0 then it converges for all larger values of y, fiom which it follows that if
converges for any value of x, -co < x < co, then it converges for all larger values of 1;. See D. V. Widder [1946; 371. This behaviour is typical of class I1 kernels. The kernels of Examples A and B belong to olagses 111 and I, respectively.
In Chapter VIII a study is made of the transforms with kernels of the form
og
(3) 1- :+-.(#(t) dl = l/e".
In this case G(t) may be aomputed explicitly,
The theory of the convolution trsmfoms with these kernels ie closely bound up with the heat equation
au, aa, -=- ah a 9
For instrtnoe it will be shown that necessaxy and sufficient aonditions for u(z, 8) to be of the form
U(Z, h) = (4nh)-* ~ - ~ r ( * ~ ~ 4 ~ p ( t ) dt - - O O < X < O O ; h > O
where 1 p(t) 15 M are that:
Chapter IX i s devoted to complex inversion formulas. It has long been known that if
roo
then 4(u) = lim (M)-l [B(-u - ie ) - F(-u + ie)],
-+ or, what is substantially the same,
I d INTRODUCTION [CH. I
See D. V. Widder [1946; 3403. If in this formula we make the change of variable described in Example 0 of 2 we find that if
a
f(z) = (W1 )-1S --a eech t ( z - t)v(t) dr,
then
~ ( t ) = lim 4 [ f ( t + iP.4 + f (t -- ip)l. ?-I -
We shell see how to associate a similar formula with each kernel G(t) of the form
where
Chapter X contains a number of shorter topics.
9. SUMMARY
9.1. In this chapter we have attempted to show a few of the principal ideas of our general theory by illustrating them with the simplest possible examples. We have chosen only a few of the many epplicationa with the hope that the reader may obtain the essential flavor of the rwults before ente* into the detailed study necessazy for a deeper undmtanding.
CHAPTER 11
The Finite Kernels
I. INTRODUCTION
1 .l. In tbhis chapter we shall be concerned with the problem of inverting the coi~volution trailsform
(1) f(x) = j';'QB(x - t A ( t ) dt,
where Q(x) belongs to a class of plcite kernel&. These are the functions G(x) which are such that
where i ( a ) is a polynomial of degree% with real zeros. The term "finite" is intended to correspond to the finite degree of the polpomia,ll(e). In a later chapter we consider kernels B(t) satisfying (2) with B(s) an entire function.
If
we shall show that (1) is inverted by the linear differential operator B(D), where D stands for differentiation with respect to x. That is,
B(Dlf (4 = q(x ) * Since the kernels (2) turn out to be frequency functions in the sense
of the theory of statistics, we begin our study with a brief discussion of , the pertinent portion of that theory.
2. DISTRIBUTION FUNCTIONS
2.1. Before introducing the frequency functions mentioned in the previous section let us define the related distribution functions.
D E ~ I T I O N 2.1. A function a(t) defined in - co < t < oo, i s a distribution function if and only if *
1. a(t) E f
3. a ( + a ) = 1. * The logical symbols usod, such an E f , are familiar. They are explained,
for examplo, in Adwnncd Calczllw, by D. V. Widder [1047a; 51. Dates in bmkete refer to tho bibliography, and the nurnbor following the semicolon is ct page number.
18 THE PINITB KERNELS
Several examples follow :
A distribution function is n d i 2 if
All three examples above are normalized. If a(t) were defined differently at the origin it would not be normalized.
In the theory of statistics it ia found uaeful at times to think of &(t) as defhing a distribution of unit masa on the i a t e traxia in ~uch a why that there are a(b) - &(a) units of mam in the interval (a, b). Hence the term.
Two distribution funotiona are e p i v a l m t if and only if they are equal at all points of continuity. From (1) i t is cl&r that equivrilent normalized distribution functions are equal at all points.
2.2. We now d e h e a statistical term, the mesa of a distribution funotim.
DEITINITIOX 2.1. The mean of a dbtribution funotion a(t), denoted by ma, is the integral
when that integral converges. For the examples of 8 2.1, m, = 0, ma = - 1, and mi, does not exist
aince the integral
diverges. If ~ ( t ) is thought of as a distribution of mass, aa explained in the
previous seotion, then ma locates the center of gravity of the distribution.
§ a FREQ UENUY P UNCTIONS 19
Bor example, if a(t) is a step-function with jumps m, at points t,, k = 1, 2 , * * * , n, fhen (1) becomes the familiar formula
1. . 2.3. Another useful concept from statistical$heory ia vam'alzce. D ~ w v r r r o ~ 2.3. The variance of a diathbution function oc(t),
denoted by V,, is the integral
when nc, is defined and the integral converges. For the examples of 5 2.1, Va = 0,
(t $ 1)8et dt = 1,
and V, is not defined. i;, ,., l
Using the foregoing physioal interpretation of a(t), we. a see ,I1 , that I I V , is the moment of inertia of the mass distribution about the center of gravity. It can consequently be considered as a measure of h& f& the maas is from its mean. Physically it is elear that Va can vanisd bn& 9 if . ,
all the mass is concentrated at a, point, as in the first example hbove: The fact is easily proved analytically.
3. FREQUENCY FUNCTIONS
3.1. A special case of a distribution function is one whiah is absolutely continuous. 1t;s derivative is then called a frequency function.
DBPIN~TION 3.1. A function* p(t) defined in -a < t < co is a frequency function if and only if the function
is a distribution function. Thus any non-negative function #(u) for which
* It will be undersfaod that dl functions considered are integrable in every eub-interval of the interval of definition.
2 19 TEE FINITE KERNELS ICE. n
is a frequency function. Using the examples of 5 2.1, we see that ~ ( t ) = b'(t), t # 0, ~(0) = 1, is e frequency function. The same is true for y'(t) = +(l + t2)-l. But there is no distribution function corre- sponding to a(t).
The mean and variance of frequency functions are defined in the obvious way:
V , = j (t - rn&# at. - O D
a 9 Thus, if q(t) = av-lI8e4 ' , a > 0, then m, = 0 and
In particular, if a = 1/G, this function p(t) ie called the n d frequency function.
4. CHAIZACTERISTIC FUNCTIONS
4.1. Distribution and frequency functions are often studied most easily by use of their Fourier or bik&eral Laplace transforms. We shall -
refer to the transformed functions ee ch~acte&tio functions in either case. Dammnor 4.1. The characteristic function, xa(s), of a distribution
function a(t) is
The characteristic function, &(8), of a frequency function p(t) is
Here, as elsewhere, a is a complex variable, s = CY + k. Notice that xa(-ir) and xP(-ir) are Fourier transforms, and it is these that are usually called characteristic functions in statistical works. No confusion will &rise by giving the functions (1) and (2) this name also.
Using the examplea of tj 2.1 we have
Q 41 CHABACTERI~STIU FUNCTXONS 2i
the transforms converging in the intervals indicated. Observe that the integrals (1 ) and (2) must always converge on the imaginary axis cr = 0 and may converge in a larger vertical strip. At the origin a cherecteristio function must equal unity.
Bs a h a 1 example choose p(t) the laat function of 5 3.1. Then
By differentiating this function twice with respect to s and setting 8 = 0 we may clearly obtain anew V,, (1) 8 3.
4.2. Operations with characteristic functions in place of distribution functions can be useful only if it is possible to pass conveniently from the former to the latter. This is indeed the cam, by use of the known inversion formulas for the bilateral hplace transform. For emy reference we quote the pertinent results here; see D. V. Widder [1946; 2411.
THEOREM A. If a ( t ) is a d i z e d function of bounded waridion 6% every Jrnite. intemd, and $? the integral
c m v e q w in, the atrip k: < a < Z , then for a22 t
TEEOREM B. If q ( t ) ia of bounded variation in some neigi6orhoodl of t = to and ij the integral
converges absolutely on the line o = c, then
cCiT lim - f(8)eato d~ - - ~ ( t t + p(to)
2 T30027* c-dT
As an example we use the first illustrative distribution function of 8 2.1. Equation ( 1 ) becomes
lirn -
This is a familiar Fourier transform. These two theorems are applicable only when the characteristic
function is defined in a, strip of positive width. This will be the case in
22 THE PINITE KERNBAS [CH.
the present chapter. However, if the strip reduces to e straight line, as it may for certain distribution functions, some modification of the above theorems is needed. We shall make this modification in 5 6.2 of Chapter III, where it will be needed. It will develop, however, that all of our chachri&ic functions are dehed by bilateral Laplace transforms which converge in strips of positive width, though this feat will not be known
5. CONVOLUTIONS
5.1. In the multiplication of characterhtic functions the corre- sponding distribution functions are combined by mvolu t im, a binary operation which is indeed the subject of the present study and wbioh we now formally define.
D~arrrrrro~ 5.1. The Stieltjee convolution of two funations ~ ( t ) and /?(t), denoted by a(t) # P(t)) is the integral
when that integral exiElh. The (Lebesgue) convolution of two functions q(t) and c y ( f ) , denoted by p(t) * y(t), is the integral
when that integral exists. I n the integral (1) it will be suffcient t o consider Riemann-Stieltjee
integrals. If a($) and @(b) are distribution functions, a # p(t) may fail to exist at a countable set of points, but it is non-decreasing in the remaining set of points and becomes a distriblition function if suitably defined at the exceptional set, D. V. Widder [1946; 2481. The convolution p, * y(t) of frequency functions is itself a frequency function. For, by Fubini's theorem
It is clear that both operations # and * are commutative and msociative. Again using the functions of $2.1 as examples, we have
Here the exceptional set consists in the single point t = 0. Of course we define the convolution (3) so as to make eqhati6ii (3) hold dt t = 0 also.
1 '
Similarly r .
6 51 UONVOLUTIONS
If q( t ) is the normal frequency function of 5 3.1, then
5.2. For our purposes i t is important to know how to compute the mean and variance of y(t) = cr # B(t) when those numbers are known for the separate functions a(t) and ,8(t). We prove that it is only necessary to add in both cases.
THEOREM 5.2. If a( t ) , p(t) ore distribution functions with mean and variance ma, mp and V,, V p , veqectively, then a # @(t) hos mean and variance ma + rnp and V , + V B , re8pectiveZy.
This theorem ia conveniently proved by use of characteristic functions, since the operation # for distribution functions corxeaponda to rnultipli- cation of their characteristic functions. For the reader's convenience we quote this result exactly; see D. V. Widder [1946; 2671.
THJZOREM C . If the i-raE8
emerge absoZutelyfw a common value of 8, the% for that value
00
f (s)B(~) = 1- dy(t) ,
where y ( t ) = a # P(t). From the definitions of characteristic function and mean it is clear
that ~ ~ ( 0 ) = 1, %;(O) = -m, with oorresponding equations for B(t) and for y (t) = a # B(t). But by Theorem C,
Hence our reeult is proved in so far as it concerns the means. Now set
Q)
A(S) = ema8&(a) = 1- ac-s(t-a) dtx(t)
B(s) = emPxB(8)
Then A(0) = 1, A'@) = 0, A"(0) = Va with corresponding equations for B(a). And by Theorem C,
e-6't-a-mpI dy (t ) ,
24 T E E FINITE KERNELS [CH. II
so that it is CD(0) = na, that we wish to compute. But
and our theorem is proved. The replult of course extends to the convolution of any finite number
of distribution functidns. As a special case of Theorem 6.2 we have COROLLARY 5.2. If pl(t), y(t) are frequency functions with mean and
variance m,, 7n, and V,, Vw, respeotively, then p * y(t) has mean and vasiance m, + m, and V , + V,, respe~tidy.
6. THE FINITE KERNELS
6.1. All of the oonvolution tramdomu to be conaidered in this book, except those of Chapter WI, wi l l have kernels which can be synthesized &om the following basia one
The finite kernels G(t) will be made up by a finite number of convolutions of h c t i o n s 1 a, 1 g(ad), k = 1, 2, - - . In the following ohapters non- finite kernels d be synthe~ized fkom the aame funotions, but by use of Mnitely many convolution operations.
As we have seen, g(t) is a frequency .function with mean -1 and vazbnce 1. Its chazrtcteristic function is
Note that I a I g(&) is again a fkequency function if a is any real number . .
not zero. Ita characterktio function is the defining integral
converging for a < a if a > 0 and for a > a if a < 0. Its mean and vmiasce are -1la and 1/a8, respectively. These facts, together with Theorem C of 5 5.2, show that the present deecription of the finite kernels is equivdent to that given in § 1 .l.
6.2. Let a,, a,, , a, be any non-vanishing real constants, some or all of which may be coincident. We introduce the fiwquency functions
8 61 THE PINIITE KERNELS 26
and combine them by convolution to obtain new frequency functions, the finite kernels to be considered. Let us introduce the following definition.
DE~NITION 6.2. The function al = al(al ,. a2, - , a,) is the largest negative a, (or -a, if all a, are positive); the function a* = a2 (a1, a,, , a,) is the smalleat positive a, (or +m if all a, en, negative).
Thus a2 = -a1(-al, -ag, - , -a,).
For example, a,(-3,6, -2,17) = -2
0c2(-3, -1) = $-a. THEOREM 6.2. If
1. akhredand#0
2. gk( t ) 6 defined by (1) 3. U ( t ) = g, * gz * ' ' * g*(t)
5. or,, a, are &Jid in De$vaition 6.2,
then Q(t) k a frequency function and
Thie re~ult is an immediate consequence of Corollary 6.2 and Theorem C. Note that the region of defmition of the charaoteristio function for g,(t) is a,(a,) < cr < al(ak), namely e half-plane which includes the imaginary axis but not the point a,. The intersection of all t h e hZf- planes, k = 1,2, - , n defines the region of (absolute) convergence of the integral ~ ~ ( 8 ) .
6.3. From the explicit formula expressing U(t) in terms of E(s) obtained from Theorem B, we can investigate the continuity properties of U(t ) .
THEOREM 6.3. If #(t) is deJSned as in Theorem 6.2, r 2 - 2, t b f i
#(t) € cn--2, - 00 < t < 00.
86 THE FINITE KERNZLS [Cw. 1.
By Theorem B,
It is unnecemaq to use the Cauchy value of this integral, as indicated irl
Theorem B, for l/E(iy) = O ( r a ) aa 1 y 1 + co when n .c. 3 2 a11tl ( 1) converges ebsolutely . Differentiating, we obtain
This operation is valid aince the integral (2) converges uniformly on any compect set of the t - d . This follows eince y'L-a/E(iy) = O(yHa) a8 1 y 1 + a. Shce the integral (2) converges uniformly, GcnM2))(t) h continuous, and the theorem is proved. To iuwtrate consider Extlmpb B, §2ofChapterI. Theren=2andG(t)=#e-ltlEO, - c o < t < c ~ .
Note also tht i f n k 3 , -
for m y constant b # 0. If 1, coincides with a root of E(s), a factor
(1 - %) wi l l cancel in the inbgrand. We thus obtain
In faat, if t # 0 we may proceed one atep further,
For, the integral (4) still converges uniformly* (though not aba~l~tely) in any ibite interval not including the origin due to the monotonic oharscter of the function y/(g + f l ) near y = fa. The value of integrd (3) is h(t) by Theorem B.
*See, for example, 8. Boahner [1932; 121
3 61 TRB FINITE KHRNELS 27
Altermtively, we may compute the integral (3) directly by the calaulus of residues or otherwise, obtaining g, * g,(t). Then applying the operator
( I - :) to thia explicitly we obtain g,(t) if t # 0. For example, if -
al = a, = 1, we found in 5 6.1 that
and (1 - D) applied to this function gives g(t) when t # 0. 6.4. Almost without exception in this book the convolution kernels
used will be frequency rather than distribution functions. However, at the present junctum let us make a brief digression to show how frequency functions could wise quite naturally and could in fact be used basically. RecaJl that we have defined our inversion funotions B(s) not vanishing a t the origin. I f we had permitted a zero at s = 0, frequenoy functions would have been introduced, as indicated in the following theorem.
THEOREM 6.4. If #(t), E(s) and as are &$d a8 in Theorem 6.2, then the fuwtitm
t u(-l)(r) = 1- d~
is a distribution function, wh8e bilateral bplace transform b [sE(s)]-l,
1 OD -- O<a < a,.
For, by Fubini's theorem we may interohange the order of integration in
#(a) a24
tQ obtain
provided that
a ( ~ ) clzl et at < a. I-' I" But the inner integral converges for 0 > 0 and the iterated integral is finite for 0 < cr < ocg by Theorem 6.2. Also by that theorem, the right-hand aide of (1) is [aE(s)]-1, so that the desired result is proved.
H 71 IN VERSION 29
integrd s ign (1) ie prevented by the fact that hf(0) does not exist. How- ever, if we write the integral (1) as the sum of two others corresponding to the intervals (--a, x ) and (x, oo) we have
hf ( X - t)y(t) dt + ht(x - t)cp(t) dt.
Since h(x) and q ( x ) are continuous and since
The differentiation under the integral signs of the two integrals above oaxmot be held in question since ht(O+) and ht(O-) both exist and the resulting integraki converge uniformly in any fkite interval of the x-axis.
If desired, the above facts may be checked by use of the explicit formula for h(t). For example, if a, and a2 are both positive, al # a,
If a, = a, > 0, h(t) is obtained from the function (6) 8 6.3 by a change of variable:
Now from equation (2) we may complete the proof at once by appeal to Theorem 3.1 of Chapter I. For,
for all x. COROLLARY 7.1. The same oonclusion holds if q(t) E 0 L. The modifications in proof needed are slight and are omitted. Hypothesis 2 of the theorem i s stronger than needed, as already
indicated. in the corollary. Later we shall give best possible results in
30 THE PINITB KIRNELS [CH. I1
this direation, but for the present the principal features of the theory are put into evidence by uss of simple assumptions on p(x). It should be observed, however, that the "local" conditions on p(Z) are already weaker than in Theorem B, 5 4.2, for example. This is brought about by the faot that the present kernel is positive in contrast with the Dirichlet kernel.
7.2. We prove next an analogous inversion theorem in which the fkequency function G(t) is* replaced by the corresponding distribution h o t i o n a(-l)(t).
T H H I O ~ M 7.2. If
1. G-l(t) a d E(s) are &fined as in Tborem 6.4
Since 0 i; G(-=)(t) < 1, the integral defining f(z) convergea absolutely for aU s. Moreover, for r > 1
We have now only to apply Corollary 7.1 to obtain the desired result. If n = 1, the derivative of (;l(-l)(t) does not exist at d = 0, but equation (1) is still true trivially. Hence the theorem is established in aJ1 cases.
8. EXPONENTIAL POLYNOMIALS
8.1. We shall frequently need to conaider exponential polynomiab. A familiax fact about the number of their zeros will be recorded first.
D E ~ N ~ O N 8.1. An exponentid polynomial of degree d is a sum of the form
where P,(t) is a polpomid of degree m, - 1, the A, are distinot real numbers and
3 81 EXPONENTIAL POLYNOMIALB
For example Q8(t) = egt f t3et - t2 + Ctt.
Note that d is a unit less than the to td number of terms if no coefficient is zero when each polynomial P,(t) is expanded in powera of t. If Q,(t) is identically zero we define its degree as -1.
TF~EORH~M 8.1. An ez(pone&iaL polyrmmicd of nm-negative degree d can haere at met d zeros.
It is to be understood rn usual that the total number of zeros is the sum of the multiplicities of the distinct zeros. We prove the result by induction. If d = 0, Qo(t) consists of a single non-vanishing term of the form BeAt. N o w assume the result true for exponential polynomia.1~ of degree d - 1, and suppose that Qd(t), defined by (I), had d + 1 zeros. The aeme would be true of e-*ltQd(t). By Rolle'a theorem, the derivative of this product would have a t 1-t d zeros. But this derivative is an exponential polynomial of degree at most d - 1, since P;(t) is of degree less than m, - 1, and the induction assumption is contradioted. T b t d zeros are possible is evident from the example (et - l)d.
8.2. We s h d show next that U(b) is a combinafion of two exponential polynomials joined together at the origin.
THEOREM 8.2. If U(t) is the kernel of Theorem 6.2, where P of t b roots al, a,, , a, are positive and N are negative, then G(t) is a% exponential polyncmiaZ of degree P - 1 in (--a, 0) Mid another of degree N - 1 in, (0, 00).
We use the explicit formula
Suppose that E(8), aa defined in Theorem 6.2, haa a, root of order m, a t A,, k; = 1,2, = , p,
Each A, is an a,, but the A, are distinct whereas the a, need not be. TO evaluate (I) let us expand l/E(s) in partial fractions. Corresponding to the root A, there will be m, terms of the form
Henoe to compute (1) we need only evaluate integrals like
38 THE FINITE KERNELS
But by Theorem B this formula inverts the familiar integrals
both of which converge on the imaginazy axis. Consequently (2) ia
when A < 0 and to
when A > 0. Hence in the computation of (1) a positive root A, of multiplicity m, contributes nothing on the positive t-axis and an exponentis1 polynomial of degree m, - 1 on the negative reel axis. When A, < 0 the contribution is nothing on (-m,0) and is en exponential polynomial of degree m, - 1 on (0, cia). Sinoe none of the roots of E(a) are at the origin A, # 0. Adding all contributio~s, k = 1,2, , p, and reoaLZlng that the A, are distinat, we obtain the desired result.
C o a o m y 8.2. If all a, me positive, U ( t ) is identically zero on (0, 00); if aU are negative, B(t) is identically zero on (-00, 0).
These results me the special culses N = 0 and P = 0, respectively, 05 the theorem.
9. GREEN'S FUNCTIONS
9.1. Let us define a Green's function for the linear differential system
(3) f ( x ) = o(eq2) x + - a .
Here B(s) is the polynomial of § 6.2 with roots al, a,, , a, and cl, a, ere the numbers defined there. If a, = +a, for example, (2) is to be understood a9 f(x) = o(eh), x + +a, for every positive E .
o 91 GREEN'S P UNCTIONS 33
DEFINITION 9.1.* The Green's function of the system (1) (2) (3) is a function G(t) with the following properties:
b. U( t ) satisfies ( 2 ) and (3)
d. a(-l)(O+) - U(n-l)(O-) = (-l)nalo,a, - a,,.
For example, if E(e) has only the two roots &1, then a, = -1, a, = 1 and G(t) = e-lti/2, the kernel of Exemple B, 8 2, Chapter I.
Let us h t establish the uniqueness of the Green's funation. THE OR^ 9.1. There is at m t one Geen's furtctim for the sy8terra
(1) (2) (3)- Use the notation of 5 8.2, indicating the diafinct roots of E(s) by
Al, A,, , A, with multiplicities rn,, m,, , m,, respectively. Then the general solution of the homogeneous equation E(D)f(x) = 0 is the function Qd(x) of equation ( 1 ) tj 8.1 with d = n - 1. If such a, function is to satisfy (2) the terms involving positive A, must drop (all such A, "e 2 a2); to satisfs (3) the terms involving negative A, must diaappem. Define
where the first sum is over negative A, only, the second is over positive A, only. Heme U,(t) is a, solution of E(D) f ( t ) = 0 on (0, co) whkh satisfies (2); Q,(t) is a solution on (-m, 0 ) which satisfies (3) . We show that the coefficients of the P,(t) oan be determined in a t most one way so that when U(t) is defined aa Ql(t) in (0, 00) and G,(t) in (-a, 0 ) then conditions o and d of Definition 9.1 will be satisfied. We must b v e
(7 1 aP-l)(o) - Qg-l)(O) = (-l)llalaa a,.
That is, the exponential polynomial G,(t) - a,($) of degree la - 1 must have n - 1 zeros at the origin. If it were possible to determine the coefficients in G,(t) and #,(t) in a second way t o satisfy (6) and (7), we would have a second exponential polynomial of degree n - 1 with n - 1 zeros at the origin. Since by (7) both must have the same value for the
* Compare M. Booher [1917; 981.
34 TRE FINITE KERNELS [CH. II
derivative of order n - 1 af the origin, their differenoe would be an exponential polynomial of degree n - 1 at most with n zeros, c o n t m d i c ~ Theorem 8.1.
9.2. The previous theorem does not show the existence of e Clraen's function for the given system. We now show its existence by exhibiting it.
THEOREM 9.2. The kernel Q(t) of Fheorem 6.2 &I th aPeen'8 fancth of the ~y8tena (1) (2) (3), fJ 9.1.
In the proof of Theorem 8.2 we showed that the kernel G(t) of Theorem 6.2 has the forms (4) and (5), 5 9.1, in the intervals (0, a) and (-o0,0), respeotively. Hence property a of Definition 9.1 is satisfied. Proparty b is satisfied trivially since Q(t), being a frequency function, vanishes at f oo. Property c follom from Theorem 6.3 when n 2 2 and is to be ignored when .n = 1. Finally, t o establish d we use equation (4), § 6.3,
Expand the left-hand member and compare the jump of the highest order derivative at the origin (the derivatives of lower order are continuous) with the jump of the right-hand side:
This completes the proof of the theorem. 9.3. In conditions a and d of Definition 9.1 the origin playa a
special role. It could be replaced by an arbitrary point t. The resulting Green's function would be Q(x - t ) . This is a consequence of the fact that the coeffioienta of the differential operator B(D) are conatants. Thue
the indicated differentiation being with respect to x. A chaoteristic property of a Green's function is tbt it enables one
to solve a non-homogeneous system explicitly by an integral. It serves fhis purpose in the present case.
THEOREM 9.3 I fq (x) E B C, -00 < x < co, the% the unique solutim of the system (1) (2) (3) fJ 9.1 is
The solution is unique since the exponential polynomial (1) § 8.1, d = n - 1, the general solution of the corresponding homogeneous equation, can satiefg (2) and (3) of 5 9.1 only if it is identically zero.
8 101 EXAMPLES 35
Thie follows sinoe no positive A, is less than a, and no negative A, is greater than a,.
That the fmtionf(x) defined by (1) satisfies (1) 5 9.1 was proved in Theorem 7.1. The boundary conditions (2) (3) 5 9.1 are satisfied trivially since f (x ) is bounded:
.. .
This completes the proof.
10. EXAMPLES
10.1. Let us give here two examples that will be of special interest to us later.
Example A. Choose a, = k, k = 1,2, - , n. Then
- 1 1 Ef(b) = - - - (n - k)!(k - l)! ?I!
.
Hence
But
so that
Note that when n = 1 this reduces to the unit kernel g(t) of 5 6.1. Example B. Choose a, = E, k = f 1, f 2, , An. Then
36 THE FINITE KBRNJLtS [CH. I1
where is the polynomid E(8) of Example A. If Gb(t) ie the kernel of that example, we must clearly compute
That is, after replacing t - z by a new variable in (1)
eD
= d 1 e-t( 1 - e - t ) + l p t ( l - e+t x > 0,
so that Q(-x) = G(x), as would be expected from the symmetric dis- tribution of the zeros of B(8). If n = 1 , a(t) = e-ltl/2, the illustrative Green's function of 5 9.1, or the kernel of Example B, 2.1 of Chapter I.
These examplee will be of interest to us because of the relation of the polpombls E(8) to the infinite product expamions of 1 /I'(e) and sin wsln8.
11.1. The ohief result of the present chapter i s that if E(8) ia a polynomial with real roots only, E(0) = 1, then its reciprocal is the bilateral Laplace transform of a frequency function U(t ) and that 1 (D){U(x) * ~ ( z ) } = p(x). The kernel G(t) of the convolution treneform waa identified with the Green's function of a oertain differential d yet em and wae uaed to solve the sptem by an explicit convolution.
CHAPTER I11
The Non-Finite Kernels
1. INTRODUCTION
1.1. In Chapter 11 we confined our attention to oonvolution transforms having "finite" kerneb, those whose bilabrd Laphoe tr&mfomu are reciprooals of polynomials with real roots. In the present chapter we enlarge the clasa of kernels graatly, including a clam whom Laplace transforms are reciprooals of entire functiom of genus one and having real roots. We shall show that the inversion theorems of Chapter I1 generake completely here, the Weierstrasa infinite product expansion of the above mentioned entire funotions leading in the expected way t o the inversion opemtors for the more general transforms. Thus a oonvolution transform with one of these new kernels is still inverted by a linear differential operator with constant coefficients. Whereas the order for the finite kernels was equal to the number of roots of the inversion polynomial, the order of the operator is now infinite.
In order to eee clearly how the enlarged class of kernels should be chosen to produce the maximum degree of generality w i t h the frame- work of our methods, we begin with a preliminary study involving limits of sequences of polynomials. In the previous ohapter the fact that the roote of the inversion polynomials had real roots only evidently played a fundan~ental role. Since we will naturally introduce the differentid operators of infinite order aa M t s of other8 of bite order, it becomes imperative to investigate the class of functions, the prospective inversion funotions, which can be the uniform limits of polynomials with real roots only. E. Laperre had shown that all such functions E(8) for which E(0) = 1 can be put in the form
where the constants o, b, a, are real and c 2 0. I n particular, the infinite pro(1uct may have only s finite number o?factore, so that the inversion polynomictls of Chapter I1 are included. We shall show that it is these
38 !?'HZ NON-.FINITE KERNELS [CH.
functions E(8) of Lagueme which we may use as our inversion operators, for each one except eb8 is the reciprocal of the characteristic fumtion of some frequency funotion U(t). The latter we t&e as the kernel of a convolution transform
f = Q * p ,
and show that under suitable restrictions on p,
In the present chapter we restrict attention to the oaae c = O. In the concluding section we specialize the results t o obtain inversion thoorem for the Laplaae, Stieltjes, and Meijer transforms together with certain iterations thereof.
2. LIMITS OF DISTRIBUTION FUNCTIONS
2.1. It will be convenient to have e notation for the claas of normalized distribution functions defined in 5 2.1 of Chapter 11.
DE-ON 2.1a. The function a(t) belongs to the class D, a(t) f D, if and only if it ia a normalized distribution function.
The limit process which we shall 5 d appropriate in the class D ia a pointwise limit at ell points of continllity of the limit function. Rather than introduce a sepamte notation for thia operation, we use inahad a symbol for the set of pointa of continuity of the limit function.
D E E ~ ~ O N 2.lb. A point t is in the set C,, t E C,, if and only if it is a point of continuity for the fknotion a(t).
The desired limit operation can now be written as follows:
The equality is to hold for each point t of 0,. For example, the functions
n = 1,2, a , all belong to D. If a(t) is the normalized function
8 21 LIMITS OP DIBTRIBUTION PUNCTIONS 30
then C, is the set t # 0, and lim a,@) = a(t) t E C,. M a
This example shows that the limit of a sequence of functions of D, though non-deoreesing, need not belong to D because (a) it may not be normalized and (b) the difference a(m) - a(-co) may fail to be 1.
2.2. We investigate the relation of the above limit process for distribution functions to the process for the corresponding characferistic functions. The result is a known theorem of P. LBvy 11926; 1951 often referred to as the "continuity theorem" in statistical studies. We reproduce it here in the form needed. .
T~EOREM 2.2. If
3. lim a,(t) = -03
then
u n i f m l y in -A < y < A for every A > 0. We must showthatGhen n + n the integral
approaches zero uniformly in 1 y 1 < A. Given an arbitrary E > 0, we choose R so that R and -R belongto Ca and so that
This is possible by the defmition of a distribution function. With this R we may now chooee, by hypothesjs 3, an integer nl such that for n > n,
Now write I&) aa the sum of three integrals, ik(y), I ) IE(y), corre- sponding respectively to the intervals of integration (-00, - R), (- R, R), (R, 00). An integration by parts shows that
40 THE NON-FINITE HERNEA8 [CE- III
for 1 y 12 A. By Lebesgue'a limit theorem, we may choose n, 2 n, - such ht
1 G(Y) 1 ==. n > r , , I Y I S A . But olearly for the same vdues of n and y
I G(3) 5 cr,(--R) + u0(-R) < 3.
I C(Y) 1 s ;_Cl - %(R)1 + [1 - a,(R)I < 3 ~ - Hence for n > a,
I I,(Y) I < 76 I s r I I A , and the proof is complete.
2.3. We turn next to the converse of Theorem 2.2. THEOREM 2.3. If hypothes 1 and 2 of T h e m 2.2 lidd, and if
thn lim a&) = a&) t E 0,.
n-beo
Since the functions a,(t) are distribution functions they are uniformly bounded and Eelly'a theorem, Widder [1946; 271, is applicable. Hence there exists a subsequence Vk(t)}: of the set {ct,(t)},w and e fnnation a(t) suoh that
(1) B,(t) = a(t) -00 < t < oo. bet,
I f a(t) is now normalized, equation ( 1 ) stdl holds for t E C,. We show first that a(t) € D.
Form the function
where T is any positive integer. The Fubini theorem is clearly applicable, so that
Ik(y) = j'-:dak(t) j"/-iut 1 - (20s y va d ~ -
Since the inner integral is a familiar Fourier trandorm, we have
8 21 LIMITS OF DISTRIBUTION FUNCTIONS 41
Applying Lebesgue's limit theorem to the integrah (2) end (3) and making use of hypothesis 3 and equation ( I ) , we obtain
Finally, we let r -+ co, again applying Lebesgue's theorem to the integral (4). Shce ~ % ( i y ) is continuous at y = 0 and has the value 1 there, we have
That is, a(t) E D m stated. But by hypothesis 3 and Theorem 2.2, za(iy) = ~ ~ ( i y ) . By the
uniqueness theorem for Fourier-Stidt jes tmmforrns, see Corollary 5.2 below, a(t) = a,,@). Sinoe @,(t)}" in the above argument may be chosen
0
from an arbitraq infinite subsequence of {a,(t)}:, and since the limit a&) is independent of the choice of subsequenoe, it is clear that
and the proof is complete. COBOWY 2.3. If hypothesee 1 and 2 of Theorem 2.3 hold for
n = l ,2 ,3 , * * - , and if
uniformly in -A 5 - y - 5 A for ~ o m e A > 0, then there exists a function a(x) of B such that
(5) lim a,(t) = a(t) n-cm
and f( y) is the characteristic function of a( t ) ,
In fact we d e h e a( t ) by equation (1 ) . Then a - before a(t) € D. Hence by Theorem 2.2 xPl(iy) -c ~ ~ ( i y ) and ~ ~ ( i y ) = f(y) b y hypothesis 3. This proves (6), and (5) now follows by Theorem 2.3 itself.
The theorems of this section show that in dealing with limits of distribution functions, one may equally well aonsider uniform limits of their characteristic functions.
42 THfl NON-FINITE KERNEL8 [CH. III
3.1. The functions which we shall use as our moat general inversion functions belong to a clam originally considered by E. hguerre [1882; 1741. They are the uniform limits of polynomials yith real roots. G. P61ya [1913; 2241 hcls designated the class by the number 11, thus oon- trasting it with the c h s I of functions whioh me the uniform limits of polynomials with red pogtive roots. We ahd be concerned here only with the former class and shall redesignate it as E to avoid confusion with the class IIalready introduced in Chapter I.
D E ~ ~ T I O N 3.1. An entire h c t i o n E(B) belongs to class E, E(8) E, if end only if i t has the form
where c - 2 0, b, ak(k = 1, 2, ) are red, and
We wish to include the case in which the product ( 1 ) hm a finite number of factors or indeed reduces to 1. To include these cmes without additional notation, we agree that from a certain point on, all a, may = oo. Examples of functions belonging to class E axe:
Observe that- the product of two fmctions of the class again belongs to it. 3.?. We show now that any function of o l ~ ~ ~ s E is the uniform limit
of polynomials with real roots. We need a preliminary result. X 1 ~ ~ ~ 4 3 . 2 . If l z I<Aanda>2A, then - -
That branch of the logarithm which reduces to 0 when z = 0 is intended. Using the &daurin expansion we have
. - &a stated.
!hmosm 3.2. If B(8) I, there exists a 8quenc-e of po~ynomid.8 Bg(8), , each with real roots only, w h that
lim E,(s) = E(s) -00
uniformly in the circle I e I < - A for every A > 0.
8 31 P ~ L Y A ' s CLASS OP ENTIRE PUNCTIONS 43
It is a consequence of the familiar Weierstrass factor theorem that when 3.1 (2) holds, the infinite product 3.1 (1) is uniformly approximated in 1 s I < A by its partial produota. Since a partial product is a poly- nomial-with real roots multiplied by an exponential eCa (which can be combined with 27 it is clearly sufficient to show that e4" and eb8, cb # 0, can also be uniformly approximated by polynomials with real roots only. But if we set z = -bs in Lemma 3.2, we have for 1 s I
Using the inequality 18 - 1 I< - elz] - 1,
this gives -
- from which
uniformly in lsl< - A/ Ibl. Replacing b8 by -a2, we obtain
uniformly in 1 s 1 < (A/c)l12. In each case the approximating polynomi.&la have red roots o& ( s h e o > 0).
3.3. It wm the converse of Theorem 3.2 which was proved by Laguerre. We prove the more general theorem of P61ya in which the uniform convergence in every circle is replaced by uniform oonvergence in a single circle. We follow the method of proof given by N. Obrechkoff [I941 ; 111.
THEOREM 3.3. If
1. akGak(n), k = l2 2, * - - , n, are reaZ
3. h E ) = ( 8 ) u n i f d y in I s I < - A for some A > 0, *00
then E(9) € E. As in the convention introduced for Definition 3.1, we may have
a, = a, for all k suffioiently large. Hence B,(s) may be of degree <n or indmd may reduce to 1. Thus we are consiclering an arbitrary uniformly convergent sequence of polynomials {lCE(s)): with real roots and such that B,(O) = 1.
44 THE NON-FINITE KERNELS [CH. III
By Weierstrase's theorem E(e) is analytic at a = 0 and
(1) lim EL(0) = Et(0), lim $(O) = E"(0). OH00 w m
If we set
equations (1) are equivalent to
As s, consequenae, the sequences (p37 and {qJy are certainly bounded, say by the constant M:
(3) I B ~ I < J ~ ~ 0 $'q, < n = l , 2 , * * *
From inequality (1) 8 3.2 with n = 1, we have
and from elementary inequalities
Hence for all s
That is, the sequence of polynomials { ~ , ( s ) } y is uniformly bounded in any circle. By Vitali's convergence theorem, E. C. Titchmarsh [1939; 1681, it converges uniformly in any circle, so that E(s) is entire.
By a, theorem of A. Hurwitz, E. C. T i t c h r s h [1939; 1191, E(8) will have only real zeros, if any. Let us denote them by orl, a,, a3, , in the order of increasing absolute value, wi th the usual convention about multiple mots and admitting a, = oo fiom a certain point on. For example, if E(e) = e8, all ak = m. By mother appeal to Hurwitz's theorem we may wsume that the roota a,(n) are arranged in such a way that (6) lim a,(n) = a, k = 1,2,3,- - -
-00
0 31 P ~ L Y A ' S CLASS OP ENTIRE FUNCTIONS 46
This is possible since in a small circle about a root a of order r, En@) will have exactly r roofs for all n sufficiently large. By (2), (3), (5) we have for m y integer p, n 2 - p, that
Now form the functions
[Both fwctions are identically zero if E(s) has no roots.] Let D be an arbitrary bounded region, say inside the circle 1s 1 < R, containing no a,. We s h d show that g,(8) tends d o d y to g(8) in D a~ ?% -r 00.
Let T be axl arbitrary number >R, and denote by N the number of the a, inside the oircle 1 a 1 < T. By Hurwitz's theorem there will also be just lV of the a,(%) in this circle for all n sufficiently large:
That is,
Hence
The first term on the right clearly approaches zero uniformly in D as n + co. Hence the right-hand side of (7) can be made less than an arbitrary positive E for all s in D first by choice of F and then by choice of n.
But
+ 2MRm T-R' (7) 1 g(8) - &3(8) 1 s N sa - 82 2
k - l a i ( 8 - ai(s-a,)
46 THE NON-FINITE KERNELS
Hence, by Weieratms's theorem
The aeries converges uniformly in I), so that we may integrate term by term f om 0 to 8,
Since a < q by (6), B(s) € B, and the proof ie complete. 3.4. ' o r our purp08es we need to genera&* the result of Pdlya,
replacing the hypothesis of uniform convergence in a single circle by that of uniform convergence on a segment of the ime-ry-axis. As a pre- limimry to thia result we prove'& lemma, somewhat stronger than needed for that result, but needed later in its full generaJity.
LEMMA 3.4. If
2. lim lCn(iy) exists uniformly in I y I - A for some A, *a0
then lim exists uniformly in 1 s I 2 - B for some B. W C Q
Set
By the definition of class B, c(n) 2 - 0, so that
Since the left-hand side of (1) approaches a limit as n -c co there edsts rt aonstant M such that
Each term of the series (2) is bounded by bl so that
inside the ckcle 16 1 < - 1/(2M), Hence we may apply 8.3 (4) tl ltafih to obtain
Again using (2) we see that 1 Q.(4 IS =_'I4
Out of each infinite subs& of integers, by E. C. 'I'itt!llnmmh [1939; 1691, we can pick a sequence Ins) such tha t Q8,(.9) rrl,~l'"'&t!llf~w IL
limit d o d g in any smaller circle I 8 I B < (2M)-1 8inc:c.r t rncl f ~ : t or.
Em($/), of Q,(iy) approaches uni formJyk-zero limit [E',,,(O) 1 fi jr f i l l m], the other factor eib(m)u must approach a, limit uniformly ill nwml interval 1 y 1 < 8 . That ia, to an arbitrary ) O there corrc .+ i~~~~~ i~ jn lLll
integer m, 8Eh that when m and la are numbers of tho Haquchn(*t! { ) ) I , }
grater than m,
1 sin [b(m) - b (p)]y I 5 - I eQ@(m)-b(p)l - 1 1 < c
I y[b(m) - b(,u)] - k~ 1 I sin-lr --a < I/ 1 4, for k: = 0, f 1, * . But this latter inequality is untcneblo fur $1 0 unless k = 0, so that when y = B
Consequently b ( a ) approaches a limit as m ca and Q,(s)c! ' b ' m j a r t i ~ ~ t b
approach a limit uniformly in the oircle 1 8 1 < - B. This limit, fir1 ~ttslyt.ic function, must coincide with lim En($) on s = dy, no mattor whitt nrilrwtI of integers {m} was aeleoted from. This proves the theorem.
TEIEOREM 3.4. If
the12 E(s) € E. I n the first instance E(s) is defined only on a, segmctrt of tht! irnrrgirlnry
axis, but the intended meaning is that this funatiorl oak& bn ctxtjcwtlc*fl analytically throughout the plane. The functions En(#) lwlt irk r?hrm E so that Lemma 3.4 is applicable. That is, B,(a) approact;tw rr litrtit, B(a) uniformly in some circle 1 e I 2 B, which of coumu uoitlc*idc*n wit li t i t o - limit of hypothesis 3 on s = iy, -A i; - y 2 - A. By Thcsorr~n 3.3, $44 E #, and the proof is complete.
48 THE NON-FINITE KERNEL8 [CH. I N
4. THE CLOSURE OF A CLASS OF DISTRIBUTION FUNCTIONS
4.1. We showed in Chapter I1 that the reciprocal of a polynomial P(8) with real roots only, P(0) = 1, is d w 8 p the characteriatio function of a distribution function. Let us say that a l l auch distribution functions belong to the subclasa Dt of D. The derivatives of functions of D, are precisely the £bite kern& treated in Chapter 11.
D E ~ O N 4.1~~. A function a(t) belongs to D, if and only if its characteristic function ~ ~ ( 8 ) is the reciprocd of a polynomid P(s) with r d roots only, P(0) = 1,
Let us next consider the closure of Dj in D, under the limit operation defined in fj 2.1.
D E N 4 . 1 A funotion a(t) belongs to Bf, the closure of D,, if and only if it belongs to D and if there exist functions a,(t), as(t), 9
of Dj such that
(1) lim a,(t) = a(t) t E C,. w m
We shall show that B, is the set of those distribution functions whose charscteristic funotions are the reciprocals of P6lyaYa functions of class E (5 3).
THZOREM 4.1. A function a(t) .& Q and a l y if a(t) E D a d ~ ~ ( 8 ) b the ~ e c i p r d of afunctkm of clast~ E.
We prove £hat the necessity of the conditions. If a(t) E D,, then a(t) € D by Definition 4.lb and its characteristic funotion
is defined at least on the 'imaginary axis and has the value 1 a t 8 = 0. Let {a,(t)}y be a sequence satisfying equation (1) end such that
where EJs) is a polynomid with rail mote only, Ea(0) = 1. By Theorem 2.2
1 lim - - - -co E,(8)
uniformly on any oloaed interval of the imaginary axis. Since xa(e) is notreroons=iy, ~ y ~ < d f o r s o m e d > 0 , -
8 51 THE NON-RINITB KERNELS 49
uniformly on a = iy, 1 y 1 < 8. By Theorem 3.4, I/&(&) is a function of class E, as we wished toprove.
Conversely, let a(t) € D and let its characteristic function (2) have a reciprocal, E(e) belonging to E. Then by Theorem 3.2 there exists a sequence { ~ ~ ( a ) } ; of polynomiab with real roots only, E,(O) = 1, such tht
lim En@) = E (8) M a
uniformly in every circle. By Theorem 6.2 of Chapter 11, equation (3) holde, where cc,(t), a,(t), ere distribution functions. Since E(0) = 1, there is en interval s = iy, 1 y I 2 - b, such that
1 1 -- lim - - -00 E,(4 P(8)
uniformly thereon. By Theorem 2.3, equation (1) holds and a(t) E n,. This completes the proof.
For the vaJidity of this theorem it ia important that the closure of D, should be taken in D. For, a sequenoe of functions belonging to D, may well approaoh a function not in D. For example, if
the limit function is constantly equal to 1. The bilateral Laplaoe trans- form of this funotion ia 0, and thie is certainly not the reciprocal of a function of class 8.
5. THE NON-FINITE KERNELS
5.1. We have shown that any distribution function which has a characteristic funotion whose reciprocal belongs to class 4 is a, member of , But we have not shown that every function l/&'(8), E(8) 3, is the chmacterhtic function of some distribution function. We shall now prove this if B(s) # t?". Indeed, we shall show that l/E(s) is the charac- teristic function of a frequency function Cf(t),
I
- = J eta(t) at. E(8) -a
The resulting functions U(t) are the ones which we shall use as kernels for the convolution transform.
5.2. Let us first establish an inversion formula which will enable us to obtain distribution function8 from their corresponding characteristic
60 TEE NON-PINITE KERNEL8 [CH. 111
functiom. It is due to Uvy [1926; 1671. We present it here in the form it takes when applied to Laplaoe transforms.
THH~OREM 5.2. If a(t) is a mddze& function of D, and (f
Note that this result comes formally from the classical inversion formula for the Laplace transform, D. V. Widder [1946; 241 1. However, the usual proof does not apply sinoe , the integral ( 1 ) may converge on a line only, 8 = iy, rather than in a atrip.
Set
By the uniform convergence of the integral ( 1 ) on the imaginary axis
I (R) = 1 1 a da(u) iR [ sin y(t - u) sin yu T -a Y + -1 Y a?4
Integration by parts gives sin R(t - u) sin Ru
du. t - u 21
Allowing R to become infinite in these familiar Dirichlet integrals we obtain
I(+m) = a(t) - a(0).
However, we give the detailm of proof since the present hypotheses on a(t) are not the most usual ones; see D. V. Widder [1946; 651. It will be enough to ahow t h t the integral
1 " O sin Rt J ( R ) = 7T - lm. ( t ) T at
tends to a(0) aa R + oo since the h t integral on the right of (2) reduces to one of type (3) after a, tra~ulation in the variable of integration. Corre- sponding to an arbitrary positive E we can determine A,, so that
sin t Iks, < e, t
§ 51 THE NON-PINITE KERNELS
when A > A,, B > A,. By the second law of the mean
sin Rt 7r t
1 - A sinRt a ( - A ) -BR sin t i LJ3 a(t) 7T I,, adl
dt=
Hence for R > 1, A > A,, B > B, we have by (4)
or allowing B to beoome insnite,
sin Rt dt
J (R) l a sin Rt 1 -;!-A
dt < 245. 1 -
< r
But this integral, over a finite range, tends to a(0) as R + co under the present hypothesea; see Theorem 7.2, D. V. Widder [1946; 661. Hence J(+co) = a(0) as desired.
COROLLARY 6.2. TWO distinct normalized distribution functions cannot have the same chrtracteristio function.
For if f(e) in Theorem 5.2 is identically zero, then a(t) - a(0) and hence a(t) (since a(t) is normalized) is identically zero.
5.3. Another preliminary result concerns the behaviour of l /E(e) on vertical lines.
LEMMA 6.3. If E(a) € E, then
1 B(o + i ~ ) 12 E(a)- For, if
then
Note that the result holds equally well if the infinite product is replaced by a finite product.
6P THE NON-FINITE KERNBLS [CH. III
-OREM 5.3. If E(8) E E and E(s) #eb8P(s), where P(s) is a p o Z ~ ~ , thn fwr any positive numbers p and R
uniformly in the &rip 1 o I< R. If E(a) has at most a finite number n of zeros then o > 0 and we have
thoughout the strip I o 15 - R. From this (2) follows immediately by virtue of the faotor tie
If E(8) h e infinitely many zeros (c 2 - O ) , choose N > p and so large that 1 ak 12 R when k > N. Set
Since EN(u) ia not zero in - R 2 - a < - R its reciprooal has a maximum in that interval. Hence
1
1E6 + i ~ ) 1 I T I + ~
Then by Lemma 5.3
uniformly in 1 cr 1 < R. This proves the theorem. 5.4. We ere &w in a position to how that every function E(s) of
alms B # $% the reciprocal of the characteristic function of a function a(t) of D. In fact a(t) will POSSeSs a derivative G(t) which is then e frequenoy funation. If I(&) is e polynomial P(s), this result was already
- c P - lblR N T ~ ) ( o + i 7 ) 1 2 e 1
-.
proved in Chapter 11, 1 --
If E(8) = P(8)eb8, then 1 ---- e - a + *)G(t) dt
eo
= j ' ; - . t ~ ( t - 6 ) dl,
- ak
e-Rllaal I E ~ ( o ) 1 .
3 61 THE NON-FINITE KERNELS 63
and C#(t - b) is clearly a frequency function. Hence we may conhe our attention to the case in which E(8) is not the produot of a polynomial by an exponential ebb.
THEOREM 5.4. If B(8) M the furzctim of T h e m 5.3, t h n there ezieta a function a(t) € D that
For, by Theorem 3.2, there exiats s sequenae of polynomiah Bn(e), B,(0) = 1, such that
lim E&) = E(8) m-+m
d o r m l y in every circle. By Lemma 5.3,
d o d y in -A 5 - y - 5 A for every A > 0. By Theorem 6.2 of Chapter I1
for some an(t) E D. Hence by Corollary 2.3 a fundion a( t ) E D must exist so that (1) holds for 8 = iy. &ally equation (2 ) h o b as a result of Theorem 5.2. Observe that no Cauchy value is needed for the improper integral (2) sinoe it oonverges absolutely by virtue of Theoman 6.3.
COROLLARY 5.4. Under the conditions of Theorem 6.4,
where Q(t) is the frequency funotion
This follows at onoe by differentiation under the integral sign in equation (2). The process is justified shoe the resulting integral (3) is uniformly convergent in -00 < t < a, by Theorem 6.3.
I t is helpful to consider the foregoing theory from the point of view of
64 TEN NON-PINITB KERNELS [OH. III
abstract topology. The clws D of distribution functions introduced by Definition 2.la becomes a topological Abelian semigroup if the group operation is defined as # (convolution) and the topology is introduoed by the limit operation
a&> z a(%
where the double mow is an abbreviation for the limit operation defined in 8 2.1 [an(t) approaches a(t) at ail points .of continuity of a(t)]. To verify that D is an Abelian semigroup we have only to recall (E. Hille [1948; 1471) that for elements a, #I, y of D
The unit function u(t) which is O and 1 in (-a, O), (0, GO), respectively, evidently satisfies the relation
for all u of D, but D is not a group since there is generally no inverse a& of a given element a:
a # a-l = U.
For example, the element a(t) which is et and 1 in (--a, 0), (0, a), respectively, haa no inverse. For, if aw1(t) existed its chmafsristia function would be 1 - a by Theorem C, 5 5.2, Chapter 11, an obvious ebsurdity.
The tranafomtion which replaces an element a of D by its character- istic function &(a) sets up a homomorphic mapping of D onto D'. In D' the group operation is ordinary multiplication of oomplex numbers m o r e m C quoted above], and the topology is set up by uniform limits:
d o r m l y in I y 1 < A for some A > 0 [Theorem 2.2 and 2.31. That the h o m ~ m o ~ h i s ~ i s isso an isomorphism follows from Corollary 8.2.
The subclass Df of De6nition 4.la is a sub-semigroup of D. This is most easily seen in the isomorphic semigroup L$ of reciprocals of poly- nomials with real roots only and equal to 1 at the origin. For, the product of two suoh polynomhla is another of the same type.
Fimlly the class B, of Definition 4.lb is also a sub-semigroup of D. The oorreaponding isomorphic class 8; is here the class of ha t ions l/E(8), B(8) E B. It was precisely the purpose of Theorem 4.1 to prove this. The semigroup property for B; is evident from the fact, already observed in 5 3.1, that the product of two fwactions of E is again in E.
8 61 PROPERTIEd? OF THE NON-FINITE KERNELS
6. PROPERTIES OF THE NON-FINITE KERNELS
6.1. Let us develop now Home further properties of the frequency funotion GT(t).
THEOREM 6.1. If E(s) & th function of Theorem 6.3, the%
where G(t) € Cm, -a < t < 00, and the integral converges in the l a r g t ~ t vertical strip which contains the origin and ia free of zerw of E(s).
A8 in 5 6.2 of Chapter 2, we define a, as the largest of the negative mots (a, = -m if there are none) of E(8) and a, as the smallest of the positive roots (a2 = +m if there are none). Then we must show that the i n t e p l (1) converges for a, < o < a,. This is an immediate conse- quence of Theorem 5.3. For, by use of Cauohy's integral theorem we may shift the path of integration in the i n t e p l (3), 8 5.4, to the line u = a 2 - e , O < e < a g ,
eiut dy Q(t) =
2n so that
a(t) = O(B(*-~)~) t - t - 0 .
Simil~xly, by shifting the path of integration to the line a = orl + E, we see that
~ ( t ) = 0(e(OL1+')0 t++00.
Henoe (1) converges for a, < CF < That it cannot converge in any larger vertical &rip is evident since l /B(s ) has poles at a, and or, (if these me finite numbers).
To show that G(t) hag derivatives of dl orders we may differentiah the integral defining Q(t),
The integrals (2) converge uniformly in -00 < t < co by Theorem 5.3. This validates the differentiation and also shows that C;r(P)(t) E C.
63. We next compute the mean and vaxrianoe of Q(t). THEOREM 6.2. If E(s) E E,
b6 TH191 NON-PINITE KERNELS [Ca. I11
then t h frequency function G(t) for which
Q)
Aos mean b and variance 2c $ 2 lla;. k-1
Here the product and sum ere e*nded over all the zeros of E(a), if any. The caaes in which E(s) has no zeros or a Gnite number of them m y be considered as included by permitting all or some of the ak to be 03. If E(s) is e polynomial, the result is already included in Theorem 6.2, Chapter 11. In m y case we have, after defining F(a) as
On the other hand
But from equations (2), (3), (4) it ia dear that F(0) = 1, F'(0) = 0,
so that the theorem is proved.
7. INVERSION
7.1. We are now in a position to develop an inversion theory for the oonvolution transforms whose kernels are the functions of ch0s E intro- duced in 5 3. For the present we limit attention t o the case in which no factor e-d is present in the inversion function E(8). That is, we assume that c = 0 in formula, ( I ) , § 3.1. We will treat this factor separately in a bter chapter sinoe it requires somewhat different methods.
0 71 INVERSION
00
1. E(8) E E and E(s) = d8 11.-1
5. P,(e) = P - ~ J ~ fi (1 - $) ealak, b-1
then,
E(Wf(0) = lim P,(D)f(z) = p(x) -a, < x < co. i t - a ~
We are wsuming that there are infinitely many roots a, of i ( 8 ) aince the case of a finite number of roots was treated in Chapter 11. Por each positive integer n set
By Theorem 5.3 this integral converges absolutely and uniformly on (-03, a). If D stands for differentiation with respect to x i t is clear that
P,(D)sf" = emP,(s), so that
Differentiation under the integral sign i s justified by the uniform con- vergence of the integral (1). Hence
This step is valid if the integral (2) converges uniformly. It does so for x in any finite interval since y(z) E B and since Q,(t) is a frequency function and hence absolutely integrable.
68 THE NON-PINITE KERNELS [Ca. 111
Sinae
we have
(3)
For an arbitrary E > 0 choose 8 so that when 1 t 14 - 6, x Gned, (4) I cp(x - t ) - cp(4 1 - € 0
Now write the integrd (3) as the sum of two others I1 and I, corre- sponding respectively to the ranges of integration 1 t 1 < - 8 and 1 t 1 2 - 6. By (4) it is clem that
Pd f a,
Suppose that M is an upper bound for 1 p(t) 1 and that n is large enough to make I e* I < 8/2. Then since (t - %)a 2 - 8914 for t on the range of I2 we have
By Theorem 6.2 the mean of %(t) is e, end
Hence
woo'
and our result is established. As in Theorem 7.1 of Chapter 11, our assumptions about p(t) are stronger than needed, and wi l l be weakened later.
COROLLARY 7.1. If
4. f (x ) = J - * m ~ ( ~ - t)tp(t) dt
then
8 81 GREEN'S PUNCTIONB 69
The infinite product of hypothesis 1 is known to converge when the series 211 1 a, 1 converges. Since 1 aa 1 > 1 for large k the series xl/ag will also oonverge and we may apply Theorem 7.1. In that theorem take
a0
Then
Hence the conclusions of corollary and theorem are equivalent, and the proof is complete.
,8. GREEN'S FUNCTIONS
8.1. In 9.1 of Chapter I1 we defined the Green's function of a certain linear differential system of finite order. Here we extend the definition to include a corresponding system of infbife order. Let B(8) E i?8 and have the speoific definition
As in 8 6.1 and earlier we denote by a, the largest negative root (or --a) of 8 ( 8 ) , by a, the smallest positive roof (or + 00). The differential ~y8tem of infinite order under consideration will be
To define the Green's function of this eystem we h t replace it by the "truncated" system in which the differential equation (1) is replaced by
where, as in Theorem 7.1,
60 THE NON-PINITE KElZATELS [CH. III
Following § 9 of Chapter 11, it is clear that the Green's function of the truncated system (4) (2) (3) should be
If the exponential factors in (6) were removed, this would be equation ( 1 ) of 4 6.3, Chapter 11. Since the exponential factors in (5) produce a translation in f(z) in (4)' it is clear that the Green's function (6) should be obtainable from that of 5 9, Chapter 11, by an equivalent translation. But,thia is preaisely what has been accornpliahed in (6) by the intro- duction of the exponential factor into P,(8).
We are now able to give our definition as follows. D ~ s w r m o ~ 8.1. The Green's function of the system ( 1 ) ( 2 ) (3) is
defined as
where U,(t) is given by equations ( 5 ) , (6) . 8.2. It is now emy to establish the existence of the Green's function.
It is, in fccot, the kernel of the convolution transform treated in Theorem
THEOREM 8.2. The Ureen's function of the ayatena ( 1 ) ( 2 ) (3) 5 8.1 exists and G equal to
From the definition of P,(4 it is evident that
The dominant function is integrable on (--a, a) and is independent of n. Hence we may apply Lebesgue's limit theorem to equation 8.1 (6) to obtain (1)' since it is known firom elementary theory that Pa($) tends to B(8) for every 8 . Since E(iy) # 0, the theorem is established. Note that we already knew from $ 8 4,5 that
We have now shown in addition that
(2) lim a,(t) = a(t) -00
in the pointwise sense.
3 81 GREEN'S FUNCTIONS 61
83. As in 9.3 of Cbpter 11, the Green's function provides an explicit aolution for the differential system under consideration.
THEOREM 8.3. V cp(t) f B - C , -a, ( t < a, the% a solution, of tha system (1) (2) (3) is
(1
That equation 8.1 (1) is satisfied by this function f(x) was proved in Theorem 7.1. Since f ( x ) E B, the boundary conditions are satisfied trivially.
8.4. Observe that in the statement of Theorem 8.3 it is not stabed that CI * p is the unique solution of the system 8.1 (1) (2) (3), as in Theorem 9.3, Chapter 11, the corresponding result for a system of finite order. If we had adopted as our dehition of E(D)
where the limit exhte boundedly or "in the mean" on (-00, a), for example, then the solution 8.2 (1) could be shown to be unique. But our deihition requires only that the limit (1) should exist for every real x, and in this case no uniqueness proof has been given. In fact, we can give an example of a, function f ( x ) satisfying 8.1 (2) (3) and for which the limit (1) is zero as n tends to oo through the even integers.
We choose the roots of E(8) as &I, &2, . By grouping the factors in the infinite product expansion we obtain
sin ~s E(8) = - - - lim Pen($)
?Ts W c o
Then f(x) = ht(x) will be the required function. It is evident that h f ( f co) = 0, eo that 8.1 (2) (3) are satisfied with a, = -1, at = 1. It remains to show that
lim l , (D)h t ( s ) = 0 -00 < 2: < a. woo
62 THE NON-FINITE KERNELS [CH. 111
Preliminary t o this result we prove by induction that (2) Bn(D)&(x) = c,[h(x)l"
C , = (2a + 111 nln!
Simple computation gives
so that (2) is valid for n = 1. Aasume it true when la is replaced by n - 1 and differentiake the equation twioe:
= cn[h(x)]*l. Thus equation (2) is estabxiahed.
By differentiating equation (2); we have (3) E,(D)hf ( x ) = on(% + 1 ) [h(x)lnhf (x). Since h(x) is an even function E,(D)hf(x) must vanish at z = 0 for aJl n. For each x # 0 the right-hand side of equation (8) is the general term of e oonvergent series and hence tends to zero with n. The test ratio of the series is
and fhis is leas than unity. Hence for all x Ern En(D)ht(x) = 0. -03
8.5. Let us now use the two examples of 8 10 in Chapter I1 t o illustrate Theorem 8.2. In Example A of that section we showed that the kernel
G,(t) = net(l - et)n-l -co<tgo = 0 O < t < o o
corresponded to the inversion function
But
and 1 00
= j-pe'~,,(t - log I ) dt. ndEn(s)
8 81 GREEN'S RUNCTIONS 63
The kernel corresponding to E(a) = l/I'(l - a) is
a($) = e-*'et
[see (A) of the table in 4 9.10 of the present chapter]. Hence we should expect that
(1)
Since
lim &(t - log n) = G(t) - ~ ~ < t < a o . D1--aoQ
q ( t - log n)= et (1 - f )"' --a < t I l o g n , -
equation (1) is equivdent t o the obvious relation
so that 8.2 (2) is verified directly in this special case. In Example B of 8 10 Chapter I1
= d l r l t ( L - rlt)-'(l - t)" dt l $ x < a .
Since &',(a) -+ a i n m/ (wa ) &s n + oo and since the latter inversion funcfion corresponds to the kernel et/(et + 1)' [see (B) of the table], 8.2 (2) will be verified if
But by the Laplace asgmptotio method, Q. Pdlya, and G. Szego [1925; $11, we have as n, + co
l t ( 1 - &).(I - t)" at rr te"tc-"' dt = r 1
tn(x + 113% ' Compare also D. V. Widder [1940; 2133.
8.6. In 5 8.2 we showed that every kernel B(t) of the type considered in Theorem 7.1 is the pointwise limit of finite kernels. Later we shall need a corresponding result involving limits of more general kernels (perhaps non-finite). In order to treat h i t o and non-finite kernels simultaneouely, we introduce the degrse of a, kernel by the following definition.
64 THE NON-FINITE KERNELS [ax- I I I
D ~ m o s 8.6. If E(s) E E and U(t ) is the corresponding kernel
then the degree of #(t), denoted by d(Q, is N if N
(2) B(a) =h8n 1 -- - k-1
otherwise d(Q) = a, (
For example, if g(t) is the unit kernel of I1 8 6.1, then d(g) = 1 ; if E(s) = e-*, U(t) = (h) -1h-tB/4, then d(G) = ca. Theorem 6.3, Chapter 11, shows that for finite kernels U(t) E CN-O, N = d(Q) ; Theorem 6.1 of the present chapter shows that for non-bite kernels B(t) E 0".
THEOREM 8.6. If 1 , ( 8 ) E B and a,(t) 69 the oorrwonding kernel, ra=o,1,2, . - * , i f
(3) lim -a < x < 00;
t h
Here, the superscript p indicates a pth derivative, and
By Theorem 2.2 equation (3) implies that l /En(iy) + l/B,(iy) u n i f o d y for every interval 1 y 1 < A. Since I En(iy) 12 1 it follows that B&y) +Eo(iy) d o d ~ i n the same intervalsand by Lemma 3.4, EJa) + E,,(a) uniformly in every oircle. If
we have by a olassiod theorem of Weierstrass that d,(m) = d,(O), E = 0 , 1 , - * . E'rom 'the definition of class E it is clear that the ooefficients in the expansion
00
are dl non-negative. On the other hand
so that D,(co) = Dk(0), k = 0 , 1 , *
Now if d(Q,) = N < m, we see from the explicit formula (2) that
DN(0) # 0. Since &(a) = DN(0), W: have for a positive constant B < DN(0) that DN(n) 2 B when n is sufficiently large, n > no. Using the fact t,hat the ooeffic~nts in (6) are 2 - 0 we obtain
Inequalities (6) show that the integraI (7) is dominated by
and this converges if p 2 N - 2. Hence for these values of p we may apply Lebesgue's limit theorem to the integral ((7) to obtain (4).
If d(Q,) = co, then Eo(e) must have the form
where either c > 0 or the product must have infinitely many factors (or both). But ,
Hence in every case the aeries (5), n = 0, has dl of its coefficients greater than 0. As before we may oonclude (6), but now with arbitrary N. Hence there need now be no restriction on p in (7) and the conclusion (4) holds for all p. Thie completes the proof.
9. EXAMPLES
9.1. In 8 5 of Chapter I we have already given several illustr&tions of our inversion theory. There we took the point of view that those examples, known from other sources, corroborated conjectures made by use of the operational calculus. Now with Theorem 7.1 at our dbposal we may specialize the kernel Q(t) in various ways, either obtaining new results or reestablishing old ones by the new method. I n particular the inversion theories for the Laplace and Stieltjjes transforms now appear as very special ceges of the developments of the present ohapter.
9.2. We record first the following simple results concerning change of variable.
66 THE NON-FINITE KERNELS [CH. III
for-m<x<oo. The h t of these identities was essentially proved in 9 5, Chapter I.
Por the second, we have
Hence if the operations (2) are performed in the order k = n, n - 1, - , 2, 1, the result is immediab. It is a familiar fact that the order of application of the factor operators is immaterial when the aoefficienta are constants.
The other identities may be established in a similar way. 9.3. The Laplace transform
(1) ~ ( z ) = edtO(f) dc
becomes after exponential change of the variables x and t
Since for a < 1 CO 00
(4) 1- m e t ~ ( r ) at = J e - ( c at = r(l- a), 0
the inversion formula, of Theorem 7.1 becomes
Using the familiar idbite produot expansion
and the identity 1 sin ns - 1 - --=
n ;ii (I - $1,
r(s) r(l - 8 ) kll
we obtain
Xn Corollary 7.1, choose
which evidently ten& to zero with l/n as required. Hence (6) may be replaced by
By equation 9.2 (1) this becomes
We have thus proved the following result. THEOREM 9.3. I!@($) E C * B m O < x < c~ a d if
then
lim - As previously pointed out, thie is the familiar real inversion of the Laplace transform; see D. V. Widder 11946; 2883.
9.4. As our next example, let us choose
sin TS 00
m8 k = l
68 THE NON-FINITE KERNEL8
If we replace a by -8 and t by -t in 9.3 (4) we obtain
But
That is,
[CH. 111
On the other hand, this kernel arises after s change of variable in the Stieltjes transform
First differentiate with respect t o x and then make exponential ahenges of variable as follows
This is a oonvolution transform with the function (1) as kernel. Hence by Theorem 7.1
- sin VD [e"F'(ex)] = Q(8)
v D
whew the prime indicates that there is no factor corresponding to k = 9. To interpret this in terms of the original variable^, we again make use
of Theorem 9.2. By equation (I), 1 9.2,
Now define a function H ( x ) by the equation
so aa t o make formula 9.2 (2) applicable to the right-hand side of (4). We obtain
Now eliminatting the function H(x) and returning to the original variables, we find that
Thia is a known inversion of the Stieltjes transform; see D. V. Widder [1946; 3501. We state the result as follows.
THEOREM 9.4. If @(x) E C B on 0 < x < a, and if the integral
mverges for x > 0, then e q 4 o n (5) holds. Note that we have proved a little more then this. For, our assumption
on Q(z) is only that it should be bounded and continuous. Thia is sufficient to guarantee the convergence of the integral
and the validity of the inversion (5). But of course it does not imply the convergence of the integral (6). As an example consider the function $(x ) = log (llx). It has no representation in the form (6) although its derivative satides equation (7) with a(t) = 1. Equation (5) is trivially aat.isfied for this function.
9.5. The Stieltjes transform arises as a special case of our general theory in another way. From equation 9.4 (6) we have
f(x) = Q * (p(x), where
Integrating by parts in 9.4 (2), the following Laplace transforma result
WS a &-at -- - -8 1 sin ns. -met -+ 1
at
77' edt - dt COB 978
- + < o < i .
Thus in the present case the inversion function ie B(s] = vol cos w, so that
77-I [coa nD]f(x) = ~ ( x ) -00 < x < 00.
70 ICHE NON-FINITE KERNELS [CH. 111
Using the identities (3) end (4) of 8 9.2 as (1) and (2) of 8 9.2 were used in the previous aeotion, we can prove the following reault.
%OILEM 9.5. If @ ( t ) l/t€ C B in (0, a), and if
fmO<z<oo. This is essentially a known result; compare D. V. Widder [1946; 3461.
As an example consider the familiar foxmula
7r -=I - at 4 Q Z/i& + t)'
When F ( z ) = 44; the left-hand side of (2) beaornes
and by Stirling's formula this i~ seen to equal z l h , as predicted. This example is the special case r = 1 of the following:
This pair results from the equation
where the integral on the right is a familiar form of the Beta-function. When v = 1 equation (3) reduces to (1).
9.6. Theorem 7.1 may equally well be applied to iterated transforms. Eeoh integration of a given oonvolution trensform, with inversion function E(8), gives rise to an additional repetition of the operator E(D) in the in~eraion. Thus, for example,
We shall illustrate by the first iterated Laplace transform. We consider the transform
where CJ is defined by equation 9.3 (3). By the product theorem the inversion function for this transform is
As in 4 9.3 it will be useful to intercept our result after exponential changes of variable to the range (0, m). Consider the transform
where K,(x) is the modified Bessel function, W. Magnus and F.
With appropriate changes of variable equation (2) reduces to the form (l), withf(x) = CP(ex), ~ ( x ) = @(e4):
-- exP(ex) = J(eeU) du ] ff (2: - u - t)ff ( t ) at. - CO -a
Since 1 II
P ( 1 - 8 ) n-rm
the inversion formula, of 5 7 becomes
a (3) lint t1(l - :) ezF(e@) I = 0 (et ) .
a+ a z-t f2 log n
By identity 9.2 ( l) ,
so as to make the same identity applicable a second time. Then
72 PEE NON-PINITE KERNaLS
h m the defmition of R(x) it is clem that
so that equation (3) becomes
then equation, (4) holds for 0 < t < 00.
Note that the integral ( 5 ) converges for 0 < x < 00, as one sees by applying hbini 's theorem to the iterated integral (2), wing the bounded- ness of 4. As an illustr~tion of formuls (Q), take Q(t) = t. From (2) we find F(x) = 1/xz and equation (4) beoomaa
x*1 (4) lim - DnxnD"F(x)
-a n! n!
This example shows, as we have remarked several times, that the con-
= 4(t). 2 = na/t
ditions imposed on @(t) at preaent are stronger than needed. For, the
We state the result. THEOREX 9.6. I f Q ( x ) E C Bin (0, a), and if
inveraion formula is valid even though @(t) = t is not bounded. It should be pointed out that the iteration of a convolution trensform
is not equivalent to the iteration of the transform produced from it by the usual exponential change of variables.
Thus, the iteration of 9.3 (2) produces (6), but the iteration of 9.3 (1) yields the Stieltjes tra,nsform 9.4 (6).
9.7. Next let us coneider the h t iterate of the Stieltjes transform,
1 2 f = U * U * q , a=- sech -
2'
By the product theorem, the inversion function is
But 1 1; - sech - * sech - =
at
4 2 2 x S" - a[e(*t)ia + e(w)/q [et/a + e- t /q
o 91 EXAMPLES 73
Hence equation (1) beoomea explioitly, using the definitions of 8 9.6 for fmdv ,
CO log x -- log t F(x) = z - t
Q(t) at . This is the familiar form of the iteralxd Stieltjes transform; compere R. P. Boas and D. V. Widder [1931; 11.
By use of the product expamion of the inversion function we obtain
Replacing f and cp by their velues in h m of the oapital letters and using the identities 9.2 (3) and 9.2 (4) as in 4 9.6, we interpret Theorem 7.1 as followa.
F(x) = -- 0 x - t
As an example we may o~nsider the equation T? -- a log x/t 6-l (z-i)did O < x < o o .
Since
we have at once, using the number c, of equation 9.2 (6),
By Stirling's formula 1 b
80 that equation (2) is vefied for this special function. 9.8. A8 our next example, let us use the Meijer transform
F(x) = C4a ~ , ( d ) @ ( t ) at,
74 THE NON-FINITE KERNELS [CH. I11
where &(t) is the mowed Bessel faction
For the latter inbgcal formula, see, for example, W. Magnus and I?. Oberhettinger [1948; 391. after exponential change of variables this becomes
00
f (x) = ex'aP(ex) = I- m e s t K o ( ~ t ) e - t ~ 2 ~ ( e t ) dt
where Q(x) = e°KO(ea), p(x) = ed2@(e+) .
The bilateral Laplace transform of thie kernel ia known, but for the reader's convenience, we derive it here with a minimum of preliminary material. Uee the product theorem to obtain the inverse Laplaoe trans- form of rqi - 4/21 :
Por the laat equation we have used (1). Hem- '"ie desired transform is
mrn the uaud product expansion for r(8) we see that
so that the inversion formula becomes
We now interpret this in terms of the original functions F and a. To make formula 9.2 (5) applicable, set
exlgF(ee) = eaR(esOP). Then
where c, was defmed in 9.2 (5). Set
and apply the above operator again:
The inversion (2) now becomes
If we set Qrzaew = z and eliminate the functions H and R we obtain the following special case of Theorem 7.1.
THIOBEM 9.8. ~f c ~ ( t ) l / t € C B on (0, a), and if
F(x) = rdii~(,(Zf)@(f) dl,
then
As an example we may apply (4) to invert
This identity follows from
which in turn is derived from (1). By me of 9.7 (3) we find that
so that (4) is verified trivially. The inversion (4) is in a somewhat different form from that originally
obtained by R. P. Boaa [1942; 211. That his differential operator is equivalent to (4) may be vefied by noting that both have the fundamental solutions
$~(lI2), +(lIa) log a k = O , 1,2, . We may treat the general Meijer transform
in exactly the same way. The kernel G(x) ia now esIlv(eP). Set
so that
and gv * g,(t) = .4etK,(2et).
Hence the inversion function is
The inversion analogous to (2) is now interpreted by the infmite product expansion of (5). We leave i t to the reader to obtain the analogue of (4) if desired. The similarity of Theorems 9.6 and 9.8 was to be expected in view of the sirnilax disposition of the roots of E(a) in the two cams.
9.9. As our final example we consider a kernel of class 111. Of course any of the finite kernels of Chapter II are of this class if all of the zeros of B(s) are taken of the same sign. However, the present example wil l be our firtst non-finite kernel of this class.
We take for the zeros of the inversion function the numbers a,= -kW, k = l ,2 , - . They are all of one sign and such that the
00
series Zl /a , converges (see definition in 8 8 of Chapter I). From the I
infinite product expansion of a-I sin e and the partial fraction expansion of cso s (see E. C. Tit8chmarsh [1939; 113]),
1 " (-l)k28 oscs =- + 2
s ~ - 1 t 1 2 - k % ? '
we obtain by change of variable s i n h d i a
E ( 4 = di = TT (1 + &) k-1
Now recall the definition of the Jacobi Theta-funation 00
$,(z, t ) = 1 + 2 2 e-XH cos 2knz k-1
(see, for example, G. Doetsch 11937; 261). Term by tern integration, easily justified, gives
Using the relations a,(&, 0+) = 0, co) = 1, we see afhr integation by parts that .-
1 -_. = l/s = rr.'o;(+, t ) at,
B(8) sinh 6 where tho prime indicates differentiation with respect to t . Thue the kernel Q(t ) in the present case is
Q(t) = @,(i t ) O < ~ C W = o -co<tso.
7'8 NON-FINITE RERNBLS [CH. III
Consequently Corollary 7.1 beoomea in this special case
For lack of an established name let us refer to (2) as a theta-transform. As a4 example take g ( t ) = t, f ( x ) = x - (1 /6) , to which the inversion (3) will apply even though q ( t ) ia not bounded. In this case the transform f (x) is easily computed from ita dehition (2) and from ( I ) , since E(0) = 1 and B'(0) = l/6. On the other hand
so that equation (3) is verified. 9.10. Let us summarize our applications in tabular form, listing the
definitions of E(e) and #( t ) in the owes treated. They were:
(A) Laplace
(B) Derived Stieltjes
(C) Stieltjes
(D) Generazed Stieltjes 1" , X y ) t ) v dt
(E) Iterated Laphce S,' 7 dt b a e - u f Q(u) du
(F) Iterated Stieltj es 1% dt I," x - t
(G) Meijer (v = 0) J - d Z ~ , , ( z f ) @ ( t ) dt 0
(H) Meijer (-1 < v < 1 ) da K v ( d ) O ( t ) dt
(I) Theta
8 101 ASSOUIATED KERNELS 79
Of course (B) and (C) are special cases of (D), ancl (U) of (H). They are listed separately since they are likely to be of more use than the general oases. The table is in no sense ehust ive .
I I
sin ns - n8
IT-l COB 7Ta
10. ASSOCIATED KERNELS
10.1. Wo conclude thix chapter by genoralidng in two respects the inversion theory of tj 7. We have hithmto demanded that the inversion function E(e) should belong to class E and, as ono consequenoe, that B(0) = 1. Moreovor the kernel Q(t ) was defined by the integral.
500 es6 (1) Q(t) = - --- &
Z n i E(a) -oo<t<oo,
80 NON-FINITE KERNELS [CH. III
the path of integration being the imagimry axis. We now allow the inversion function to vanish at the origin and permit the path of inte- gration in (1) to be any vertical line not passing through a zero of B(8). In this way (1) gives rise to a whole series of associated kernels corre- aponding to the intervale of the a-axis which are free of zeros. Only one of these will be a fiequenoy function but all of the corresponding con- volution trmaforms will be inverted by the same inversion operator E(D) .
10.2. Let us begin by proving two preliminary results. LEW 1 0 . 2 ~ ~ If
1. T ia a non-negative integer 00
2. a$=a, -c#O, k = 1 , 2 , - - ; C l / a i < o t , 2-1
9' O0 C 3. h * = b + - - 2 -
c i-1 a d
then E(8 + c) = E(c)E*(s).
It is clear that E(8 + C) and Z*(s) have the same zeros. That their non-vanishing quotient is constant, and hence equal to E(c), is easily verified algebraically by use of the identities
L- 10.2b. If l ( 8 ) and B * (8) are defined as in the previous lemma, then
E (D)f (x) = E (c)eoxE * (D) [e--f ( x ) ] .
AB always the differential operstors are to be interpreted as limits corrqonding to the limits defining the bhib products involved. Hence the result ia true if established for each of the corresponding faotors of these produots, and follows from the differential identities
(C + D) [e-mf (x)] = e+"Df (x )
(a; - D) [e-ea(z)] = ema(a, - D)f(o).
10.3. We turn now to the generalized inversion theorem.
3 101 ASSOCIATED KERNEL8 81
1. E(s)saT E E for eonze ideqer r 2 - 0
then
E(D)f(x) = ~ ( 4 -00 < s < 00.
Let I ( s ) be the funation of Lemma 10.2~3. Since B(o) # 0 end c # 0 hypothesis 2 of that lemma is satisfied. By the oonclusion and by change of variable
(1) where
Since E*(s) E E, #*(t) is a frequency function to which Theorem 7.1 is applicable. In terms of this kernel hypothesis 4 becomes, using (l),
00
J ( C ) ~ - ~ ~ ( Z ) = J'_ U * (x - t)gctpl(t) dt.
Hence by Theorem 7.1
(2) E (c) E * (0) [e-vf (x ) ] = e-cxf(x) -00 < x < a. But by Lemma 10.2b the left-hand side of (2) multiplied by em ie E(D)f(x), and the theorem is proved.
For example, choose E(s) the function l/r(l -- 8 ) of Example A, 5 9-10, with 1 < o < 2. Since the residue of r(l - 8) at 8 = 1 i8 -1 we have
t ~ ( t ) = - dtI'(l - 8 ) de = ti?-. et -- et.
After the usual ahanges of variable we fmd that the transform
(3)
is inverted by the operator which inverts the Laplaoe transform:
82 NON-.FINITE KERNELS [a. III
provided that @(t)td E B C on (0, co). More generally, if n < c < n + 1 the kernel of (3) may be replaced by
and the same inversion (4) applies.
11. SUMMARY
11 .l. The principal result of the present chapter is the inversion formula,
Poa
where
As justification for this choice of kernels the chapter began with a proof of Laguerre's important result (somewhat generalized) to the effect that the uniform limit of a sequence of polynomials with real roots only and equal to 1 at 8 = 0 is dways a function (1) multiplied by a factor em&, o 2 0. [A discussion of the opercttor ecD' is deferred to Chapter VIII.] ~ y u s e of this theorem of Laguerre we were able to show that the kernels Q(t) considered in this book form a semi-group and hence from a certain point of view constitub all of the "natural" kernels. Finally, in an effort to make the theory as concrete as possible a large number of special kern& waa treated.
CHAPTER IV
Variation Diminishing Transforms
1. INTRODUCTION
1 .I. It is natural to ask whether there exists an intrinsic description of our class of kernels U(t),
or whether theso kernels have been selected only for reasons of convenience. Theorem 4.1 of Chapter I11 may be thought of as giving an aihnative response to this question. In the present Chapter we shall show in an even more striking fmhion that this is true.
Let us denote by Y [ u ( l ) , , a(n)J the number of ohanges of sign of the real sequence a( l ) , - , a(%). Thus, for example,
Iff ( x ) iis a real funotion dofined for --a, < x < m then "YLfx)] 7 1.u.b. Y [ f ( x , ) , - , f (x,)], taken over all set8 -a < x1 < 4 < < x, < +a. It is of course possible that T[f (x) ] = a.
DEPINITION 1.1&. A frequency function ~ ( t ) is said to be variation diminishing if T [ h * 91 < V [ h ] for every h € B C(-m, a).
DBFINITION 1 .lb. rfrequenay funotion p(t) is said to be totally positive if and only if x1 < x , < < x,, t , < t , < < t, implies that
D, = dot [p(x, -- t,)], 2 - 0. This inequality should hold for n = 1,2, *
We shall pmvc that overy kernel (1) is both variation diminishing and totally positive, anrl we shall show that every variation diminishing
84 VARIATION DIMINICSHINO TRANSFORMS [CH. IV
fiequenoy function, and every totally positive frequency function is of the form (1). These remarkable reaults are due to I. J. Schoenberg [1947, e t seq.].
From these fundamental properties other important facts concerning U(t) can be derived. For example, we will prove that for each value of n = 0, 1 , , f f ( . ) ( t ) has exactly a changes of sign, and that -log Q(t) is convex. Such properties play an important role in later chapt,optcr.u.
2. GENERATION OF VARIATION DIMINISHING FREQUENCY FUNCTIONS
2.1. In this section we shall prove that B(t) defined by 1.1 (1) is variation diminishing. Let pr(t), y,(t) (n = 1, 2, - ) be frequerlcy functions. We shall write ,
if, for every tl and t,,
thus extending the aymbol of p. 54 to inolude frequency functiutm. Let sgn s be 1, 0, or -1 as x > 0, z = 0, or x < 0. We my that x
and y are of opposite sign if sgn (xy) = -1 and of the samu ~ i p i if sgn (xy) = 0 or 1,
LEMDU 2.1 a. I f { ( t ) } are variation diminishing frcqtiellc!y functions and i fp = ql * 9% * + * p ? ~ then p is a variation tfiminixhiry~ frequency function.
We proceed by induction on N. Our lemnns ip t - * t~o for N := 2 . Suppose that it has been established for n = 1, , 2V - 1 ; wc? ~ht t l l show that it is true for N. Let p, * tp, *. * (p,, = p'. Wo have
n * p = ( n * ~ ' ) * ( ~ ~ . Since p~ ia variation diminishing -Y[h * p] 2 I;[h * (o']. By ooltr ila iuc- tion assumption Yp * (p'] < Y[h]. Our desired conc1.lnsion B I ~ ~ O W H .
LEMMA 2.lb. Let ( t ( t ) be real functioll~ dt?fi~lwl for - oo< t<oo . If
1. w [ ~ k ( t ) l k~ 1,2, @ . , 2. lim ~ k ( t ) = y ($1 --co <. t < a,
k c 0
then 1CP[y(t)] I n. I f ~ [ y ( t ) ] = N then there exist points -a, < to < t , %< t , ~i: m.
auch that y(tf-J and y(t,) are of opposite sign, j = 1, * . , N . I f k i~ sflciently large then sgn [cp(t,) y,(t,)] = 1 for j = 0, , N, and t h u ~ for k sufficiently large v [ y k ( t ) ] Y [ y ( t ) ] , which proves tho ltmmr.
5 31 LOGARITHMIC CONVEXITY 85
LEMMA 2.10. Let* (p(t), {tp,(t)}y be frequency functiom. If 1. q,(t) is variation diminishing, n = 1, 2, - , 2. vn(t) S ~ ( t ) n 3 a,
then q(t) is variation diminishing. Since (p, is variation diminishing Y [ h * (p,J i; 'Y[h], n = 1, 2, .
We have h * (p = lim h * p,. It follows from emm ma 2.lb that V [ h * tp] -00
< V [ h ] . - - As in Chapter I1 we set
LEMMA 2.ld. If a is any real number not zero then 1 a 1 g(at) is a variation diminishing frequency function.
Since if pr(t) is a variation diminishing frequency function and if a # 0, then 1 a \p(at) is also a variation diminishing frequency funotion, it is sufficient to prove our lemma for a = 1. Let h(t) € B - C (-GO, m) and consider
We must show that P [ f ( x ) ] < - T [ h ( t ) ] . We have
h(x) = ( 1 - D) f ( x ) = -eaDe-2f(x).
Appealing to Rolle's theorem and remembering that e-@f (z) vanishes at + oo we obtain our desired result.
Combining our lemmas'and Theorem 4.1 of Chapter 111 we obtain THEOREM 2.1. U(t) deJined by 1.1 ( 1 ) is a variation, dimin6hing
frequency function.
3. LOGARITHMIC CONVEXITY
3.1. In this section we shall show that if ~ ( t ) is a varihtion diminishing frequency function and if q( t ) = i[q(t+) + q(t-)] for all t then -log q(t) is convex. h t us recall the formal definition of convexity.
DEFINITIOX 3.1. Let ~ ( t ) be defined for T < - t < - T' ( ~ ( t ) may assume the value +GO). ~ ( t ) is said t o be convex if for T 5 t l , ta 5 T' and 0 - < 0 - < 1 we have
86 VARIATION DIMINIBRIN# FRANSFORMS [CH. IV
LEMMA 3.la. A necessarg and sdcient condition that ~ ( t ) be convex is that for every T < - - - t1 t , < T' and every a, -00 < o < m,
(2) 1-u-b- [ ~ ( t ) - at1 $ max [&) - at,; - at,]. hatat, ,
Nemttity. Suppose thatl (2) is not satisfied for some t,, t,, and a. There would then exist t, tl < t < ta such that
We may write t in the form (1 - O ) t , + Bt, where 0 < 0 < 1. Multiply (3) by (1 - 6) and (3') by 0 and add. Simplifjing we obtain
so that ~ ( t ) cannot be convex. 8u&iency. Let t, < t , and 0 < - 0 1 - 1 be given. If t = (1 - B ) t l
+ et, then t , < - t - < t,, and hence by (2) we have for every a that
( ~ ( 0 - at< - max [ ~ ( t , ) - at,; pr(t,) - at*]. ChoosQ a = [ ~ ( t , ) - v( t l ) ] / ( t , - t,) and simplifying we h d that
5 (1 - 0 ) ~ ( t l ) + Oy(t,) aa desired.
Lmmu 3.lb. If
2. y(x) = *[v(x+ + Y(X-11 then
J-a0
where
I1 = -17 -112rv(~ - O I ) L - ~ ' d ~ ,
I , = n-mc tp(z. + .u)c-f a ~ . Now
8 31 LOCTARITHMIO UON VEXITY
Applying Lebesgue's limit theorem we find that
lim 1, = t y (TI, d+
1i.m 1, = *y(x+), d+
aa desired. Let us define
Anv(x) = Y(X + H) - y(x - h)
1. ~ ( x ) is a variation diminishing frequency function,
then for any real a and h > 0, Aheaxy(x) has a t most one ohange of sign. We apply Lemma 3.lb with y(x) = e4"A,eatq(x) to obtain
e-azAheax~(x) = lim [eahq(x + h - t ) €-PO+
n o m this we deduce that 03
e4aA#ay(z) = lim 1- =tp(x - t)n -112,-1 [e c r ~ - ( t + h ) ~ l @ - , - ~ - ( t - h ) a ~ # ] y €40+
Since e&-(t+h)m/l$ , e-di-(t-h)"leg
has one change of sign and since a, is variation dimi&hing it follows that for each e > 0
has at most one change of sign. Applying Lemma 2.lb we obtain our desired result.
LBMU 3.ld. If 1. ~ ( t ) is a variation diminishing frequency function,
then -log p(t) is convex. In view of Lemma 3.la it is enough to show that
or equivalently that
g.1.b. [q(t)ea" 2 - min [p(t,)eatl; rp(t2)eab] ti st s1,
88 VARIATION DIMINISHINB TRANSFORMS [CH. IV
for every t,, t , and a. Suppose that for Borne t,, t, and a thie were not true. Noting that v(t)eat v d h e s at +XI or -00 (or both) and drawing pictures it is easily seen that this would imply that for h sufficiently small Aheaxp(x) has two or more ohanges of sign, which we know to be impos~ible.
!I?HEQREM 3.1. If U(t) is dejfned by 1.1 (1) then -log Q(t) ia convex. This is an immediate corollary of Theorem 2.1 and Lemma 3. ld.
4. CHARACTERIZATION OF VARIATION DIMINISHING FUNCTIONS
4.1. In this section we shall establish the converse of Theorem 2.1. L E ~ U 4.1. If
1. cp(l) ia a variation diminishing frequency function,
then ^Yth * (PI 5 V [ h ] .
Let UE set ~ ( T , t ) = l for l t l < T, 2 - I ~ I / T for T < It1 - < 2 ~ , and 0 for I t I > 2T. By ~ebes~ue'slirnit theorem
Let na 2 - V [ h * TI. It follows fkom Lemma 2.lb that for T sufficiently large, T = T',
The function x(T', u)h(u) € B C (co, m) and thus since g, is variation diminiehing
Y h ( T ' , u)h(u) l2 m* Since
Y[h(t)l h Y[x(T' , t)hpl we have that
Y [ h ( t ) l & Our deaired result follows.
MORU 4.1. If y(t) i 8 a variation diminishing frequency fumtirm then there mhts a k e d U(t) hjaned by 1.1 (1) such that p(t) = G(t) ahnost everywhere.
8 41 VARIATION DIMINISHING BUNOTIONS 8 9
The fuaotiong-?e-" is variation diminishing and thus cD(t) = y(t) *T-* e-t' ia variation diminishing. Further @(t) is bounded, infinitely differen- tiable, and positive. By Lemma 3.ld -log @(t) is convex which implies that Ot(t)/@(t) E $ . We define
PI = lim Qt(t ) /@(t) , t*+
Pa = lim ar(t)/O(t). h-a,
(It is possible for4B1 to be -a and for to be +m). It is easily verified that because @ ( t ) is a frequency function
and that if 8; > PI, < pn then
It follows that the bilateral Laplace transform
converges absolutely in the strip @, < Rl s ( p,. Since
the function Q(a) is analytic in some circle (of radius R) about s = 0. We set
The relation defining Q(8) can be put in the form
Let p, , (x) = x[x + E] [ x + (n - I)€]. We have
$0 VARIATION DIMINISHING TBANSFORMS [CH. IV
then P,,(t) is a real polynomial in t of degree n. The formula above may be rewritten in the form
This impliea that P,,(t) haa only r e d seroa. Let
By a theorem of A. H d t z , E. C. Titchmmsh [1939; 1191, P,(t) has only real zeros. If we set
then Qn(8) has only real zeros. An easy computation gives
from which it followa that lim Q,(s) = 0 (8 ) a-3 cO
uniformly in every circle about 8 = 0 of radius less than R. By Theorem 3.3 of Chapter I11 R(s) is of the form
where a,, b, c' are r a l constant8 and
Now by the convolution theorem
at lemt for R1 8 = 0. Thus
8ince ~ ( t ) E U (- m, 00) the function e-8a14/ln(e) must be bounded on the line Rl s = 0. This is possible only if o = o' - f 2 0. If
i m ~ ( 1 ) = (2rri)-I J - i L ~ ( a ) l - l r a,
then
(2) J-;*(t )e-sl a = e-z/4/i2(~) c ~ , < R ~ s < Q .
It follows from (1) and (2) that q(t) = #(t) ahnoat everywhere.
5. THE CHANGES OF SIGN OF G'"'(t)
5.1. If E(s) is of the form
then we say that the corresponding kernel Q(t ) i s of degree N. I f E(s) is not of the above form the degree of Q ( t ) ia said t o be infinite. Let Q(t) be of degree N then U ( t ) has N - 2 continuous derivatives (see Chapter
N 11); the N - 1 st derivative is continuous except a t t o = b + 2 a;',
L where p - ' ) ( t 0 + ) and U(N-l)(t ,-) both exist. Let ua set #(N-l)(to) = t [Q(N- l ) ( to+) + Q(N-l)(t ,--)] . With this agreement a ( t ) of degree N has (for N finite or infinite) N - 1 derivatives. THEOREM 5.1. If Q ( t ) &e$ned by 1.1 (1) i s of degree N then (d/dt)nQ(t)
has exuctlynchangesof sign forn=O, 1, - 9 X - 1. Applying Lemma 3.1 b to we have
After rt integrations by parts this beoomes
The function (d/dt)me-(tlc)' has exactly n ahanges of sign. Since U(t ) ie varia;tion diminishing
92 VARIATION DIMINISHING TRANSPORMS [CH. IV
haa, for -ah E > 0, at mod rc changes of sign. Applying Lemma 2.lb we see that G'R)(x) has at most n change0 of sign.
Ginoe lim I G(x) 1 = lim 1 G(x) I = 0, U(l ) ( t ) has a t least one change - - -+ 03 m- a0 of sign (by Rolle's theorem). This together with lim - I #(l)(x) 1 = lim I G( l ) ( z ) I = 0 implies that G'a(z) has at least two changes of s i p , -
x 3 - 0 0
etc. Thus B(")(x) hm at least n changes of sign. This result for n = 0 , 1 , 2 ahom thrtt the graph of #(t) is bell-shaped. 5.2. It is possible to give en alternative proof of Theorem 5.1 which
is of some interest. We begin with the case in which G(t) is of finite degree N . We may
suppose that N > 1. Let to = b + Z ~ U ; ~ . By Theorem 8.2 of Chapter 11 a($) can be written in the form
where Q,(t) and Q,(t) are exponential polynomials of degrees a, and de, respectively, and
d l = - 1 + x 1 d B = - l + z l . ak>O ok<0
If G k ) ( t ) had more than E chsnges of sign then ~ ( ~ - l ) ( t ) would, by suooessive app1ioa;tiona of Rolle's theorem, have more than N - 1 changes of ~lign. We shall show that this is not possible. We must distinguish three cams: (i) d l > - 0, d a 2 0; (ii) dl& 0, d2 = -1; (ii) dl = -1, d2 2 0.
(i) Q(N-l)(t) h, not more than d , zeros for t < to, not more than d, zeros for t > to, and not more than 1 zero at t = to. Thus G(*-')(t) haa at most dl + d, + 1 = N - 1 zeros.
(ii) Here O(t) = 0 for t 2 - to. GN-')(t) haa no changes of sign at t = to, and has at most d, = N - 1 zeros for t < to.
(iii) Here U(t) = 0 for t 2 to, eta. . . . If U(t ) is of infinite degrezhen by 55 4.1,8.6 of Chapter I11 there exists
a sequence of kernels of finite degree Q,(t) r = 1,2, , such that
It follows from Lemma 2.lb that #(')(t) has at most k change8 of sign. 5.3. If a continuous function f(t) has m changes of sign in (-00, co)
then there exist points t,, t,, . , dm such that f ( t ) is of alternate sign in the s~ccessive intern& (-a, t l ) , (t l , tP) , , (tnrl, tm), (tm, +a). The points t,, 9 t Tn me zeros of f ( t ) ; however they need not be of any definite order and indeed (since f ( t ) can vanish in an interval) they need not be uniquely determined.
!&XEOREM 5.3. If G(t) dejned by 1.1 (1) & of degree N then t k p&t.s wociated with the changes of e&gn of G(k)( t ) (k = 1, , N - 3) are &mpk zeros of G k ) ( t ) .
We recall that x is a simple zero of f( t) if f(x) = 0 while f ' ( x ) exists and is not zero. Consider the case k 3 1. Let t , be associated with the one change of sign of B'(t). If our theorem were not true then t, would be
a zero of Q'(t) of order greater than 1, or Q'(t) might conceivably vanish in a11 interval containing t,. I t is quite easy to see by drawing pictures that this would constrain #(s)( t ) to have ti or more changes of sign, which however we know to be impossible. Similarly if the changes of ~ i g n of Uck)(t) are not effected by simple zeroa Q(x+P)(t) would have at least k + 4 changes of sign, etc.
6. INTERSECTION PROPERTIES
6.1. ht
94 VARIATION DIMINIBHINQ! TRANSPORMrS [CH. IV
where c', o", h', b", a;, a;, are real a d c'> - 0, c" 2 0, 2 (a;)-' < co 9
2 (a;)-2 < a, and let k
k iao i a o
Hl( t )= ( zn i ) - l I -4 00 [ E l ( a ) ~ e 8 i d a , B , ( t ) = ( 2 r r i ) - l ~ - r a ~ 2 ( ~ ) ] - l e ~ & .
If E(a) and G(t) am defined by 1.1 (1) then the relations
and
( 2 )
are equivalent. It is convenient and natural to express these relations by writing
t h fw any a, -m < a < a, Q(t) - uHl(t) iacGs at m a t two ckmge8 of *- Let ue set A(t) = 0 for l t l > l , - A ( t ) = l - l t l f o r l t l < l . I t is
easy to verify that
a 1 ( x ) = lim j" :~ , (x - t ) { ~ - I A ( ~ P ) } a. h 4 +
haa at most two changes of sign, and therefore since H1 is veriation diminishing the funation
has (for small h) at most two change0 of sign. Appealing to Lemma 2.lb our theorem is established.
§ f l UENHRATION OF TOTALLY P08ITIVE PUNCTIONS 96
then Hl(t)/#(t) has ad most two changes of trend. If G(t) = 0, then Hl(t)/U(t) must be dehed (possibly as +OD) by
continuity considerations. I t is easy to see by &awing piotures that if H,(t)/G(t) had more than two changes of trend, then there would exist a, constant a such that G(t ) - aAl(t) would have at least three changes of sign, and this is not possible.
7. GENERATION OF TOTALLY POSITIVE FUNCTIONS
7.1. The remaining sections of this chpter are devoted to the study of totally positive functions. We begin by showing that Q(t) defined by 1.1 ( 1 ) is totally positive. LEMMA 7 . h . Let g(t) be defined as 2.1 (1). I f a is a non-zero red
constant then I a I g(at) is a totally poaitive frequency function. It is evident that if q(t) is a totally positive frequency function end if
a is a real non-zero constant then I a 1 p(d) is e, totally positive frequency function too. Thus i t is sufficient to consider the case a = 1. If
then D, = det [g(x, - t , ) ] , is of the form exp [z xi - 2 t,]E, where 1 1
the i - jth entry being 0, ) or 1 as xi - t, is positive, zero, or negative. It is easy to see that E, = 0 unless
v being the number of equalities in (2).
96 VARIATION DIMINISHING TRANSFORMS [CH. N
LEMMA 7.lb. If cp(t), p(t) ED(-m,a) and if ~ ( t ) =q(t) * 'p(t), then
By the multiplimtion theorem for determinants we have
We read that
is equal to
Let N be the claas of functions k(j) fiom 1, - , n t o 1, , n. Repeated application of the above identity gives
Let N' be the subclass of N consisting of those fnnctions k ( j ) such that k( j , ) # k(j,) if j, # j,, and let N u = N - N'. If k ( j ) E N" then two (or more) rows of the d e t e h a n t
ere proportional, so that it is zero. If k ( j ) E N' then
The class N' contains nl functions k ( j ) . Our lemma ia a consequence of these results.
0 81 MATRIX TRANSFORMATIONS Q?'
LEMMA 7.10. If {+,(t)}Fml are totally positive frequency funations and if q = cpl * vt - * ( p ~ then cp is a totally positive frequency fmotion.
It is enough to prove this for rp = pi * q,, the convolution of 2 functions, since it then can be extended by induction to the convolution of any finite number of functions. By Lemma 7.lb we have
(3) det[q(x, - tt)l, deb xi - ~ J l n det ip)a(% - f,)l, d u ~ * * durn-
Let z, ( X , < * * - < x,, t , < t, < - , * < i,. If any two ukY8 are equal then det [pll(x, - and det [cp2(uk - t,)ln &re both zero. If the uk's ~ J X distinct then det [(pI(zi - and det [q,(u, - t,)], am both non- negative if e permutation of even order is needed to amnge ul, - , u, in increasing order, and are both non-positive if e permutation of odd order i s needed. Thus the integrand on the right side of equation (3) is non-negative. Our lemma follows.
TE~EOREM 7.1. If U(t) i s deJimd by 1.1 (1) Wen U (t) is a total2 y pmaive frequency functh.
If U(t) is a mte kernel then our conclusion follows from Lemmas 7.la and 7.1~. If @(t) is not a finite kernel then we can choose a, sequenae G,(Q of finite kernels such that
lim Q,(t) = Q(t) C a m
See 5 8, Chapter 111. This implies that det [(?(xi - t,)], = lim dot [Ur(x, - t,)]..
*oQ
If the inequalities (1) are satisfied then ilet [%(xi - t,)], 2 0, r = 1,2, , and thus det [G(x, - t,)],Z 0.
8. MATRIX TRANSFORMATIONS
8.1. In order to establish the oonverse of Theorem 7.1 we must show that every totally positive frequenay funotion is also variation diminish- ing. This necessitates (I preliminery study of matrix transformations and it is to this that the present section is devoted. The re~~ults which we shall prove are due to Schoenberg[l930], Motzkin[l936], and Schoenberg and Whitney [I961 J.
DEFIPTITION &la. A red matrix A = [a(i, j)], (d = 1, , m, j = 1, - , n), is said to be variation d h h h h i n g if
98
YW = 2 4 i 2 j ) x ( j ) ( i=l ,==*, m) 3'-1
implies that "Y[y(l), + , y(m)] Y[z(l), , x(n)] .
98 VARIATION DIMINISHING TRANtSPORMi3 [CH. rV
Consider the following matrix operations :
(i) Multiplication of a row or column by a non-negative constant; (ii) Addition of a row or column to m adjacent row or column; (iii) Omission of & row or column.
It may be verified that these operations applied to a variation diminish- ing matrix yield a variation diminishing matrix. The unit n x n matrix is variation diminhhing. By repeated application of the operations i and ii, we may, for example, deduoe that the n x n matrioes
are variation diminishing. Here C; = n!/[(n - j)!j!]. Let us put
where 1 < - il < d2 < = < - my 1 djl <ja < <j&n*
A f ~ : ' sk) is the minor of A formed from the rows dl, * , ik 3 1, 9 3 k
and the columns j,, 9 h a The order of this minor is k. The rank r(A) of the matrix A is the lasest value of k for which there exists a minor of order k which is not zero.
DEBWITION 8.lb. A real m&frix A = [a(iy j)], (k = 1, , rn, j=l, . , n), is said to be minor definite if minors of the same order R < r(A) have the same sign and if minora of the same order R = r(A) have the same sign if they belong to the same combination of columras.
The operations i and ii and iii applied to a, minor debite matrix yield a minor definite mahix. The unit matrix ie evidently minor definite; thus the matrices (1) are minor dehite. These examples suggest that the properties of being variation diminishing and of being minor definite may be equivdent. This is indeed the case. It is convenient to f i s t prove an intermediate result.
3 81 MATRIX TRANl3FORMATIONS 99
Given s sequence x(l), , x(n) of real numbers we define .W[x(l), - , x(n)J as the number of x(i)'s which am not zero.
DE~RNIT~ON 8.1~. A r e matrix A = [a(i,j)J, (i = 1, , m, j = 1, , n), is said to be variation limiting if
implies that
D E ~ ~ O N 8.ld. A real m ~ t r i x A = [a()], (i = 1, . , m, j=1,==- , r) ia said to be column definite if minors of order v belonging to the same oombination of columns are of the same sign.
THEOREM 8.1. A red matrix A = [a(& j)] M variation limiting if a& only if it is coZumn de$n,ite.
Let us prove that if A is column definite then it is variation limiting. We first note that it is no restriction to prove this under the assumption that no x, = 0, or, what is equivalent, that#'[x(l), , x(n) ] = a. If some x, = 0 we set
.'(j) = ( x(') j < k ~ ( j + 1) j z k ( j= l , = - - , n - l )
at(i, j) = ( a($, j) j<& a( i , j + 1) j 2 b . -
We have
The matrix [a'(i, j)] is column definite, and if we can show that Y [ Y ( ~ ) , - . , ~ ( m ) ] s m x ' ( l ) , , xr (n - l)] - 1 then since &'[x(l),
, x(n)] =&'[xt(l), a , x'(n - l)] it will follow that Vk(l), , y(m)] #[x( l ) , , x(n)] - 1. Repating this a x p e n t , it
follows that we may assume no x, is zero. We next assert that if A is column definite then "YCy(l), - , y(m)] ( A ) - 1 We proceed by induction on r(A). The result is clearly
t&e if r(A) = 0 or 1. Suppose that it has been established for r(A) = 0, l , - - , R - 1 ; we will show that it is true for R. We may suppose that m > R since if m = R then "Y[y(l), - , y(m)] 4 R - 1 trivially. It is enough to show that if 1 i, < i, < < iR I. m then "Y[y(i,,), * . . , ( ) ] - B - 1 If A' is the submatrix of A oozisting of the rows i,, i,, - = , iE, then A' is column definite. There are two came : r(A') < R and r(A') = R. If the f i s t caae is realized then Vw(i,), , y(iR)] I - r(Af) - 1 by our induction assumption and r(A') - 1 < R - 1.
100 VARIATION DIMINISHING TRANSFORMS [CH. IV
Suppose the second case holds and let j,, , jR be a selection of columns of A'. We set
, - , , iWl, - ' , (k = 0, , R). . . . 2 Y , jR
Since A' is minor definite no two u,'s are of opposite sign, and because r (Af ) = R we can so choose j,, - 3 jR that not all ~(k) ' s are zero. We have
It ia easily seen that V@(io), , y(iR)] = R ie not compatible with the identity (2) so that in this case too, Vb(i,), - , y(iR)] I. R - 1.
We can complete our proof. We suppose, as we may, that dEP[x(l), , x(n)] = n. We have Vb(l), , y(m)] r(A) - 1. Since
r(A) < - n, it follows that Yb(l), - , y(m)] 5 .%'[%(I), , z(n)] -- 1. Weshallnow showthat ifA = [a(i, j)], (i = 1, , m; j = 1, * * , rt)
is variation limiting then it is column definite. Comider the non-zero minors of order R oontained in the columns j,, - , jR. Two such minors a' and a' are laid to be adjacent if they contain R - 1 rows in common. We begin by proving that adjacent non-zero minors are of the same sign. Both d and or' are contahed in R + 1 rows i,, - , in. I;st
then a' and a" a m among u(O), , a(8). A necessary and suffiuient aondition that y(io), . , ~ ( i ~ ) be represented in the form
is that
If sgn (a' a") = -1, then it is possible t o choose y(io), , y(iB) none of whioh are zero, which alternate in sign, and which satisfy the relation (3). We would then have V[y(i,,), , y(iR)J = R >dE P [%(A), - , x ( j E ) ] - 1, a contra&ction.
- ~ -
Let a' and a" be two non-zero minors of order R contained in the columns j,, , j , We define the distance A(ocf, a") between a' an.d a" to be the number of row in which they both lie less R. If A ( d , a") = 0 then a'= a"; if A(af, a") = 1, then a' and a" are adjacent. We assert that if A(a', a") = d, c l > 1, then there exists a non-zero minor a (contained in the same aombination of columns) such that A(a, a') = 1,
8 81 MATRIX TRANSFORMATIONS 101
A(a, a") = d - 1. Let a' lie in the rowa i,, , iR, where the rows d,, i,, , iRWd also belong to a". Let i represent the vector whose components are [a(i,j,). , a(i, j,)]. Since d is not zero the vectors al, , i , are linearly independent. Choose i such that the row i is contained in a* but not in a' and consider the set of veotom i, i,, - " 0 . Proceeding from left to right we strike out the first vector which iFl linearly dependent on its predecessors. We obtain in this fahion a linearly independent set of vectors and from the corresponding set of rows e non-zero minor a. It is clear thet B(a, cz') = 1, A(a, a"} = d - 1.
We may now show that if a' and a" are non-zero minors of order R contained in the columns j,, , j, then a' and a" are of the same sign. We proceed by induction on A(%', a"). We know our result to be true if A , a ) = 1 suppose fhat it has been estabhhed for A(a', a") = 1,
, d - 1; we will prove it for A(a', a") = d. We have shown that there exists a non-zero minor a such that A(=, or") = 4 - 1, A(a, a') = 1. By our induction amumpt,ion both a and a' and a and or" have the same sign and so a' and a" have the same sign.
8.2. We now proceed to the demonstration of the oonjecture xnade at the beginning of the preceding section.
'IBEOREM 8.2. A red matrix A = [a(i, j)], (i = 1, - , m; j = 1, . , n), is variation diminishing if a d only if it i s m i w definite.
Let us prove thaf if A is minor definite then it is v e t i o n diminishing. We assert that it is sufficient to demonstrate this under the wumption t h t QE(l), , x(n)] = #[x(l), , x(n)] - 1. If any z ( k ) = 0 we set
then we have
The matrix [a'(i, j)] is minor definite and if we can show V[y(l), . , y(m)] 5 - "v[zl( l) , , x f ( n - l)] then since V[x'(l), , x'(n - l)] = V[x(l), - y x(n)] it will follow that Y[y(l), * , y(m)] 5 7"[~(1),
, z(n)]. Repeated application of this algorithm shows that we may
102 VARIATION DIMINISHING TRANSPORMS [CH. rV
aasume that no x(k) is zero. Next we may suppose that x(k) and x(k + 1 ) are of opposite sign. If this is not true then there exists A > 0 auoh that x(kS.1) = Ax@)). E w e set
The matrix [a'(;, j ) ] ie minor definite, and if we can show that Y @ ( l ) , , ~(m) ] s V [ x J ( 1 ) , , x - 1 then Bince V [ x ( l ) , , x(m)] = V[x' ( l ) , . , x'(n - l ) ] it will follow that Y b ( l ) , - , y ( n ) ] $ V [ x ( l ) , * , x(lt)]. This shows that we may assume V [ x ( l ) , , x(n)] = .#[x(l), , x(n)] - 1. Since a minor defmite matrix is a, fortiori column definite we have from Theorem 8.1 that Y[y(l), , y(m)] 2 &'[x(l), , x(n)] - 1, and our desired result follows.
Let us prove that if A ia variation diminishing then it is minor definite. We know from Theorem 8.1 that minors of order R are of the same sign if they lie in the same combination of columns. It remains to show that if R < r(A) then minors of order R axe of the same sign without restriction as to position. Let a' and a" be two non-zero minor0 of order 12. Let ue euppose that a' and a" lie in the columns j,, , j B and the rows i,, , i, end that if
then a' = oc(pI, v l ) and a" = a(p,, v*). We know that a(p, v ) , (p = 0, , R), are of the aame sign. Thus we may suppoee that v, # v,.
* Let us solve the system of equations
Q 91 TOTALLY POSITI VE FREQUENCY FUNCTIONS
We have R
a4vl) = ( - 1 1.1 2 a(p, v1) P-0 rz
= ~ v P ) = (--1)v=2 M(P, up) r-0
and thus sgn [x(vl)x (v*)] = ( - 1)'l -'a sgn [a' a"]. Since A is variation dimjnishing we must have Y [ x ( O ) , , x(R)] = R end hence sgn [x(vl)x(ra)] = (-l)'l-'a. This shows that a' and a" have the same w-
Now suppose that a' and a" are oontained in R + 1 columns J = (jo, - . , jR) of A and that these oolumns considered ae vectors are linearly independent. We can then choose R + 1 rows I = (i,, , 6,) so that the minor
Let us choose a non-zero minor 8' of order R oontained in the same aom- bination of columns as a' and in the rows I. Similarly let 8" of order R be contained in the same oombination of columns as a" and in the rows I. We know that a' and /3' and also uw and are of the same sign. We have just proved that B' and 8" are of the same sign; it follows that a' and a" are of the same sign.
Let 6(a', a") be defined aa the number of columns containing or' and a" less R. Guppose first that 8(af, a") = 1. Let J' be the set of columns of A containing a' and J" the set of columns of A containing a". If J' U J" are linomly independent then we have shown above that a' and a" are of the same sign. If J' U J" are linearly dependent then there is a column j such that J' U j and J" CI j are linearly independent. Let us choose a, non-zero minor f l contained in (J' n J") U j. We have shown above that a' and and also a" and ,8 are of the same aign ; it follows that a' and a* am of the same sign. Our proof may now be completed by an induction on 6(a', a").
9. TOTALLY POSITIVE FREQUENCY FUNCTIONS
9.1. We can now esta1)li~h tho corlverao of Theorum 7.1. Txrnn~ar () . la. If ~ ( t ) is a totally podtiwe frequency Junction then,
~ ( t ) i8 variation dimini.97kBng. Lot us fir& prove our tlloorem nn(Ic~ tho tul(litio11al assumption that
(p(t) E B C(-m, co) . Wc must R ~ O W that if q(t) E B C(--a, a) and if
104 VARIATION DIMINISHINU' XRA1VSFORM8 [CR. IV
then m x ) ] < - Vb(f)]. If "Y[f(x)] 2 m then we can choose n + 1 points z, < z, < - < zm :,so that Y'~(x0), - , f ( z ~ ] = m. We w n then take T so large that if
rT
then sgn [ f ( ~ % ) f i ( ~ d ) l = 1 ( i = O y 0 * - , m )
and thus V&(zo), - , fi(x,)] = m. Let us now subdivide the interval [-T, TJ into n equal parts each of length h = 2F/lt; t j = -2" + jh being the points of subdivision. Consider the Riemann sums
n
f2(x*) = a 2 90, - t,)g(t,) (i = 0, - , m). j-1
If .n is laxge enough then a@;n [fi(x,)f*~x*)l = 1 ( i = O , 9 m)
and thus vf,(z0), = , ,f2(x,)] = m. By Theorem 8.2
and our theorem is established for q ( t ) € B C(-co, m). In the general ease we set y,,(t) = ~ ( t ) * ? r - ' P ~ - ~ e - ( ~ / ~ ) ' . Sinae
7r - l / zE- le - ( t l~ )~ is totally positive ao h (p,(t), and (p,(t) is bounded and continuous for each E ) 0. Thua if g(t) € B C(-a, a) and if
then T[f,(z)J I; - V [ g ( t ) ] . We assert that if f(z) is given by (1) then cO
f ( x ) = - lim j'" ar.(x - t ) t ( t ) dt=
It is easy to see that
where g&) = g(t) * m-1/2s-1e-(t/')'. We have, making use of Lemma 3.lb,
Applying Lebeague's limit theorem we obtain our desired result. By hmma 2.1 b " Y l f ( x ) ] lim V L f E ( x ) ] < - v[s( t ) ] and our theorem is proved.
e q + We require the following result which was first proved by Sierpinski
[1920]. I f A is any measurable set on the line then we denote by m4 the Lebesgue measure of A.
4 91 TOTALLY POSITIVE PREQ UENCY FUNCTIONfl 105
THEORBM 9.lb. If 1. -00 < f ( x ) 2 +a, a < x < b , 2. f(x) i s nzeaezardk, and $nite almost everywhere,
$hm f ix) it3 continuma fw a < x < b. We firat assert that f is everywhere finite. Suppose to the contrary
that f ( x ) = co. 1st ( x - c, x + c) be a neighborhood of x contained, in {a, b) . If x' is any point of this neighborhood and if x" = 2 2 - x' then x" ailso belongs to this neighborhood. 8ince
2 f ( x ) I f @'I + f(z") one of the relations
must hold. It follows that if 4 is the set of points where f = co then mE 2 c; but this is contrary to assumption 2. Thus f (x ) must be finite.
next assert that iff ia not continuous at z then
lim f(y) = +a. rn
f discontinuous at x implies that there exists 8 > 0 such that in every neighborhood of x there is a point y for which If(%) - f (y ) I > 8. Let ( x -- c, x + c ) be a neighborhood of x contained in (a, b). There exists a point x' in this neighborhood such that either f (x ' ) 2 - f (x) + h r
f ( 3 ~ ' ) 5 f(x) - 8. If the second inequality holds let xu = 22 - x'. We have x" E (z - o, x + c) and using sgsumption 3 it is easily seen that f(z") 2 f ( x ) + d. Thua (z - c, z + o) oontains a point s, such that
Let xi be a point in ( x - Bc, x + t o ) for which f ( x i ) 2 - f ( x ) + 8, and let I x, = 22, - x. It is clear that x, f ( x - c, x + c) ; further
Repeating t h i ~ argument n timos wo find that there is a point x, E ( x - c, x -I- c ) R U C ~ that
We have thus verifioci our assertion.
106 VARIATION DIMINISHING TRANSFORMS [Ca. IV
Iff is not oontinuous at x then, as we have seen, there exist a sequence of points h contained in (x - ic , x + h) such that f(&) 2 - k. If x' is any point of the interval ( x - fc x + f c ) and if x" = 2& - sf then xf and xn belong to (x - c, x + c). Since
art least one of the inequalities
must hold. It follows that if 1, is the set of g E ( x - o, x + c) for which f (y) > 7% then mE, > ic. Since this is incompatible with assumption 2, f must be aontinuoua art x.
BXEIOREM 9.10. If p(t) iS a totaZZy positive frequency fuwt im, then there ex&& a kernel Q(t) of the form 1.1 (1) sw;h that i j Q(t) is of order greater t h n 1 then p(t) = Q(t) everywhere, and if #(t) ia of mder 1 then, ~ ( t ) = Q(t) except poess'bly at the diamtinuity f of Q where we have ody that
It follows from the fact that q ( t ) is totally poaitive that
The second of these relations may &o be written in the form
Theorems 4.1 and 9.la show that there exists a kernel Q(t) such that q ( t ) = Q(t) almos$ everwhere. There are three oases:
Let na conaider case (iii). We first assert that y( t ) = 0 fox -00 < t < 5. If t h i ~ were not true then there would exist q < 6 such that (p(q) > 0. For all z and 7 < x we have q ~ ( x ) ~ 2 cp(q)pl(2x - 7 ) . From the fact that g(2x - 9 ) > 0 almost everywhGe for &(C + q) < x, we would have ~ ( x ) > 0 almost everywhere for i(E + q) < x < which is impossible. Thus p(t) and B(t) coincide for -XI < t < C. Theorem 9.lb applied t o -log p(t) ~hows that -log q(t) is continuous for < t < co.
CHAPTER V
Asymptotic Behaviour of 'Kernels
1. INTRODUCTION
1.1. This chapter is devoted t o a detailed study of the behaviour of B(t) a8 t + -&a In the case that G(t) is a finite kernel,
G(t) can be expressed as one exponential polynomial for t 2 b + 2 q1 m 1
and as another for t < - b + 2 &c1, and from these expressions the - 1
behaviour of U(t) at f CCI can be eesily read off. In the general case where
1: dt = I/E (a), i ( s ) = G - ~ ~ ~ + ~ l~ (1 - k ) e ~ / a k , E a the behaviour of Q(t) at &a, may be more diilicult t o determine.
2. ASYMPTOTIC ESTIMATES
2.1. Let us reod that
aI = max [aka -a], as = rnin [a,, +a]. a,tO ak> 0
We further d e h e p, + 1 as the multiplicity of 8 - a, as a zero of i ( 8 )
and p, + 1 as the.multiplicity of s - a, as a zero of E(s). As we shall see the behaviour of B(t) at +a(--m) is largely determined by a,(a,).
T H E O ~ M 2.1. If U ( t ) i s no%-jhite then: A. a, > -a implies
la = 0,1, , where k is a real nuder <cr, and p(t) ia a real poZynomia2 of hgree p,;
B. or, = -a impliee CPn)(t) = O(ekt)
208
§ 21 ASYMPTOTIC ESTIMATES 109
n = 0,1, , where k is an arbitrary (negative) real nurnber; C. a, < $03 implies
n = 0, 1, , where k is a real number >a, and q(t) is a red polynomial
n = 0, 1, - = , where k is an arbitrary (positive) real number. We have
Suppose f h t that orl > -a. Choose a real number k smeller than a, but greater than any other negative zero of E(s). Let T > 0 and define D aa the rectangular contour with vertices at &iT, k iT. Integrals about D proceed counterclockwise. The integral
ia by Cauchy'a residue theorem equal to the nth derivative of the residue of eEt/E(s) at s = al. Let B(s) = (s - a,)'~+'B,(s). The expamion
is valid in some circle about a,. Thus
It follows that the rosicluo of eat/E(s) at a, ia
By Theorem 5.3 of Ohaptor 111 wc have
lim13 = lim I, = 0. T 4 T-+m
Hence
110 A8YMPPOTIC BEHA VlO UR OF KERNELS [CH. V
A second application of Theorem 5.3 of I11 gives
We have thus establiihed conclusion A. To prove B our argument proceeda in the same fashion except that k is chosen as any negative number. Here, of courae, there is no residue. Similaz arguments serve to establish conclusions C and D.
2.2. The argument used in the proof of Theorem 2.1 can be made to yield additional information. Let 0 > A, > A, > be the distinct negative zeros of l ( s ) , the multiplicity of A, being Mi + 1. If in the demonstraton of Theorem 2.1 we choose k, < k < A,, then
where P,(t) is an (ordinazy) polynomid in t of degree Mi for i = 1, 2 9%
Similarly, if 0 < B, < 8, < are the diatinct positive zeros of IC(s), the multiplicity of B, being N, + 1 and if B, < k < B,, we may show that
where &,(t) iEl analogous to kernel as two
a polynomial of degree N,. T h e fomulaa (1) and (2) are the expression, obtained in 8 8 of Chapter 11, for a finite exponential polynomiale joined together.
2.3. Let us coneider several exemplea referring for the necessary information to the table of Chapter 111. If E(a) = coslrs and if G(t) is the corresponding kernel then we find from Theorem 2.1 that
1 where E is any fired number greater than -312. Since act) = - eecb (g )
2n thia relation may be verified directly.
If I(&) = l/I'(l - 8 ) and if U(t) i s the corresponding kernel then, applying Theorem 2.1 again, we. find that
for every (negative) k. Since G(t) = etee-6* it is evident that this relation is correct.
Let E(4 = fi (1 - b) and let G(t) be the corre8ponding kernel. 1 ke
4 31 ASYMPTOTIC ESTIMATES CONTINUED 112
Theorem 2.1 implies that a(r) = O(ekd) t + + m
for every (negative) k; however, we know that G(t) = 0 for t ) 0. Let E(a) = e - ~ * and let #(t) be the corresponding kernel: Theorem
2.1 shows that G(t) = O(ek$) t++m
1 for every k. Here G(t) = - e - t s k . 2/G
3. ASYMPTOTIC ESTIMATES CONTINUED
3.1. Theorem 2.1 fails in the cases al = -a, (a2 = +a) to give precise information as to the behaviour of Q(t) EM t + +a (t -+ -00). It tells us only that U(t ) = O(eXt), t + + ca (t -t -00) for every k, which leaves open a wide range of possibilities ea to the actual behaxiour of Q(t), and aa we have seen, e very great range of behaviour does take place. In this section we shall obtain precise information for these caaes. Let
4 r ) - - [2c + ; (a, + *)) 1": THEOREM 3.1 a. If U ( t ) ie a m-Pf i i t e kernel then:
A, a, = -m implies that
#(n)[A(r)] - ( 2 ~ ) - ~ @ ( - ~ ) ~ h ( r ) f o ~ V b = O , l , .. .
Let us consider first the case E, = -a. We have
Making the change of variable a = a' - r we obtain
112 ASYMPTOTIC BEHAVIOUR OF KERNELS [CH. T
It is evident fiom Theorem 6.3 of Chapter III that we can deform the path of integration back to the imaginary axis. Let us make the change of variable a' = e / ~ ( r ) . We now have
then it m y be verified that
See 5 10.2 of III for a similar computation. Thus
If we set u = A(r) we 6nd that
ecn'( l (r ) ) = ( - ~ ) ~ A ( r ) l , , where
i c o 8 " = (%)-lJ -i co [I - -1 Br(8)-l h.
We note that P Q ( ~ )
The first of these relations is obvious. The second follows from the fact that m(r) 1. co as r 4 + 00. Equivalently one may show that r % ( ~ ) ~ 1. co aar++co. wehave
Clearly r2c(r)l E t , and since either c > 0, or there are infmibly many a,'s, or both, r2cr(r)= .o' a as r 4 + m.
We assert that
0 31 ASYMPTOTIC E8TIMATEtS CONTINUED 113
for all 8, uniformly for e in any compact set. By a familiar inequality, see E. C. Titchmarsh [1939; 2461,
I log [(I - ~ ) e ~ + ( ~ ' / ~ ) ] 1 1 2 1 s 13 1 s 1 1 112- Here that branch of the logarithm is taken for which log 1 = 0. It follows that if ( s I 5 .fA(r), where
4) = d v ) (v + a3) then
I log ~ ~ ( 8 ) c . " ' 1 5 2 1 s I " A , ~ ( ~ ) a-
since, by equation @'),
We have lirn ZA,(r)-3 = 0; r-cm k
for 2Ac3(r) 5 A (r)-lZA k(7)-2 2 A@)-1, k k
and A(r) + 00 88 r + +GO. Thus equation (4) is verified. By Theorems 2.3 and 8.6 of I11
Since n
I,. = (2n3)-I 2 ( - r ~ ( r ) ) - ~ '))ln=O
and since ru(r) -+ as r -+ f co it follows that lim I,. = (28) -11%. W c O
The proof of our theorem ia thus completed in the oase a, = -a. The case at = $ a can 110 decllt with similarly. The reader will recognize that the mechanism of the latter pert of the
proof is exactly that of the central limit theorem of probability. 00
If a, = -03 arid if c = 0, 2 a;' < a, then we know that B(t) ;= 0 k-1
aJ
for t 2 h f 2 a,;'. In t h i ~ oaso canobion A tell^ us the hehviour of I
a
G(t) as t -) h + 2 a;1 - . Similarly if a, = +m and if c = 0, L
DD QD
za;' > - oo, then G(t) = 0 fort 5 b $- 2 ~ ; ' . In this cese conclu- 1 1
00
sion R tells ua t,hc hshaviour of C(t) a~ t 4 b + 2 GI+. 7
114 ASYMPTOTIC BEEAVIOUR 03' KERNELS [CH. V
w COROLLARY 3.1. If act) ia non-finite and if c = 0, 2 agl is fmite,
then : 1
A. a, = -00 implies that
B. a, = +a: implies that
A
This follows from Theorem 3.1 and the fact that V(t) has one and G(t) no change of sign for -00 < t < 00. See 8 6 of IV.
3.2. Let us consider as an example the kernel ete-6t oorresponding t o ( 8 ) = 1 - 8 ) . We have, see Titchmmh [1939; 1501,
4 4 - l /2e~, . - r - l /2
It i e now wily men that
( 2 ~ ) -112e-rqr)[o(r)E(-r)] - re-r-l/g
thus verifying, in this special caae, Theorem 3.1.
8 31 ASYMPTOTIC EBTIMATIES UONTINUHID 11 5
3.3- Let U( t ) = e - x o ; then Theorem 3.1 implies that
~ ' [ A ( r ) l ~ r r + +a (a, = --GO),
X'[A(T)I -- r r + - a ~ (as = +a).
These relations are simpler in that A(r ) no longer appears. The above formulaa can be rewritten in a more advantageous fom. If c = 0 and
1 2 - is finite, we define M as a function of t by the equation k QIk
1 O < t < c o i f a r , = - c o (2) t = z
k M + a k -oo<t<Oi fa ,=+m.
1 If c > 0 or if 1 - is infinite we define L as a function of t by the equation
k a k
L b < t < ~ i f a , = - t ~ (3) $ = 2 c L + b + Z
k %(a, + L) - m < t < b i f a , = + m .
TIIE~OREM 3.3. If U(t) is non-Bnite, then:
1 A. a, = --a, c = 0 and 2 - jinite implies that
k Qk
1 B. rr, = -a, c > 0 or 2 -- infinite impliee that
k ak
1 C. a, = +a, o = 0 and 2 -jEnite implies that
k: Ok
X D. a, = +a, c > 0 or 2 - injFnite impliu, that
k: ak
x'PI L( t ) t + -a. Let us provo A, sinoo B, C, end D are entirely similar. We suppose
00
that al = --a, o = 0 and 2 g1 < co. Using (2) we have 1
116 ASYMPTOTIU BEHAVIOUR OF KERNEL8 [&I. V
It follows from (1) that x ' [ ~ ( ~ ( t ) ) l -- M ( t ) M-,+rn,
which is what we wished to show. 3.4. A result very aimilar to Theorem 3.3 may be obtained by a
different argument. Actually it is the theorem of the present section which we shall require in subsequent chapters.
~ E Q R ~ 3.4. If ~ ( t ) i s deJined oa dove and L ( t ) as in 3.3 (3) then:
Let us establish conclusion A. I f in equation 3.1 (2) we set n = 0 and multiply through by eru we obtain
where
a,(u) = (24-1
Differentiating with respect to we have
Let z(r) be the zero of Hi associated with its one change of sign; since H: (z ( r ) ) # 0, z(r) is uniquely determined; see 8 5 of IV. By Theorems 2.3and 8.6 o f I I I i f O s r , < co
lim 1 1 El:)(@ - H?~(u) 1 1 = 0 m = 0 , 1 , --•
v-w, V
we &ssert that
(3) lim Z(T) = z(r0), -0
that is, Z(T) is a continuous function of r. This is an immediate conse- q u h e of (2) and the fact that ET"(z(r,)) # 0. A second application of Theorems 2.3 and 8.6 of III yielh
$31 . ASYNIPTOTIC E8TIMATES CONTINUED 117
where H ,(u) is ( ~ v ) - ' l ' e - ~ ' / ~ . Since z(m) = 0, H L ( 0 ) # 0, i t follows, aa above, that
lim z(r) = 0. r-c+ aJ
Setting u = l ( r ) + cr(r)z(r)
in (1) we obtain x'D(s) + a(r)z(+)l = T=
Since z(r) is continuous every large positive value of t can be represented in the form
possibly in several ways. Let r ( t ) be for each large value of t some one solution of ( 5 ) ; then
~ ' [ t ] = r( t ) . Note that r ( t ) -+ 3-00 aa t + + oo. Now
= L[t - a(r(t))z(r(Wl, and thus
~ ' ( t ) = A[t + 4111 #++a, aa desired.
Theorems 3.3 and 3.4 are so similar that one would think it possible to deduce one from the other. This does not however seem to be the case.
3.5. In this seation we shall conaider in detail the tlsymptotio behaviour of the kernels KJt ) corresponding to the products
The pasametor a may be either in the range + < o: < 1 (alms 11) or 1 < a < a (class 111).
Let UR first slippose that + < a < 1. We consider the integral
where c < 2, ca > 1. Here & ) is the Riemann zeta function. We require horn various elementary properties of [ (z ) : { (z ) is andytic in the complex plane except at z = 1 whero it has a pole of order 1 and residue 1 ; kg+ 1 [ ( x + iy) I = o(l y 1) as y + &co uniformly for x in any finite
03
interval ; [(z) = z r z the series converging absolutely, boundedly, and 1 A
uniformly in every half plane Rl z 2 1 + r . These result8 may all be
118 ASYMPTOTIC BEHA VIO UR OF KERNELS [CH. V
found in Titchmarah [1951]. We suppose that 0 < r < 03. Thus 9 + r v = O(1) as y + &co uniformly for x in any finite interval. Further 11 I sin s(x + iy) 1 = ~ ( e - ~ l v i ) y + f 03 uniformly for x in any finite interval. Applying the Lebesgue limit theorem we find that
It is easy to see that (812)-I-(a
rkp) = (%)-I 1 - d ~ ; ($12) -i oo sin n~
here v = log r - cx log E. If we set z = (312) + w we obtain $a e W V
= -e Jv"("~)-1 1 aw. -4ao cosnw
By the reeults of 8 9.5 of I11 v
I&) = -esvlz(2?r)-1 aech - 2 '
Let us now deform the line of integration in the integral de*g I(r) from Rl z = c to Rl z = -n - i. We obtain
In the case 1 < a < CQ we choose c so that uc > 1, o < 1 and proceed as before. Here we obtain
7r 1 0 0 . - -1 7r *" + 2 - m - ) (1 < cr < 03).
a sin - m-0
§ 41 SUMMARY
If we define LJt) end Ma(t) by the equations
the asymptotic developments (1) and (2) imply that1
(More preciae estimates could, of coarse, be obtained.) Applying Theorem 3.3, we have
a
- a a-1
- log ~ ~ ( [ ( a ) - t) .r [ q t sin (:)I ( t+O+, 1 < a < oo). at
Integrating these relations we find that
- a 1 [ ; (;)I1----- log ~ , ( t ) - - (1 -- a) - - sin ra ( t - t + a , + < a < l ) ,
4. SUMMARY
4.1. In the present chapter we have studied the asymptotic b e h i o u r of G(t) for large values oft. This behaviour varies widely according to the structure of G(t) and this foreshadows the separation of kern& into olasses, described in I, which will prove necessary in later chapters.
CHAPTER V I
Real Inversion Theory
1. INTRODUCTION
1.1. In the present chapter we shall make use of the results of Chapters IV and V in order to obtain in its sharpest f o m the inversion theory for the kernels B(t),
Bs one might anticipate from the results of Chapter V it is necessary to clsssifg these kernels on the basis of the properties of the a,'s. See 5 8 of Chapter I.
DEITINITION 1.1. Let G(1) be defined by (1). Q(t) is said to belong to class I if there are both positive and negative a,'s; to class I1 if there are only positive a,'s and zarl = m; and to olsss I11 if there are only
R positive a,'a and 2ar1 < co.
k Either G(t) or a(-t) belongs to one of these classes. It is necessary
to treat separately the kerneh belonging to alas8 I, class 11, and class 111. However, although the details differ from class to clasa, the main outline is in every case the same.
2. SOME PRELIMINARY RESULTS
2.1. We develop here several elementary lemmas which it will be convenient to have at our disposal.
LEBIXA 2.la. Let ~ ( t ) be continuous and a(t) of bounded variation on every bite aubintervd of a I: t < co. If -
1. rB(t) da(f) converges (conditionally),
2. y(x, t)/p(t) is continuous and has no change of trend a < t < co - for eaoh x € I,
'g CD c t ' o
4 t h t , E y f : d - 8 ti' P , g h l 2s; e: %% & Q - w.
IM g - 0 4
A s 8- B g &
G E 0 &
zg g
REAL INVERSION THEORY [Cx. VI
B. 0 < lim q(t) < oo implies that a(+w) exiet~l and that, h m
a ( + a ) - a(t) = o(1) t -+ +m;
C. lim y(t) = a, implies that a(+co) exists and that t-c CQ
Let us demon&a,te only conclusion C; A and B may be established in a, similar manner. By the mean value theorem
where t, < < t,. Thus if t , > t,'
Since l/y(t) + 0 ei, t + a, we have
from whioh it follows that a(+oo) exhts. Just as we wtablished (1) we may show t h ~ t
where t < 6 < a. It follows that
aa desired. LEMMA 2.ld. If
1. k(t) belongs to L on every finite interval,
3. k(xo - t)r(t) € L( - co, co) for some x,,
then
6 31 CON VERUBINCE
We have
where we suppose x - 2 xo. (We define 010 as 0.) We have
Applying Lebeague's limit theorem in (2) we obtain our deaired result.
3. CONVERGENCE
3.1. We can now determine the aonvergenoe behaviour of the 00
transform 1- *0(2 - t ) d.(t).
THEOREM 3.1. If
3. u(z0 - t ) da(t) umuc*gca (mmi i twy) ,
then
/ - ' m ~ ( x - t )
eonveyles un@mdg for x i~ any infinite intend; i.e. jiw
Let I be the interval [x l , xz]. By Theorem 3.1 of Chapter IV ,
a ( x - t)/G(x, - t ) is, aa a function of t, non-increasing for x 5 xo, and non-deorwaing for x 2 so. Thus G(z - t ) /G(zo - t ) has no rbnge of trend for each s € I- his together with Theorem 2.1 of Chapter V shows that 0 - < G ( x - t)lQ(xo - t ) 4 - M for x E I, where
Appealing to Lemma 2.la our theorem is proved.
124 REAL INVERBION THEORY [CH. VI
As a consequence of this theorem we see that if G(t) € class I then Q # a m y be referred to as either convergent or divergent. The reader will recognize here the familiar convergence behaviour of the Stieltjea transform.
3.2. THEOREM 3.2. If
1. Q(t ) € class 11, 2. a(t) is of bounded variation in e v e y finite intend,
Q)
3- I-= Q(x, - t ) dcc(t) mveTgges (CrmditimEIy),
then
j'-'mQ(x - tl W t )
wnvergee unifmly for x in any finite interval bolunded on the left by x,; i.e. f o r x O < - XI - xl < a.
Let I be the interval [x,, xl]. By Theorem 3.1 of Chapter I V U ( x - t)/Q(x, - t ) is non-decreasing as a function of t for x E I. Thia together wi th Theorem 2.1 of Chapter V shows that 0 < - U ( x - t ) /U(x , - t) < M for x € 1 where - -
jJf = e%(%-so).
Applying Lemma 2.1a our theorem follows. As a consequence of Theorem 3.2 we see that if G(t) € cless 11 then there
exists a constant yo, which may be + co or - co, depending on a ( t ) such that the tramfonn G(t) # a(t) converges for x > yo end diverges for x < y,. For x = y, it may converge or diverge. The number ye is cdled the abschsa of convergenm. The reader will recognize here the oonvergenoe bebviour of the Laplace transform.
3.3- If Q(t) E class I11 then B(t) = 0 for t > b + 2a;l. It follows E
that we need not suppose a(t) defined for all t; it is enough t o assume a(t) defined for T < t < m and of bounded va,ria,tion in every subinterval provided that we consider only those z for which z > T + b + 2ac1.
2. a( t ) is dejlned and of baud varhtion in e'ueqy interval
§ 41 TH.E SEQUENCE OP KERNELS 125
convergm uniformly for x in any $nite interwl tying to the right of T+ b
Let 1 be the interval [x,, xe]. Choose u
and let
Since Q(t) vanishes for t 2 - b + za;', J,(z) is uiform1y convergent for a
x E I . On the other hand #(x - t)lG(x, - t) h a as a function of t no changes of trend in r < t < a for each x E I. In addition G(x - t) l Q(r, - t ) is easily seen70 be uniformly bounded for u t < m, s E I. Appealing to Lemma 2.la we see that J , ( z ) is un i fodY~nvergen t .
4. THE SEQUENCE OF KERNELS
4.1. We suppose that we axe given a sequence {b,}: of real numbers such that b, = b, lim b, = 0. As in Chapter I11 we set
and we define p,(m) + l (pp(m) + 1 ) as the multiplicity of a,(m) (a,(m)) as a zero of Em($). The unique zero of #,(t)' we denote by 5,.
The {~ , ( t ) } : form a sequence of frequency functions. The mean of G,(t) is b, and the variance is &. Each #,(t) is bell-shaped and two different Q,,,(t)'0 cut in at moet two points. Since as m + co b, converges
126 REAL INVERSION THEOBY [CH. VI
to the origin and dm decreases to zero, it aeems natural to wppose that the a,$) appear somewhat as in the following illustration.
Further verification of this picture ia oontahed in the following result.
L m 4.1. If {~ , ( t ) } : , and dm are defined as above then:
1 > 1 -- - - 3 1;- tBam(t + a,) at = -
8, 4'
Prom this it follows tht
We will prove that I [, - b, 1 < 8 #i2. Suppose the contrary. Then aince #,(t) haa a unique m & m ~ at 5, it is monotonic in the interval fkom b, to 5,. If I is the internal consisting of those pointe between 6, and 5, whioh lie at a distanoe 2 die or more from b, then the length of I i s at least 6 8z2. In I Om(t) is not less than ,9,-l'le. This would imply that
a contradiction. Thus conclusion A is vaXd.
5 61 THE INVERSION THEOREM 127
Let t, be im arbitrary number different from zero and choose m so l q e that I b , I < I to 112. Then
Hence
But if m is so large that 1 [,I < I to 1/2, then @,(t) is monotonia over the range of this integral and takes its smallest value at to; i.e.
l b m this inequality conclusion B is immediate.
5. THE INVERSION THEOREM
5 .1 In this aedion we shall treat the cage G(t) € class I. We begin by considering the application of the differential operator.
T F I E O ~ M 5.la. If
1 . U( t ) € class I , .
2. P,,,(D), Qm(t) are &fined as in 5 4,
3. a(t) is of bounded variation, in every finite intervd,
ths integrd converging u n i f d y for x in any finite intervat. Let us choose numbers x' and a". The function U,(x' - t)/G(x' - t )
has a function of t at most two changee of trend. We have either a,(m) > al or a,@) = a,, p,(m) < pl. It follows from Theorem 2.1 of Chapter V that
128 RZAL INVERSION THEORY [Ca. VI
Similazly since either orl(* < a, or al(m) = czl, pl(m) i ,ul, We have
h - o o
Applying Lemma 2.1 b we b d that the integral
(1)
converges for x = x X . If G,(t) E clam I then (1) is by Theorem 3.1 uniformly convergent for x in any finite interval; if Q,(t) E class 11 then ( 1 ) is by Theorem 3.2 uniformly convergent for x in any f i t e interval bounded on the left by x" a d since x" is iwbitrary (1) is uniformly con- vergent for 2' in any finite interval; if Q,(t) € alms I11 then by Theorem 3.3 (1) is uniformly convergent for x in any f i t e interval. It remains to verify that
This follows from the fact that a linear differential operator may be applied under the integral sign if the resulting integral i s u;~XoM]aly convergent in the small.
TEEOB~N 5.1b. Ij"
2. y (t) is integrable on eve y jnite internal,
6;. q(t) i 8 w ! & ~ U O U ~ X ,
then lim Pm(D)f(%) = v(4* -00
By Theorem 6. la we have
8hoe Om($) ie a frequency function 00
p m ( ~ ) f ( x ) - p(x) = [-a am(. - t ) [ ~ ( r ) - p(z)l dt.
It. is therefore enough to show that rc0
Given r > 0 let us ohoose 6 > 0 ao small that
Q 51 THE INVERSION THEOREM
we set a0
/-ao,(x - t ) b(f) - ~ ( 4 1 at = I1 + 1, + 1 , s
corresponding to the ranges of integration (-a, x - a), (x - 8, z + a), and (1: + 8, +a). We have
x-8
= J- a [%(x - t ) / Q ( ~ l - t)l La(., - t ) - v(x)}I a, where x, is any real number. By Theorem 6.lb of Chapter IV a,(x - t ) / /Q(x , - t ) has at most two changes of trend. If m is aufficiently large then G,(x - t)/#(xl - t ) is increasing for t near -00 and decreasing for t near +a. Thus Q,(x - t)/Q(x, - t ) haa an odd number of changes of trend and hence one change of trend. By Lemma 4.1 this must lie within ( x - 8, x + 8 ) if m is aufficiently large. It follows that Q,(x - t ) / G(x, - t ) ia increasing for -a < t I; - x - 8 if m is large. By the mean value theorem.
where -a < < x - 8. By Lemma 4.1 we have lim I, = 0. Similarly m-+a
we may show that lim I, = 0. Thus m+a3
Since Q is arbitrary our theorem is proved. 5.2. In this seotion we consider the case Q(t ) € class 11. THEORBM 5.2a. Xf
2. P,(D), G,(t) are defined as in tj 4,
3. u(t) is of bounded variation in every finite interval, 00
4. J" - t ) da(t) = f(z) converges for x > yc,
then rco
n o REAL ~ V B R B I O N TEBIORY
the i w r d converging u n i f d y for x in anyfmite intererd m
y s - b + ~ , - ~ a ~ 1 < x 1 1 . x ~ ~ 2 < ~ . - 1
Let us ohoose numbers z', x" such that x' > yo, x" > xf - b m + b - %. The function Q,(x" - t)/Q(z' - t ) has, a9 a function of
1 t , at most two cbnges of trend. We have either a&) > as or 02(m) = a, and ,up@) 5 pa. It follows fmm Theorem 2.1 of Chapter V thak -
lim U,(x" - t)/B(x' - t ) < a. t+f aJ
Prom Theorem 3.4 of Chapter V we have
t --t --a, where L(t) and L,(t) are defined by the equations
" L,(t) CO
t + b - b , + t : = b + 2 J,(t) 1 ak(ak + Lm(t)) 1 ada% f Lm(t ) )
from whioh we obtain
Thus a m - log Q,(xX - t)/G(x' - t) = L(x" - t + b - bm + 2 a;1 + o( l ) ) at 1
- L(xf - t + ~(1)). Shoe L(t) is non-decreasing it followe that log U,(zW - t)lU(xt -- 1) is non-increasing ae t + -a and thus that;
Applying Lemma 2.lb we see that f a
is oonvergent for x = 2". By Theorem 3.2 the inteed is uniformly oonvergent for x in any finite interval bounded on the left by xC, etc.
§ a THE INVERSION THEOREM
1 . U( t ) € class 11, Q(t ) h rum-Jinite,
2, y ( t ) ie integrable on every Jinite interval,
then lim P m ( D l f ( 4 = 944.
ds in the proof of Theorem 5.lb it is enough to show that
Given E > 0 we cho08e 6 > 0 80 smdl th&f
and we set, as before,
corresponding to the ranges of integration (-a, x - d), ( x - 8, x + d), and ( x + 6, a). Arguing as before we find that I I , I < - E . We h v e
where xl is any number > y,. By Theorem 6.lb of Chapter IV a,(% - t ) /U(x , - t ) has at moat two changes of trend. If m is sdcientlp large then Q,(x - t)/U(xl - t ) is increming for t near -00 and dearwing for t near f co. (See the argument used in the proof of Theorem 6.2a.) Thus a,(x - t ) / U ( x x - t ) has one ahange of trend and this must lie within [x - 8 , x + d] if rn is sufficiently large. It follows that U,(x - t ) /@(xl - t ) is inoreasing for -ca c: t < - x - B if m is euficiently large. By the mean value theorem
where -a < 8 < x - 8. By Lemma 4.1 we have lim I, = 0. Similarly we may show that lh I, = 0, etc. -00
m 3 *
132 REAL INVEB8ION THEORY [Ca. VI
5.3. We now turn to the case a(£) E aleas 111. The demon~trations of the following results follow in the pattern of aections 5.1 and 5.2. T ~ o ~ M 5.3a. If
2. Pm(D), U,(t) are defined as in 5 4,
3. u(t) is defined for T < t < co and is of bouded variation in every internal T < t , < t < t , < - - a,
then
the inhqral converging uniformly for x in any Jtnite interval of the f o m 00
2. ~ ( t l de$md for T < t < co and is integrdle over every intend
5. ~ ( t ) is M i n w at t = x(x > T), then
lim Pm(D)f (2) = p(x) m--+ Q
6. STIELTJES I N T E G W S
6.1. In these sections we ahall consider the inversion formula for tho Stieltjes convolution C7 # a rather than the Lebeague convolution CJ * 9. We begin with U(t) € class I .
!l?mom~ 6.la. If
3. a(t) i s of bouded variation in evey JEnite interval,
4. f (z) = /-:~(z - 1) e c t h ( t ) canuerpea,
§ 61 STIELTJES INTEGRALS
then for m szlflciently large:
A. al < c < a, implies that
B. c 2 - a, implies that er(+oo) e x i 8 ~ 9 and that
C. c < - a, implies that oc(-a) exists and that
The funation #(x - t)ect has as a function oft either one or no change of trend. It is therefore monotonic near +oo and near -m. By Lemma 2.10 if x,, is any real number we have
CON~LUSION A. By Theorem 6 . h we have
the integral converging uniformly for x in any finite interval. Integrat- ing by parts we obtain
The estimates given above end Theorem 2.1 of Chapter V show that the integrated term vanishes uniformly for xl < - x 2 - x2. Thus
134 RBAL INVERSION THEORY pi. VI
the integral converging d o d y for x, - < x I; - 2,. We have
Because of the uniform convergence of the inner integral the order of the integrations may be inverted to give
Using Theorem 2.1 of Chapter V end the estimate (1) we see that if rn is sufficiently lmge the integrals
will oonverge absolutely. Q.E .D. CONOLUSION B. Proceeding exactly aa in the previous case we obtain
+ I-: am(%, - t)e-o(a~-t)[a(+co) - a(t)] at.
We must show that
w e define r(t) = [G(x, - t)ect]-1,
We have, using (2), that
The ssaumptions of Lemma 2.ld we satisfied so that
Thus equation (4) haa been eatabliahed.
$ 6 1 STIRILTJE8 INTEGRALS
1 . U(t) E class I , G(t) it? non-finite, I
2. u(t) is of bounded variation in every finite intemzl,
5 . a(t) is continuous at x,, x2, then:
A. a, < o < a, implies that
C. c 2 - a, implim that
Let us demonstra;te conolusion B. The other conclusions follow h m jimihr mgurnente. We have shown in Theorem 6.la that there exists an integer rn, such that if m 2 - m, then
Thus if
then
where
Appealing to Theorem 5.lb, 5.2b, or 5.3b as Bm0(t) belongs to olasa I, alas 11, or class I11 we obtain our desired result.
6.2. G(t) E class 11. Shoe only small changes in argument are necessary for this case we shall forgo detailed demonatrations of our results.
136 REAL INVERSION THEORY
1. #(t) Eclsss 11,
2. P,(D), Q,(t) are &$ined ae in 1 4,
the% for m w & W y large: A. c < a, implie8 that
B. c 2 x, implies that a(+oo) &ts a d tM -
1. Q(t) E claaa II, G(t) is non-Jinite,
2. a(t) i a of , f w d variatim in every $mite intervat,
4. P,(D) is defined as in 5 4,
6. a(t) is continuow at z,, x,, then:
A. c < or, invplics that
6.3. U(t) OE class III. Bs in the preceding section it is left to the reader t o supply proof.
2. a(t) is CEeflned for T < t < m and is of baud war&ion in everyJ;nite i n t e n d T < t l < - t - 2 ta < c ~ ,
3- P,(D), C;r,(t) are d e $ d ccs d n 8 4,
then for x,, x, > T a d m mfi ien t l y h g e :
A. c < a, implies that
B. o 2 - a, implies that a(+co) d t a a d that
2. a(t) i s de$ned for T < t < co a d i s of b o z c ~ v a S m b every jinite interval 2' < t , 5 t t ,,
6. a(t) is c 0 n t . b ' ~ ~ at x,, x, (xl, x2 > T ) ,
7. RELAXATION OF CONTINUITY CONDITIONS
7.1, It is mturd to suppose h r n the known examples of our theory, e.g. the Laphce asd Stieltjes transforms, h t if
f(d = I;?(. - t)p(t)
oonverges then Jim Pm(D)f(x) = 943) -00
almost everywhere. We shall show that this is true if the constants bm approach zero not too slowly. L n 7.1. If (~,(t)};, 8, am dehed as in 5 4 then
There am two oases we must oonsider:
- 112
Now
the higher term having positive coeffioient~. Shoe (8;, - ac8) 2 $9, - for k = m + 1,m+ 2, w e b v e
Hence 00
%(t) s (2n)- lLJ1 + t ~ ~ 8 m l - '
2 11 (28,,,)112. - If m e B obtains there exists km > m auch that 214 > rS,, 1 uk I < (2/Sm)1'2.
m m If we set
then we have
Um(t) = j=-:B(ahm. ~l)a:(t - U ) au.
8 TI RELAXATION OF CONTINUITY CONDITIONS
It follows that
am(t) 5 [I*u-b- g(akms @I Gg(t - U) du, CC
It is interesting to note that Si1B is the true order of the maxim- of am@), Bince it was shown in § 4 that the maximum exceeds A8;1/2 where A is an absolute constant. THEOREM 7 . h . If in T h m - e m 5.lb, 6.2b, and 5.3b t h m q t i o n ,
that p(t) is continuow at x i s replaced by the weaker as82crwption
than the c m c l h of t W e t h e m , that
still hold. In parti&r equation ( 1 ) holh at all points of t k Lebwgtce set of p(t) and therefme almost everywhere.
We must e how that given c > 0 there exists 8 > 0 such tht if
then
If 1, is the point where G,(t)' changes sign then by Lemma 4.1
where A is an absolute oonstant; thus
I 1" 1 = O C ~ + ! ~ )
Let M = 1.n.b. 1 Ern I Gm(l;,). This is 5 i t e because of Lemma 7.1. We m-0,1,. . .
now choose 8 so smaJ that
J y ( t ) 1s 6 it1 (2M + 1)-l l t l l where
r(t) = j b ( Z - U) - +)I dug 0
140 REAL IN VERSION THEORY
Integrating by parts and using Lemma 4.1 we have
NOW 1 t 1 1 t - [,I + I [, 1 from which it follows that
We have
I . 2 I t IQ(US2B. - Combining these results we obtain
It interesting to note that Theorem 7.lb depends information concerning the mode of O,( t ) .
7.2. In this section we shall discuss the conditions inversion formula for the Btieltjes convolution is valid continuity of the integrator function a(t).
essentially upon
under which the at points of die-
. .
00
THEom~ 7.2a- Let C,,, = 2 1 ak 1-11 If b,,, = 0 ( , 9 ~ ) 1 / 9 , ~ ~ .. o(~,)812
then m + l
unifomdy for t in any finite i n t e n d . Let A&) = adze and let
Fm(z) = e a m dm)
If in the fornula
we set 8 = all2 and t = ~ 8 2 ~ we obtain
We will show that
lim Fm(z) = e-812 m+ao
uniformly for 1 z 1 - R for every R < a. Let us first note that
If R is given then we will have I A,(m) 1 - 2 2R if rn is sufficiently lasge. The following inequality is well known
Ilog(l - z ) e ~ e ~ " I 5 2 - I I ~ (1~1s t); here that branch of log z is taken for which log 1 = 0. See Titchmamh [1939; 2461. Thus R being given and m being suEciently large we have
It follows that
I log ~ , ( z ) e * ~ 12 B I bml 8;;;'12 + 2 ~ ~ 2 (~,(m)l-~, m + l
We have thus established (1). Applying Theorem 8.6 of Chapter III we obtain our desired result.
WOREM 7.2b. If in Theorems 6.lb, 6.2b, and 6.3b, we make the a&ditionaZ a a ~ p t i m thd 0, = o(B,)U2, hbnr = o(&)'", and if a(t) = f[a(t+) + ~ ( t - ) ] then the c d h m of these theorem hoId for dZ 31, x g .
We have
By Fatau's lemma and Theorem 7.2a we have
142 REAL INVERSION THEORY
Since
we have
(4') - J ) J , ( i ) dt + G j';*Gm(t) dt = I. -00 -00
The relatione (2), (3), (4), (4') impIy that
0
and our theorem follow^. It iB to be noted tht thie result depends upon information concerning
the m&n of G,(t).
8. FACTORIZATION
8.1. Lef E(8) = 3?,(8)8,(8) where
d let H,(t) and OJt) be the kerneh oorregponding to FJa) and &(a) respectively, fro ~JuL~ @ = a, * 8,. If (p is aufliciently restricted, for example if cp E U(-GO, m), then fibini'e theorem may be applied to show th&t
* y = 0, * (H, * p).
In this section we will prove that if it is merely supposed that * p, is defined, then U, * (H, * p) is defined alao and they are equal. The aonverse is falae; Un * (En * y) may be defined even though (9 * 9 is not. This result, which is related to the material o f 5 3, is needed in later ohaptem.
WOREIM 8.la. Let B(t) E class I , let GJt) and H,(t) be deJIrzec2 as d o v e , and let a(t) be of bouded, variation in every $nib interval -- co < t s t s t a < XI. Iftheintegral
§ 81 PACTORIZATION 143
Let ua suppose, for the sake of defbiteness that a, > 0. If x ia fixed the function e-"lt/@(x - t ) ia non-increming. It follows from Lemma 2.lb that the intepall
A ( t ) = Jlmc-al~da(u)
is convergent. Employing the mean value theorem we find that
It follows from this that
(1) A ( t ) = ~ [ e - % ~ / U ( x - t ) ]
Let t , be arbitrary and t < t,. We have
Given e > 0 we can choose t , so Large aad negative that
for a, b < - t,. Thus ( A(t)G((z - t)ealt I < - E, or since E is arbitraw, e - a
(2) A(t) = o[e-st/O(x - t ) ] t 4 --a.
We have
03
= [ -d( t ) e.ltO(z - t ) ] - 0 0
By (1) and (2) the integrated tenn is zero and
144 REAL INVHRSION THEORY [CH. VI
Here g,(t) k = 1, 9 n is defined as in 86 of Chapter 11. This ha9 been established under the assumption that a, > 0, but it is of aourse true if ol < 0. We may apply this same argument to GI and a,, and then again to Q, and a,, and so on until after n such steps we have
If x, xo are fixed' then E n ( x , - t)lU(x - t ) has at most two changes of trend and is bounded. It follows from Lemma 2.lb that
converges for -m < x < co. Applying the same argument to 8, that we formerly*applied to U we find that
Combining (3) and (4) we have our desired result. THEOREM 8.lb. Let U(t) E clam 11, let OJt) and Hn(t) be dejincd os
&we, and let a(t) be of bounoled varidim in every finite i n t e n d -m < t, 5 t I- tr < CQ. If tlre i-al
converges f' y, < x < 00, then
THXOREM 8.10. Let a(t) € O~WSIII, let an($) and Rn(t) k &$d M
above, and let a(t) he of hounded variation in euey interval F < tl l; - t 4 t l < a. If the integral
These theorems are proved in the same manner aa Theorem 8. la.
SUMMARY
9.1 In this chapter a substantially complete development of the operational inversion theory of the convolution transforms with kernels
haa been effected. It is to be noted that the re~ults proved here are no 1008 precise than those obtained in special cases by particular methods.
Representation Theory
1. INTRODUCTION
1.1. It haa been shown, see Widder [1946; 3101, that necessarg end auBoient conditions for +(x) to be represented as a Laplace transform
with Act) non-demeesing are that
In this chapter we shall see how to associahe such a theorem with each convolution trmsfom
with kernel U(t) ,
If G(t) € class II then, ae we shall show, f (x ) can be represented in the form (1) with a(t) E f for y < x < oo if and only if
If we put E(8) = l/r(l - 8 ) then we obt& as a special case the result cono8rning the Laplaoe transform which we quoted above.
146
8 21 BEHAVIOUR AT INPINITY
2. BEHAVIOUR AT INFINITY
2.1. It is a familiar result that if the Stieltjes tramform
converges, then f(x) = o ( r l )
= o(1)
Similarly if the Laplace transform
converges for some value of x, and therefore for all sufFiciently large x, then
f(x) = o(1) ( 5 3 3-00).
These results are special caaes of e general theorem which plays an import- ant role h the represontation theory.
TFIEIORH~M 2.1. If the tramfm
converge8 fw m e value of x then :
B. G(t) € class I1 or I11 impZim
Let us suppose for the sake of definiteness that U(t) E clesa 11. Let x, be any value for whbh the integral (1) is oonvergent. We write
f(x) = I l ( t 9 4 + I&, x ) where
148 REPREBENTATION THEORY [CH. rn
If z > x, then G(x - t)/G(zo - t ) is non-decreasing as a function of t and we have
by the mean value theorem. It follows from thia that
Similarly since lim B(x - t ) /U(xo - t ) = e%(x - xo) we have t+ m
I,([, x ) = eas(+ - Q(xO - t ) h ( t ) ( I < 6 < 00).
Thus I"
W c O
where
Since s can be made arbitrarily small by taking 5 large our theorem is established for G(t) € class 11. The other cases may be dealt with similarly.
2.2. We require certain elementary Tauberian theorems which will be used in conjunction with the preceding result. See Boas [1937].
THEOREM 2.2. If
2. f ( x ) = o(em) a > O ( X + t -4 ,
Let 9 be any real constant not equal to zero. The identity,
may be verified using integration by parts. Assumption 3 implies that there exists a non-negative consfant A euch that
8 21 BEHAVIOUR AT INFINITY
Using equation (1) we may establish by elementary estimatione that
f ' ( x ) < - o(eda) + OAea(s+B) (s++m, e > o ) , f (x) 2 o(em) + OAem (a+ +a, 8 < 0).
Since 8 may be chosen arbitrarily small these inequalitiee imply that
f ' ( x ) = o(em) ( X --+ +a), w desired.
THEOREM 2.2b. Let Q(D) be a Zineur differential operator of degree
n with conatant coe&ie& Q(D)=q,Dn + q,lD"+l+ - +qo(q,#O). If
then
f@) ( x ) = o(ePZ) (x++oo; k = 1 , - O , n - l ) .
It will be suffioient to prove our theorem for k = 1. For, suppose that it has been established in this case and that we have fr (x) = o ( P ) as x 4 +a. Let Q(l)(D) = q.,Dhl + qn-lDh' + + ql. Since & ( I ) (D)f'(x) = Q( D)f(x) + pd(x) we have Q(1)(D)Jlf(x) 2 O(em) as s + + m . Applying our theorem with k = 1 to f'(z) and Q ~ ( D ) we find that fM(x) = o(em) as x + +a. Proceeding in this way we may show that f ( k ) ( x ) = o(eW) aa x + +a, for k = (2, 3, - , n - 1).
Let us establish our theorem for k = 1. We set
Integrating by parts we may show that
where n ( x ) is a polynomial of degree n - 1. Assumption 2 implies that
F(x) = o(em) On the other hand
hence by assumption 3
F"(x) 2 - O(eW)
I60 REPRESENTATION THEORY [CB. VII
Applying Theorem 2.28 to 8 ( x ) we obtain I " (x ) = o(eca)
But F f ( x ) = (n - l ) l q j ' ( x ) + o(eM)
and so since qn # 0, we have f ' ( x ) = o(eW)
as desired.
3. AN ELEMENTARY REPRESENTATION THEOREM
3.1. Throughout the present chapter we sh11 write
[r' ( - o 0 < t < 1 ) (1) ~ l ( t ) = i (t = 1 )
(1 < .t < a), (2 ) g(aY t ) = 1 a ( q(@* This represents a slight departure &om the notation of previous chapters.
Let {a,) k = 1, - , n be real numbera not zero. We set
(3) M, = 1.u.b. [a,, -a], a, = g.1.b. [a,, +m]. ak<O a,>O
The following simple result lies at the foundation of our representation theory.
! ~ E O B E M 3.1. Let a, and a, Be cZe$d os dove. If
l a . a, > -a, a, < +a, 2a. f(x) €On(-oo < x < a), 38. f(k)(x) = o(ew) (x++oo; k = 0 , 1 , , n - 1 )
= o ( a Z ) (x+-03; k = 0 , 1 , - - - , n - l ) , then
lb. al = -a, a, < +a, 2b. f(z) €Cn(a < x < a), 3b. f ck)(x) = o ( e v ) ( x + + ~ ; k = O , l , - " - , n - l ) y
the%
6 31 AN BLENENTARY REPRBfJENTATION THEOREM 151
2c. f ( x ) € Cn (-00 < z < a),
then
(-oo < x < a).
Suppose that assumptions la, 2a, and 30, are satisfied.. Let us write
Integrating the linear differential equation
we see that 1
There are two possibilities, a, 2 or, or a, 2 eel. We shall consider only the first of these which is typicz. By assumption we have
Since an 2 - a, equations (4) and (5) imply that
this equation may be rewrittan as
Repeating this argument s times we obtain conclusion A. The proofs of conclusions B and C axe entirely similar.
152 REPRESHINTATION THEORY [Ca. VII
4. DETERMINING FUNCTION IN 0
4.1. Let &*(I 2 p < a) denote the class of functions p(t) such that IIpI 1, is finite where
and let L" denote the class of functionsp(t) ~uch that I I p I 1, is finite where
I I p(t) I 1, = essential upper bound 1 q(t) I. - m < t < co
There corresponds to each index p a conjugate index q defined by the equation
1 1 - + - = I . P Q
With each kernel
G(t) = -
we associate a sequence of fmite kernela
Here {b,); is any sequence of real numbers suoh that lim 6, = 0. WQJ
L E ~ U 4.1. If G(t) and H,(t) &re defined as above, then
It was shorn in 8 8 of Chapter I11 that
It is now easy to see that
lim 1 1 Q(t) - H,(t) 1 = 0. n+co
5 41 DETERMINING PUNCTION IN L*
we have I I Q(t) - 11, = 0.
woo
l?or a demonstration of the following well known result see Banach [1932; 1301.
THEOREM 4.1. Let 1 < p < oo a d let cpn(t) E L* (-a, oo) n = 1,2, . If t h e ezists a-collstant M independent of a such that
then there &ts a subsequence nl < ne < ns < * and a function q(t) E LP, I I p) 1 ID 5; M, mch. tihat for every y ( t ) E Lq we hiwe
The functions h l ( t ) , tp%(t), are said to converge weakly to tp(t) in A*. 4.2. We are now in a position to establish our first representation
theorem. THEOREM 4.2a. Necursay and s u f l c h t cond i t im that
with 1 1 ~ ( t ) 1 ID 5 MY 1 < P 1 0 0 9 are:
The necessity of A ia obvious. To establish the necessity of condition B we write
roo
P*(D)f (4 = J a,(% - t)pl(t) at, -a
m
= 1- ;un(x - t ) l l h [ ~ . ( x -- t)lllgy(c) at.
By Holder's inequality
1 P n ( D ) f ( x ) I P 6 [ / - *m~n(x - 1) dt]: /:a~(m~ - t) / pl(t) . at,
BBPREBENTATION THEORY
We now turn to the sufficiency of our conditions. A simple application of Hijlder 's inequality yields
which implies that
(1)
where G is a constant and since by Holder's inequality
The relatione (1) and (2) together with Theorem 2.2b imply tbat
for k = 0, 1, , a - 2, or since n ie arbitrary for all k. (If a, = --a or a, = + oo the corresponding relations are to be interpreted vacuously .) By Theorem 3.1 we have
s 8inw TT (1 - E) eDkf(z) E Ls by B thia iterated integral is rbaolutelg
1
84 DETERMINING FUNUTION IN Lo 156
convergent and may be inverted to give, after an adjustment of the
for n = 1, 2, . Let P,(D)f(t) =vn(t). By B and Theorem 4.1 there exists a set of indices n, ( n, < n3 < - and a function q(t)
with I 1 p, 1 1, M , ~ u c h that rp,%(t), qna(t), converge weakly to p(t) in Lp. Eor fixed x G(x - t) belongs to 1;Q and hence
By H6lder's inequality
Lemma 4.1 implies that
Combining relations (4) and (5 ) we obtain our theorem. It is c l w that the preceding theorem would remain true if the con-
ditions B were replaced by the weaker conditions
A alight generalization of the preaeding reeult is given in the following theorem.
%OREM 4.2b. New8ay a d aumient conditim that
186 REPRESENTATION THEORY
The neoessity of oondition A is obvious. We have
Using Holder's inequality we obtain as before
Thus the necessity of condition B i~ established. The proof of the sufficiency goes exactly as before.
5. DETERMINING FUNCTIONS OF BOUNDED TOTAL VARIATION
5.1. The following theorem is essentially demomfrated in Widder [1946; 261.
THEOREM 5.1. If P,(t), n = 1, 2, , aye uniformly bo21whd and of total variation lesd thun or e p u a l to M, then there exists a aet of indices rt, < n, < n, < , a fzcwtion j(t) o f total variation not exceeding M , and two camtanti3 B, and B, m h thut i f p(t) i s any Minuow fumtim of t for which y(+a) and q(-m) e x i 8 t then I
TEEOREX 5.2a. Necessary a d mficient d i t i m that
f(4 = /ImoO~(x - 1) dB(t)
with p(t) of tot& variation not exceeding M are:
A. f ( x ) € C" (-00 < x < a); B* 1 1 Pn(Qf (XI 1 11 5 M m = 0 , 1 , a . o .
8 51 FUNCTIONS OF BOUNDED TOTAL VARIATION 267
It is clear that condition A is necessary. We have
Since the integrand is non-negative the order of integrations may be inverted to give
which establishes the necessity of condition B. To establish the sufficiency of our conditions we begin by showing,
just as in the proof of Theorem 4.2a, that
By Theorem 3.1 we have -
where r x
Condition B implies that the B,(x) are uniformly bounded and of total variation not exceeding M. By Theorem 5.1 there exists P(t) of total variation not exceeding M such that
for any continuous function (p(f) vanishing at &aco. Since for each z U ( x - t ) is a continuous function of t and vanishing at & co we have
168 RBPRESBNTATION THEORY [CH. V I I
Now 00
Xlemrtm 4.1 implies that P W
Equations (1) and (2) yield our theorem. Theorem 6.2e would remain true if oonditions B were replaced by the
wea3rer oonditions
TEIEOREM 6.2b. Neemay d eujtficient & i t h thut
w h ~ e a, < o < ap a d where B(t) i s of tola2 variation not exceeai~ng M are:
The demonstration of this result k left to the reader.
6. DETERMINING FUNCTION NON-DECREASING
6.1. We must here vary our argument as Q(t) belongs to oleas I, 11, or III. . We begin with G(t) E class I.
THE OR^ 6.1. Let Q(t) E: class I. Nece88a y and m@ient caditions that
where p(t) € f are:
8 61 DETERMINING FUNCTION NON-DBCREASING 159
The neoessity of condition A is evident and the necessity of condition B follom from Theorem 2.1. The equation
and the fact that @(L) E f implies that
which establishes the necessity of condition C. We now turn to the suffioiency of our conditions. Conditions B and
C together with Theorem 2.2b imply that
f 'k) (x) = o(e'-) x-, +a,
for every k;. Using Theorem 3.1 we obtain
?&
Since (1 - :) f (z) 2 0, this iterated integral is absolutely 1
convergent and may be inverted to give, after an adjustment of the translations,
I(.) = J'->.(X - t )P,&(D)f(t) dt-
Take any x', -a < st < co. We may rewrite the above formula ea
where
yn(x, t ) = H, ( X - t)/Hn(z' - t ) . Let us set
Y ( X , t ) = B(X - t ) / a ( x t - t ) .
Becauae o fC qn(t) f t , n = 1,2, , and since
I60 RDPREBXNTATION THEORY [CH. V I l
it follow tbat the p,(t) are uniformly bounded and of uniformly bounded total variation. For each x, p(x, t) is a continuous function o f t and
It follows &om Theorem 6.1 that there exist indices n1 < a, < n, < I
a function q(t) of bounded total variation, and oomtante gl and pa suah that
If n is sufficiently large then y,(x, t ) is a continuous function of t and
By Lemma 4.1 1 . h y,(z, t ) = yJ(x, t )
W a O
uniformly for t in every finite interval. The functions ~ ( x , t ) , y,,(x, t ) , n = 1,2, - , are non-deoreasing if x - 2 x' and non-increasing if x 2 x'. - It follows that for each x
lim I I v ( x , $1 - yn(x9 t ) 11, = 0, W C O
We have
and from this we obtain
(2) i4 00
Combining relations (1) and (2) we obtain
where
By Theorem 2.1
8 61 DETERMINING PUNCTION NON-DEOREASING 161
Making use of tohis and condition B we see that pp, = p, = 0, so that we have
Theorem 6.1 remains true if condition C is replaced by the weaker condition
THEOREM 6.2. ]Let G(t) € class 11. Nemsary and suficient conditions that
where B(t) E t are:
The necessity of our conditions may be established just as in the proof of the preceding theorem. In order to establish their sufticiency we show precisely as before that
for every k. There i s of course no corresponding result as x + -00. By Theorem 3.1
A
for z > y,. Since the functions g(al, z), , g(o., z), ( l - :) d'laY(z) 1
are non-negative the order of the integrations may be inverted to give
f ( 4 = J -00
a n ( % - t )P , (D) f ( t ) > Y e -
Choose any z' > yo; for x > x' we have
162 %BPRESdCNl'ATION THEORY
where
yn(a t ) = Hn(x - t)/Hn(zt - t ) , Here 0/0 is dehed es 0. The functions y,(q t) are continuous and non- demeaahg, and for n sufficiently large
'/',(XI -00) = 0, yn(x, +a) = e%(*-R. If we set
Y ( X , t ) = G(X - t) la(xf - t ) , then y(x, t) is continuous and non-decming sad
~ ( z , -00) = 0, y(x, +a) = e.l(s-d). Further y,(x, t ) converges to y(x, t) as n + a, d o r m l y in every finite interval by Lemma 4.1. It is now esay to deduce that
lim 1 1 ~ ( z , t ) - yn(x) t ) 1 loo = 09 -03
The proof fkom this point on can be completed as before. Theorem 6.2 is still true if we replace condition C by condition
THEOREM 6.3. Let U(t) E class 111. New8ay a d mmient d i t h that
f(x) = j"*ao(x - t ) #(t) (z > T + + 2 ') 1 a,
where D(t) E f (T < t < a) are:
The proof of this theorem is left to the reader. In Theorem 6.3 condition C could be replaced by the weaker condition
Cl- PnJDlf (4 2 0
6.4. As we remarked-in 8 1, the fa&r Hausdorff-Bernstein-Widder theorem is a particular caae of the results of the present seotion. It i s (3urious that the demonstration given here does not spechlize to any of the ]mom proofs of this theorem.
BBPRESENTATIONS OF PRODUCTS
7. REPRESENTATIONS OF PRODUCTS
7.1, Let
where we assume of courae that Za;2 < a, za&B < co (j = 1, , a), and let
eat R,(t) = - S'" - th (j = 1, . , a).
2& --w Ej(8)
We wish to @ve conditions which wi l l insure that if, for example,
with /?, E ' t (j = 1, , n), and f (4 = f1(x)f*(x) - -fn(z)
then f (x) will be repmentable in the form roo
J - w with B(t) E 1. . In order to do thia we rewrite the produots E(a), E,(s) in the form
* " ~ A , , - , ~ A ,,-I < O < A , , , ~ A , , g * - - a
If necessary A& and A,, may be +a or -a. D o 1 We write G(t) N [Hl(t) , , H,(f)] if
7b
2 4 3 ", s Av j-1
n
for v j 2 l(j = 1, = , n), z v j 2 v f n - l s a n d i f j-1
164 REPREBENTATION THEORY
LEU 7-1- If u(t) [Hl(t), , If,($)] and if
then
4 where g - 2 0, q 2 - 0 and the summation extends over all values of the indices p,, ,pny B ~ Y , qnforw&chpj> - O,qj< O ( j = 1, -, tq), - P I + P I S . * ~ ~ + P ~ S P , ~ ~ + q ~ + * * * + ~ ; > q . - Moreover, we have
A@. ~ 1 s Pss 9 P q, q,, , p n ) 2 0, ~ A ( P , PI, Pas ' 9 P n ; q, q ~ s 9 qn) = 1,
the range of indices in the summation being as above. This may be established by induotion making use of the identities
(q1 - 1) + + (qn - 1) g (p - 1) - n + 1, 80 that by our aseumptions the coefficients in equations (1) and (2) are non-negative.
B 71 REPRESENTATIONS OP PRODUCTS , 165
where q(t) E Lr (-03, GO). More precisely we have
l l l l l l l l l l l l I ' ~ n I I r n * Let f (x) = f,(x) f ,(x) fa(.). By Theorem 4.2a we have for every
P ~ Z 0, (I& 0
By Lemma 7.1 we have
Applying Minkowski's inequality we obtain
- .. The following inequality is well known
IIQ~(x) g n ( ~ ) 1 I,$ 1 I * I I g a ILn' see Titohmarsh [1939; 3941. Thus
Making use of (3) we find that
Here we have used Lemma 7.1 again. The inequality (4) together with Theorem 4.2a yields our desired result.
7.2. THEOREM 7.2a. Let U ( t ) E olass I and let U( t ) - [H,(t), , Hn(t)] .
166 REPRBrSENTATION THEORY [CH. VII
where &(t) E t (j = 1, , n) tAen
where p(t) E f . I t is easily verified that U(t ) E class I implies H,(t) € clsas I ,
j = l s a g e n. By Theorem 6.1 we have
f r(x) = 0(eaj@)
f ,(x) = o (eajl')
for j = 1, , n whem
an = 1.u.b. (a,, -m), %kc 0
4 a = g.1.b. (a,,, + 00). 4 h > O . ~
since a,, = Ajs-l, ~ $ 1 - Ah+,? and since a, = A_,, as = A+1 our assumptiom imply that
It ~O]~OWEI that i f f ( ~ ) = fi(x)f2(x) f,(z) then
f (x) = o(e".3
f (3) = o (eaix)
For every p, 2 - 0, p, 2 0 - we have
Using Lemma 7.1 we see h t
(-a < x < m).
Appealing again to Theorem 6.1 we obtain our desired result. O R 7 2 Lct U(t) € clasa 11 or III and let G(t) - [E (t), . . .
#
If
where p(t) 6 1.. Here T = ma,x (T,, T,, , Tn).
0 rl RXPRESENTATIONS OF PRODUCTS 167
The demonstration of this result is exactly like that of the preceding theorem except that we Theorems 6.2 and 6.3 in place of Theorem 6.1.
7.3. We note that if 0 < 1, < 1, j = 1, , f i , and if rll + r Z I + + A, < - 1 then our =sumptiom will be satisfied if (f** any4)) we set H,(t) = I ,G(A,t). Since if
then
we obtain as corollaries the following results. Ta~oasad7.3a. L e t ~ j > O , j = l , ~ - o , n , ~ , + t , + = - = + A ~ ~ l .
If
J -00
where p(t) € L" (--a, m). More precisely we have
T I I E o R ~ 7.3b. Let O(t) € claae I and let 1, > 0, j = 1, - , n, A , + A , + --• +A, ,2 ,%1. If
where bj(t) E f j = 1, - , n, then
where p(t) E f' , and X = may [T JA,, T , / l , , Ta/A,].
168 REPRBSENTATION TEEORY
7.4. As an example let
U(t) = - r ($2
where ,u > 0 and let
w h e r e p J > 0 , j = 1 , * * - , n. See 5 9 of 111. Here we have
The conditions of Definition 7.1 are satisfied if p = p, + p, $ + p,. After a logarithmic change of variables we find that if
then
Ae a second example we set
for j= 1, , n. See $ 9 of 111. Here we have
A,, = k (k > O), A,,=-co (k<O).
The conditions of Defmition 7.1 are satisfied. Mter a logarithmic change of variables we find that if
then
This is a familiar result and may be proved directly. The present methods although not direct are quite general. See also H. Pollard [1946a].
CHAPTER VIII
The Weierstrass Transform
I. INTRODUCTION
1.1. In Chapter I11 we were led in a natural way to see that the moet general invemion function E(s) commensurate with our method0 should be a function of class I1 (P61ya-Laguerre) :
We have thus far treated only wch funotions for which o = 0 and b v e proved that E (D) b indeed an inversion operator for a suitable convolution tirandorm, namely fhat for which the kernel hae a bilateral Lapleoe transform equal to l /E(e) . In the present chapter we shall t r a t the complementary a s e B(s) = em&, o > 0, the one factor of (1) hitherto neglected. Since this fanction haa no roots an interpretahion of E(D) analogous to that wed in Chapter III is denied us, and aa a ooneequence the earlier methods must be modified here.
As a baais of attack consider the known Lapleoe trenaform
Thie shows that if i ( s ) is the special fwaction e-02, then ite reciprooal is still the bilateral Laplace franaform of a hquency function
If we adopt this fanotion as the kernel of a convolution transform,
we may hope, from previous experienoe, that the transform will be inverted by the operator e - a p if properly inferpreted. This hope may be further strengthened by the following formal oonsiderations.
I70
In view of equation (2) it is natural to interpret eoDaQ,(z) as
or by equation (4)
eO'(p(x) = f ( x ) .
Then if D is treated as a number we obtain the symbolio equation
the predicted inversion formula. Observe that if x is replaced by 16% and y by 4Fy in equation (4)
it becomes
That is, f(l/cCz) is the convolution tramform of t p ( d Y ) hving kernel equal to the function (3) with c = 1. Hence there ia no loss in generality in supposing c = 1. This we do throughout the remainder of the chapter. Following an aocepted terminology, E. Hille [1948; 371 1, we refer to the resulting transform aa the Weierstrms transform.
The chief purpose of the present ohepter will be to interpret and prove the inversion formula (5) and then to use it for the purpose of developing representation theory. That is, we seek necessary and sdc ien t con- ditions upon a funotion f (x ) in order that it m y be the Weiemtraas transform of a funotion p, of preaoribed class (such ae (p 6 LP). It will develop that ewtp(x) is e solution of the heat equation
and much use will be made of this fact. M.a,ny results, of intareat in themselves, will be proved about suah solutions. In particulair, we wi l l obtain a chmac~rization of solutions which are positive in a half-plane, t > 0. Th% will be analogous to a c1aasioa;l theorem of Herglotz concerning positive harmonic functiona.
1.2. Since the Laplace trsnsform 1.1 (2) is no longer valid when o < 0 it does not provide us with an interpretation of edD'. However, the complex inversion of that tranafonn, D. V. Widder [lW; 2411,
173 THE WXIERr9TRASS TRANSPORM [Cs. VIII
~uggeata an a;ltsm~tive procedure. Replace o by 0 < t - < l in 1.1 (2) and invert:
We me thus led to write
eWiDtf(Z) = - J K(8, t)dDf(z) ds h* a-ico
By a change of variable this becomes
where the constant d is arbitrary and will be chosen ao that the vmtical line cr = d will be one on which f ( s ) is defined. It will be soen later that it is not appropriate to define e-D' by setting t = 1 in (l), for the integral (1) would then diverge for certain Weierstrass transforms f(x). We: chooso rather to set
and this operator will serve to invert evexy convergent Weierstrass tramsform. If we set
k(x, t ) = (4mt)-11ee-Z'la 9
the above statement becomes
1.3. In order that the reader may see clearly that the methods of tho present chapter are in essence the same as those employed earlier, wo set forth here in juxtaposition the corresponding functions and operations of Chapters 111 and Vm. The chief point of contigst is that the inversion operahor of the former
0 21 THE WEIERSTBABS TRANSFORM 173
involves the disorete parameter n, whereas 1.2 (2) involves the continuous parameter t. The following table will bring out the analogies in detail.
lim P,(8) = B(a) n+cO
ps+l(m 8. -- P O ) Qn(x) = Q n + 1 ( 4
lirn I?&) = B(s) t-+l-
4. = E(4lPt(8)
7 . lim k(x, 1 - t ) = 6(x) 1 t-1-
These analogues will become abundantly clear as the analysis of the chapter progressee. I n 7 the Dirao 8-function is intended. To establish 8 from 6 in the continuous case, differentiate with respect to t
a aa at k ( ~ , 1 - t ) = - --
ax2 k(x , 1 - t ) .
2. THE WEIERSTRASS TRANSFORM
2.1. We now make our formal definition of the Weierstrass transform. DDFINITION 2.Xa.
k(s, t ) = ( k t ) -112e-z'#t O < t < m, -00 < x < (23. This is the familiar "sourco ~olution" of the heat equation, and B(x, 1) will bo the kernel of our transform.
174 THB WEIERSTRAS+9 TRANSFORM [Ox, VIII
D m m m o ~ 2.lb. The Weierstrass transform of a function p(x) is the function
whenever the integral oonverges. DIU~NITION 2.1 c. The Weieratxasa-Btieltjes transform of a function
a ( x ) is the funotion
whenever the integral converges. Here ~ ( y ) i a of bounded variation in every finite interval. If, for
example, a(y) is constant exoept for a, single unit jump at the origin, f(s) = (P.) -ll2e-$l4.
We turn next to a theorem of Abelian character which fl prove useful t o us later. The proof requires an elemenhry lemma.
LEMMA 2.1. If a(%) is of bounded variation in a < x 5 R for every - - R > 0, a(m) exists, p ( x ) ie positive, continuous and non-increasing in a<x<ao,then -
For if E is given we can determine R so that
2 d ( a ) This proves the lemma.
THEOREM 2.1. If the integral
8 21 THE WEIERSTRASS TRANSFORM 176
wnverges to o value A when x = xo, t = 1, then it &o wnverges for - m < x < o o , O < t < l , a n d
For fixed x, x,, t set
Then
This integral converges, by Lemma 2.1, shoe p(y) is decreasing for large y and a*(m) exists by hypothe~is. If we replace y by -y we may prove by use of the result just proved that
converges. Finally, note that
The integral on the right is a Laplace integral which oonverges at t = 1 by hypothesis. Since it represents a continuous function oft in 0 < t 2 - 1, D. V. Widder [1946; 661, the theorem is proved.
2.2. Let us investigate next the relation of the tranaforrn 2.1 (1) to the Laplace transform. It may eaaily be put into the following form
and this is clearly e bilateral Laplace integral. Hence we may infer many of the properties of the Weierstrass transform from known theory, D. V. Widder [1946; 237). For example, the region of convergence is an interval, as noted above. Or if s is replaced by the complex variable a = + i ~ , the region of convergence is a vertical strip of the complex plane. Also we may deduce at once a complex invemion of the Weierstrass transform.
I76 THB W E I 1 S T R A S S TRANBPORIM [CIL. VIII
THEOREM 2.2. If q ( x ) is of bounded variation in a neigMmluK,d of a point xo, 76a8 th t r a n $ m f(x) defined by 2.1 (I), and
Thia is an immediate consequence of Theorem Sa, D. V. Widder p946; 2411 applied to equation (1). InequaJity (2) asserts that the integral 2.1 (1) converges absolutely at- the origin and hence on the line cr = 0. Iff(%) is the tramform of pl(y), then f(x - o) is the transform of (p(y - C)
for any constant o. Hence if 2.1 (1) is known to be a,bsolutely convergent at some other point than the origin, Theorem 2.2 may still be made to apply by a, tramslation.
COBOLLARY 2.2. If (p(x) satisfies the conditions of Theorem 2.2 and if for some positive constant c
then
This corollary follows from the theorem by s eimple change of variable. As an example of the theorem consider the pair f ( x ) = ~ ( x ) = 1. Equation (3) becomes
The Cauchy value is not needed since this integral converges absolutely for all x. It reduces to
or by endytic continuation t o
We thua have a proof of the formula (2) 5 1, there ass~uned. In par- ticular, this result give8 the following familiar integral expression for the source solution
8 $1 THE WEIR8TRASS TRANSF0BM 177
2.3. W e shall need a few simple properties of the s o m e solution. We collect them in THEOREM 2.3. If k(x, t ) id the function of DeJinition 2.la, thm
Here A is a suitable constant and s = a + i ~ . Conclusion A follows by differentiation of the equation
1 xa log k ( ~ , t ) = - - log ( h t ) - -
2 4t' Conclusions B and C follow directly from the definition of k(x, t).
2.4. Another property of k(x , t ) that will be essential to us is contained in
THEOREM 2.4. If k(x, t ) is the function of De$nit ia 2.18, and i f 0 < t,, 0 < t,, -03 < Z < CO, then.
For, the integral ia equal to
where
Sinco the integral appearing in (1) ia equal to &/A, the deaired result follo WR.
2.5. A companion rosult, useful when the variable x in k(x, t ) is complex i~ tho following.
TJ~EORXM 2.5. If k(x , t ) i s the function of DejEnition 2.111, and y O < t , < t , , - a , < x < c o , - m <v<co,tfien
To prow this note first that by Theorem 2.4
k(z - y, t $ o - V , t2 - t,) dy = E ( x - V , t,).
Now apply Corollary 2.2 to this equation with
178 TEE WEIER~~TRASS TRANSFORM [Ca. VIII
The reault is (1). Hypothesis 2.2 (2) is evidently satisfied trivially here sin- for fixed t,, t , and v
k(y - 3, t , - t,) = O(1) I Y I+ a* 2.6. We conclude this section by a brief table of Weierstrass trans-
forms. It wiIl serve the double purpose of illustrating the t y p e s of functions which can be Weiestrtlss transforms and of providing conorate enmples as checka for our later theory.
For all of these paira except 9 the integral (1) converges in -GO < x < Q). For 9 the internal of convergeme is -1 < x < 1. Peirs 1, 2, 3 am apecid case8 of 4, and 6, 7 an, inoluded in 8 (easily proved by (2) 0 1). The pair 0 results at o m from the equation
which may be verified by direct integration or by use of the funotion g(t), 1 6.1, Chapters. E the formula
8 31 THE IN VERSION OPERATOR 179
is expanded, the pair 6 is proved. Finally, to establish 4 we must ahow t h t
where H,(x) is the Hermite polynomial, defined by the equation
Uaing (3) in the integral (2), the latter equation reduces, after i n b p t i o n by parts, to
This proves the desired result.
3. THE INVERSION OPERATOR
3.1. We have already indicated in 5 1.2 the heuristic considerations which lead us to the following definition.
DDFINITION 3.1a.
(1) K(8, t) = (~/t)'l~e.'l~ = 2nk(ia, t).
D~xmmoa 3.lb. The operator e - 3 % ) ie defined to be
whenever the integral converges for 0 < t < 1 and the limit exists. At first sight thia operator seems to depend on d, but for all functions
f(z) to which we shall apply it the result will be independent of d by virtue of Cauchy's integral theorem.
As an example, take f (x) = ee-0'1'. The intepal (2) diverges for all x when t = 1 but converges for all x when t < 1. Simple computations show that the limit (2) is zero when x # 0, fails to exist when x = 0.
Note that ififf(8) i8 defined on the imaginerg axis, 0 = 0, the 'conahnt d may be chosen equal to zero and e-tDy(x) takes the form
180 THE WEIERSTRASS TRANSFORM [Crr. VI=
3.2. Let us introduce e notation for a clam of functions to edD' will be applicable and to which aJl Weierstrsss transforms will belong.
DEFXNITION 3.2. A function f(x) belongs to class A in an interval a < x < b if and only if it can be extended analytically into the complex plane in such a way thitt -
1. f(x + iy ) is analytic in the strip a < x < b 2. f ( x + iy) = o ( l yleY'"), 1 y l+ CO, uniformly in every
subinterval of a < x < b. For emmple, e-* E A in -03 < x < m. That every Weierstrass
transform convergent in a < x < b belongs to A in that interval folio ws from equation 2.2 (1) and t-he known order of Laplace tramforms on vertical lines, D. V. Widder [1946; 921. If f ( x ) E A m -00 < x < 03 a n o h usefd form of e - D y ( ~ } is available.
T H ~ ~ O R E M 3.2. Iff(%) € A in -a, < x < a, then OD
(1) e-tDaf(x) = j'- ak(Y, t)f(x + iy) dy,
the integral converging ab8oludely for -oo < x < oo, 0 < t < 1. Since the function
e'"-"alGf(z) -m<a<oo, O < t < 1, is an entire function of x, we see by Cauchy's integral theorem that the integrals of this function along the two vertical lines x = ;I and x = d (any d) are equd if
But this is evident by assumption 2 of Definition 3.2. Hence
dSi m
2 ~ i d-im Replacing I by x and referring to 3.1 (2) we obtain our result. Note that this theorem cob our earlier remark that e-='f(x) will generally be independent of the constant d appearing in Definition 3.lb.
COBOUY 3.2. I f f ( % ) E Bin -00 < x < GO, then
J -co where
" cos yDf(x) = 1 - f ( * ) ( x ) Y B L . k-0 (2k)l
For, by Taylor's theorem
6 31 THH INVERSION OPERATOR 181
The imaginary part being an odd funotion of y disappears when substi- tuted in the integral (1) so that the corollary is proved. This latter form of the operator puts into evidence the fact that e-tD'f(x) is real when f ( x ) is real.
3.3. We show next that dDapl(x) is a solution of the heat equation when it is defined. Since we shall have frequent use for such solutions let us introduce an abbreviation.
DEFINITION 3.3. A function u ( x , t ) E H in a domain D if and only if u ( x , t ) E C2 and u,(x, t ) = zl,(x, t ) there; u ( x , t ) E dl in a region R (perhaps closed) if R can be enclosed in a domain in which u ( x , t ) € H.
For example, u(x, t) E X in xa + t P < - 1 implies u(x , t ) € H in s2 + tg < pa for some p > 1.
THEGEM 3.3. ~f
the integral converging in the atrip 0 < t < o, then, u ( x , t ) E H there. For fixed t this integral is the product of the entire function e-d14t
by a Laplace transform whiah converges for all x. Hence u ( x , t ) is an entire funotion of x, and differentiation under the integral sign is per- mitted, D. V. Widder [1946; 671. Hence
To prove that
(2)
it will be sufficient to show that for any fixed x the integral (2) is uniformly convergent for B 2 - t I; - c - 8, where 8 is an arbitrary positive number less than c. The integral (2) is the sum of two others corresponding to the intervals of integration (0, a) and ( -oo,O). By Theorem 2.3 the h t is
Since the latter integral is the sum of two Laplace i n t e p k (in l / t ) we may appeal to known theory, D. V. Widder [1946; 641 to verify the desired uniform convergence. The integral over the range (--a, 0 ) is treated ~imilasly. Another appeal to Theorem 2.3 shows that the integrals (1) and (2) are equal, and the proof is complete.
182 THE WEIER8TRASS TRAHSFORdd [CH. mu
COROLLARY 3.3. If the Weierstrass transform of ~ ( z ) converges for some x? then dDav(x) E H in the strip 0 < t < 1.
By hypothesis the integral eDaV(x) will converge for some z,. By Theorem 2.1 the integral etpq(x) will oonverge for all x, 0 < t < 1 and Theorem 3.3 is applicable.
3.4. We prove the following companion result. momma 3.4. If f(a + i y ) = o( 1 yl e*'14), I Y I+ co,for sonzcfi~mber
d, then the furactim
a(%, t ) = e-(l -+D(f(Z) = - 9 ( 8 -- X , 1 - q f ( 8 ) ds
wiZZbelongto H i n t h e ~ t + O < t < 1. By the defjnition of K(8, t ) and by Theorem 2.3 we have
provided that differentiation under the integral sign is valid. We show that for every B > 0 the integral ( 1 ) is uniformly oonvergent in I x I 5 23, b - t 2 - 1 - 8, 0 < d < 1. By C of Theorem 2.3 the following be a dominant integral independent of x and t in that region
Thia integral converges since 0 < c9 < 1, and the proof is complete.
4. INVERSION
4.1. Let us ahow in this section how the operator eWD' inverts the Weierstrass transform in the apecid cwe in which (p(x) &I bounded and continuons. The aeaentiala of the method will thus be put clearly into evidence in the, absenoe of the technical diEcultiea which more general hypotheses will introduce into later work.
O M 4.1. Ifcp(y) is bounded and oontinuozce in --co < y < a, and ij?
o 51 UNIQUBNESS THEOREM
By Definition 3.lb, with d = 0,
e-D8 f(r) = lim J - kc9 + i x , t ) f ( i y ) d ~ k l - -a
03 ca
= lim I-:(y + i x , t ) tiy J- / ( iY - z, 1)92(z) dz. t-tl -
By B of Theorem 2.3 and the boundedness of q ( z ) Fubini's theorem is applicable, so that
03 03
e-Ds f ( x ) = lim 1 Dv(z)dz 1- :(y + ix , f )k ( iy - z, 1) dy t-1-
k(x -- z , 1 - t)tp(z) dz. t-41-
We have here used Theorem 2.5. Finally, by a change of variable 1 0 0
(1) e- Da f ( x ) = lim z e-#tp(~ + 2241 - t ) a ~ . 1+1-
Now we may use Lebesgue's limit theorem, since the integrrtnd is dominated by a constant multiple of the integrable funotion em#. Hence the limit (1) is ~ ( x ) , and the theorem is proved.
5. TYCHONOFF'S UNIQUENESS THEOREM
5.1. We have seen that etDgq(z) and e-(l-t)D' f ( x ) satisfy the heat equation. To capiblize on this fact we need to know a few elernentaxy facts about solutions of that equation. In particulax we ahall need to know to what extent a solution ~ ( x , t ) is uniquely determined at later times by its values at a given time t . One form of this uniqueness theorem is due to A. Tychonoff [1935; 1991 and is the form prmented here.
5.2. We show h t that a function of olass H, like a, haxmonio function, casnot take on its minimum (or maximum) value in the interior of a,
region where it belongs to H. We need in fact a slightly more general result. We shall not asume that u(x, t) belongs to H on the boundary of our region nor that it approaches a limit as we approach the boundary. We shell -thus be able to apply the theorem to functions like the source solution k(x, t ) which approaches no limit as (x , t) approaches the origin. We will need to consider only rectangular regions.
To save writing let' us introduce the following notation. Denote by D the set of points ( x , t) for which I x I < R, 0 < t o. That is, it is the interior of a rectangle plus its upper boundary. shall be the closure of D, the interior of the rectangle plus all. of its boundary. And B shall be fi - D, the lower side and the two vertical sidea. The top points of the vertical sides form a part of B.
184 THE WEIERSTRASS TRANSFORM
1. u(x , t ) El? D
In hypothesis 2 it is aclaumed that (x, t ) + (x,, to) through points of D. As mentioned above, the theorem avoids an assumption about the existence of a limit on B. By the debition of "lim," for each E > 0 and each (z,, to) of B there is a 8, such that u(x, t)> -E at all points (2, t ) of D within a distance do of (x,, to). Since B is a closed set we may use the Heine-Bore1 theorem to show that there is rim of points Bd in D aJ1 a distance <8 from B where u(x, t ) > -e.
Let us now make a n assumption contrary to (l),
and deduce a contradiction. Xorm the equation
where T is a positive number t o be determined so that v(x, t ) will take on a minimum in D. Whatever it is, v ( q , t,) = -1. Hence we need only determine r so thak v(x, t) > -I in Bd. But there
Hence if we choose r so that rt, < l and then E < I - rt,, we have v(z, t ) > - I in Bdr w desired. Hence V(X, t ) must take on a, minimum in D, in fact in D - B8, say at (x,, t,). But at such an interior minimum ve(x2, t l ) = 0, vt(xg, tg ) 0 (the inequality being possible if t , = 6) and V @ ~ ( X ~ , t l ) 2 0. That is, vaa(x,, t,) - v,(x,, t,) 2 0. But since u(x , t ) E H, equation (2) gives v,, - u, = -r < 0 for all (x, t ) in D, and the desired contradiction is at hand.
Of course it follows from this theorem that if u(x , t ) E H in d, then it has a minimum value for all points of b whioh is taken on at a point of B.
5.3. We can now prove the useful theorem of Tychonoff, OR EM 5.3. If
2. lim u(x, t ) = 0 for aU x,, -m < xo < a, 3
3. f ( x ) = max I u(x, t ) ( O < t b c
4. f(x) = 0(ed), I x I + a, for some a, thtn u(x, t ) = 0 throq#wtd the strip 0 < t < C. -
where k(x, t ) is the source solution and R ia an arbitrary constant >O. Using the notation of $6.2, let us show that the two functions (4nc)'l2~,(x, t ) & u(x, t ) sat- the hypotheses of the previoue theorem. They belong to H in D and as (x, t ) approaches the points of B on the x-axis
lim - ( ~ C ) ~ / ~ U ~ ( X , t ) f u ( x , t ) - 2 lim r ( x , t ) = 0, a iz?
- B 2 - xo < - R. Next consider the vertical sides of B. We have
U,(f 3, t ) Zf (f R)k(O, t ) - 2 f(f R) (kc)-lb
for 0 < t c. By the definition off(& R),
I u ( f R, t ) I Sf(& R) s (kc ) ' l 2U , (&~ , t ) .
This shows that the two functions in question are 2 0 on the vertical sides of B. By Theorem 6.2 they must be - 2 0 throughout D,
-(4rro)'l2u,(x, t ) 5 - u(x, t ) 2 - (4nc)l"ua(z, t ) .
Now hold (x, t) fixed and allow R to beoome infinite. By hypothesis 4 UR(x, t ) will tend to z&o if t < 1/(4a). I f 4ac < 1 the proof is oomplete. Otherwise, repeat the above argument with u(x, t) rephced by U(X, t + 1/(4a)), and eto forth.
It is important to note that the approach of (x , t ) to (xo, 0) in hypothesis 2 is a two-dimensional one. The theorem would be false if that hypothesis r e d u(x, 0 + ) = 0, as the example k(x, t )x / t shows. Thie function belongs to H for t > 0 and approaches zero along every vertical h e aa t + Of . But it is not i d e n t i d y zero!
6. THE WEIERSTFUSS T M S F O R M OF BOUNDED FUNCTIONS
6.1. We can now beady characterize those functions which are Weierstr~s transforms of bounded functions. Here again, as in 5 4, the technical difficulties whiah will confront us for other olasses are laoking, so that the essence of the method will not be clouded by detail.
6.2. We need next result oonceming solutions of the heat equation. LBMMA 6.2. If
186 THE WEIERSTRASX TRANSFORM [CH. VIII
Denote this integral, which converges for all x and all positive t, by v(x, t + 6). It will be enough to show that v - u satisfies the hypotheses of Theorem 6.3. As in $4.1
and since u(x, t) is bounded, Lebesgue's limit theorem enables us to show that v(x, t + 6) + ~ ( x , , a) as (x, t) + (xO, 0+). By continuity u(x, t + 8) approaches the same value. Both functions belong t o H and both satisfg- condition 4 of Theorem 5.3 (a = 0) since they are bounded. Hence v - u is identically zero, as stated.
THEOREM 6.2. CMUIdtiom 1 and 2 of Lemma 6.2 are 1zecesr8ary and wficient that
- o o < x < m , O < t < c ,
where I < M , -a < y < oo* f We prove first the necessity, assuming the re$'rasentation (1). That
condition 1 is satisfied we see by Theorem 3.3. Condition 2 follows from equation (2), 5 1.1. In fact, both conditions are satisfied in the half-plane O < t < o o .
In proving the converse we begin by use of Lemma 6.2. Then allowing 6 to approach zero, we have
CO
(2) U(Z, t ) = lim 1' 2(x - y, t)u(y, 6) dy. m+
In the first instanae t < c - d, but since 6 + 0, t may be taken as any number < c. We may now use a weak compactness theorem, D. V. Widder [1946; 331, to conclude the proof. By that theorem there exists a function p(y), 1 p(y) 1 < M, and a sequence d,, n = 1,2, , tending to 0 such that
This limit, through a subset of the values of B used in (2), must have the same value, u(x, t), as given by (2). This completes the proof.
6.3. We turn next to the principal result of this section. ' T H E O ~ M 6.3. The conditions
5 61 THE WEIERSTRASS TRANSFORM 03' BOUNDED FUNCTIONS 187
are necessay and suficient that
where lq(y) I < M , -00 < y < a. The conditions are necessary. For if (1) holds
ss waa proved in 4 4.1. Since the integral (1) converge8 absolutely for all x it is clear that f (x) € A in -00 < x < m. Condition 2 followa from (2) and from (2) 5 1.1.
Converselyy condition 1 insures that the function
€ H in - co < x < oo, 0 < t < 1, by !t%eoxem 3.4. Moreover, u(x, t) is bounded there so that Theorem 6.2 is applicable t o conclude that
where 1 q(y) I < M, -m < y < a. But this integral converge8 for aU t > 0, so by continuity
QO
U(X, 1-1 = j ' w z - y, ~)v(Y) dye
Hence our result will be proved if we show u(x, 1 -) = f ( x ) . By equation (1) 5 3.2 and by a change of varieble
Since f(x) E A the last integral is dominated by
for a auitable constant N. Hence another application of Lebesgue's limit theorem shows that
t+l - This completes the proof of the theorem.
THE WEIERSTRASS TRANSFORM [CH. TAII
7. INVERSION, GENERAL CASE
7.1. In tj 4 we showed how to invert the Weierstress tramform of a function q ( x ) under simpl5ed hypotheses. We turn next to the general case. We observe that the inversion provided by Theorem 2.2 required ~ ( x ) to be of bounded variation in a neighborhood of every point where the inversion was to be effective. The present method, with no local condition imposed upon rp(x), will provide inversion for almost all x. We assume, aa d w a p , that ~ ( x ) is absolutely integrable in every finite intervd. Note that formula (3) 2.2 is closely related to our definition of emD'f(x). It adopts a different method of summability.
7.2. Let us prove first the following preliminary result. h w ~ a . 7.2. If
lm J_9. k(x, - y, l)y(y) dy converges for smne x,
2. [[v(v) - ~ ( 4 1 dv = - a) then
Choorte an arbitrary positive 8 and write the integral (1) as the sum of two others, i , ( t ) , I&), corresponding respectively to the intervals of integration (a, a + d), (a + 6, a). Set
Then a(a + 8) = 0, and a(+ W) exists by hypothesis. Moreover, P'(V) < 0, a + d < v < co for all t sufficiently small, so that B(v) is deorewing and /3(+co) = 0. Hence for amall t
If M is a constant not larger than ( a(t) I,
Me- dalqte(a+d - ~ ) ~ / 4 s = o(1) (4Tt)'12
!in IN VERSION, GENERAL OASE 189
To I,@) we apply Theorem 2b, D. V. Widder [1948; 2781, with k = l/t, h(v) = -(v - a)4/4, h'(a) = 0, hW(a) = -112.
We conclude that
Thia completes the proof. By use of this lemma, we can now prove TKEOREM 7.2. If
m
l- 1. k(x, - y, l)pl(y) dy converges for sonae x,
then, 00
(2) lim I-.. k(a - Y, t)g(y) ay = ~ ( 4 . M+
For, write the integral (2) as the sum of two others corresponding to the intervals of integration (-co, a), (a, 00) The second of these approaches p(a)/2 by the lemma. The fksf does also since it can be reduced to the integral
to which the lemma may again be applied. COROLLARY 7.28. If hy-potheais 2 is replaced by: ~ ( a f ) and p(a-)
exist, then
C O R O L ~ R Y 7.2b. If hypothesis 2 ia omitted, (2) holds for dmost all a. For, it is known, E. C. Titchmarsh [1939; 3621, that hypothesis 2
holds automatically a t the Lebesgue set for q ( x ) , and hence almost evewhere.
7.3. The previous theorem is the basis for the inversion of the Weierstr~s transform in the most gonerd case. However, to apply it we need the following prelimhazy result, symbolicaJly equivalent to the equation
e-tD'eDaq(z) = e(l-BD'q(z)
T ~ O R E M 7.3. If ties Weierstram t ransfm
i90 THE WEIBR8TRASS TRANSPORM [CH. VIlZ
Choose two conatants 6 7 such that a < < d < 7 < b. By Lemma 2.1~ of Chapter VI
g(u) = O ( ~ ( U - - V ) ' / ~ ) zt-, +oO
Now integrate (1) by parts. The integmtea pa& vanishes by (2) when E<x<q,sotha; t
where
Hence
(3) e-tD%x) = kl(s - u, l)cc(u) du.
Set a = d + Sy and note that the resulting integral is dominated by
by virtue of Theorem 2.3. But this converges for 0 < t < 1 by (Z), SO
that we may apply E'ubini's theorem t o (3) and obtain
By Theorem 2.6 we have for -a < x < a, -a < w < CQ,
so that 03
e-tsf(~) = /Smmkl(z - u, 1 - t ) s ( u ) du = J'_ k(x - u, 1 - t ) da(u).
The h a 1 integration by parts is again justified for -a < x < by (2) 7.4. The inversion of the Weierstrass-Lebesp transform is now an
irnmediaite consequenoe of the foregoing resulta.
8 71 1N VERSION, GENERAL CASE
! ~ E O R E ~ M 7.4. If 00
k(x - u, l)q(u) du, the integral wnvwging in
a < x < b ,
then
For, by Theorem 7.3
k ( ~ - u, 1 - t ) ( p ( ~ ) a%.
Theorem 7.2 is applicable to this integral (replacing t by 1 - t ) , and the proof is complete. As in Corollary 7.2b, hypothesis 2 may be omitted, in which caae the conclusion (1) is valid for almost all x.
7.5. Our next conclusion enables us to invert the Weierstrba- Stieltjes integrd. THEOREM 7.5. If a(u) ia a nanalized function of bounded variata'o~n
k every finite interval and if cO
f(x1 = j _ W X - u, 1) d.(u).
the integral c r m v e i n g for a < x < b, then for a < a < b and any two real numbers xl , x2
By Theorem 7.3
The relations 7.3 (2) not only justify. this integration by parts, but show, in conjunction with Theorem 2.3, that the integral (1) converges uniformly for x, < - x 5 - x,. Hence we have
m
= j'- - U, 1 - t)a(u) a~
- [ * k(zl -- u, 1 - t)or(u) du.
192 THE WEIERSTRASS TRANSFORM [CH. VIII
A n application of Theorem 7.2 to each of these integrals now yield8 the desired result .
7.6. In 5 7.2 we diaowsed the behaviour of e-tD'f(x) aa (3, t ) approached (x,, O), the tlpproach being dong a normal fo the x-axis. For certain purposea (compare 15.3 and the example at the end) it is important to let (2, t) approach the boundery in an arbitrary two-dimensional way. We shall need the following result.
THEOREM 7.6. If p(z) E C at x = a and if
t ? ~ dntegrd converging at (x,,, l), then lim u(x, t ) = q(a) . s-w l44+
Choose an arbitrary B > 0 and make the usual decomposition of (1) into Il(x, t ) , I,(%, t ), Is(%, t), these functions corresponding respectively to the i ~ ~ f e r v l a (--CO, a - d), (a- 8, a + d ) , (a + 8, GO). Define a(v) as in 5 7.2, but write %
B(v) = e-(~-G'/ue(+-")'/4 9
80 fh8t
If 1 5 - a 1 < 612 and t is so small that t(a - x, + 8 ) < 8/2, then P'(V) < 0 for a $ d - < v < a, and we have, as in 5 7.2, for a suitable aonstanf M
s M(*) -l]2,-B/1Sbe(a+d-~o)sj4
Hence
A ~ W W aqpment applies to il(x, t) . lllso
I Is(z, t ) - ~ ( 4 I s 1 i . b . j ~ ( y ) - ?(a) I + Iv-al d
when 1 x - o I < 612. That is,
By our hypothesis of continuity at a, the right-hand side approaches zero with 6, 80 that the proof is complete.
COROLLARY 7.6. If I ~ ( x ) 1 2 M , 1 z - a 1 2 - d and if ( 1) converges at (x,,, l), then
& 1 u(z, t ) 1 5 M. x + a
t-tO+
For, under the modified hypotheses
8. FUNCTIONS OF fl
8.1. This section is devoted to, known results from red-vaxiable theory (compare 8. Bochner and K. Ch~ndrasekharan [1949; 981). Proofs axe included for the reader's convenienoe. We recall that f(x) E L* (in -m < x < oo) meam that
rtnd that the !ruwm off ia 11/(4 11. = 11"
We are concerned here with real p 2 ,l. We shall frequently omit the subscript p on the notation for the norm, when no ambiguity results.
8.2. The first result needed is THEOREX 8.2. If f(x) LP,p> - I., andaf
~ ( 4 = I 1 f(x + h) - f(x) 1 19,
then A. ~ ( h ) i s bmded --c0<7L<a2
Conclusion A ie a result of ows ski's inequality,
7(4S IJf(2. $. b) 1 1 + f ) 1 = llf(x) 1 1 - Let us first prove B under the additional assumption that f (x ) f C.
Then for arbitrary positive R
and by Mjnkowski's inequality
194 THE WEIER8TRASS TRANBPORM [CH. VlCU
If ] s l > R a n d O < h < 8 , t h e n - - l x - h 1 2 ~ - 8 . G i v e n ~ > O , we can detsmine R, using 8.1 (I), so that I, << for 0 - 5; h < - 8. But with this fixed R, I, tends t o zero with h by Lebesgue's limit theorem. The inkgrand has a bound independent of h in 0 - < h < - 8 by the continuity off(x) in - R < x 5 . - - R + d. Hence
and I@+) = 0. Since the class of continuous functions is dense in L9, E. W. Hobson [1926; 2501, the general case is eady reduced to the special case jwt colzsidered aa indicated briefly by the following inequalities:
I lf(4 - BC) I I < c f E L D , g E Q
Ilf(" + 4 - g(x + h) 1 1 < I I B(X + h) - g(x) 1 1 < e 0 < h < ho*
These imply Ilf(x + h) -f@) 1 1 < 36 0 < h < h,,
as required. Thia completes the proof. 8.3. Our principal result, the one needed in later sections, is TH~~OREM 8.3. If f(x) (2 L p , p - 2 1, and if
then,
lim Ilf -fnII = 0. M+
Denote by R(y) a function which ia 1/2 for I y I < 1 and 0 for ( y 1 - 2 1. The integral of R(y) over (-a, ao) is 1, so t h t
By Hiilder's inequality ( p > 1) or directly ( p = l),
J-00
provided that f ibhi ' s theorem is applicable. Since the inkgrand of the double integral is 2 0 we have only to check that (1) converges. It does so since ~ ( y h ) is boGded and R(y) is integrable.
Finally let h + O+ in (1). We may apply Lebeegue's limit theorem by virtue of A, Theorem 8.2. Hen= the limit i8 zero, a;nd the theorem is proved.
5 91 WEIERcSTRASS TRANSFORMS OF FUNCTIONS IN L9 185
9. WEIERSTRASS TRANSFORMS OF FUNCTIONS IN LD
9.1. Thia section is devoted to the derivation of necessary and sufficient conditions on (I function f(x) in order that it may be the Weierstraas transform of a function 47 € L', p > 1. The result is derived from another, of interest in itself, concerning temperahre functions which arise from initial temperature functions belonging to L'.
9.2. The theorem needed about solutions of the heat equation is T~EOREM 9.2. The d i t i o n s
are necemary and suflcient tisat
Assuming (1) we see by Holder's inequality that for fixed x and t
Since the right-hand side is finite, (1) converges absolutely for --a, < x < co, 0 < t < oo. By Theorem 3.3, u(x , t ) EBthere.
To prove condition 2 we have
Fubini'a theorem i e applicable since k 2 - 0 and p, f L' by (2). The necessity of the oonditions is established.
To prove the ,sutIiciency consider the function
196 TEE WB)IERSTRAS&' ITRRNSFORM [CH. VIII
which belongs to class H where u(x, t ) does. This follows by direct computation of the partial derivatives of uh:
Moreover, by Holder's inequality
for -m < 2: <m,O < t (c . Hence b y h m m a 6.2
for O < S < c , O < t < c - 6 , - - c u < x < o o . Let h+O+. By the law of the mean uJz, t + il) -+ u(x, t + 8). By weak convergence, E. C. Titchmersh [1939; 3891, the integral (3) approaches the same integral with u, replaced by u. The weak convergence theorem is applicable since k E La and sinoe uh + u "in mean of index g" by Theorem 8.3. Hence equation (3) alao holds if the subscript h is omitted throughout.
Now let 6 -+ 0,
u(x, t ) = lim
Using waak oompactness, D. V. Widder [1946; 331, we complete the proof as in $0.2, obtaining equation (1) as desired.
9.3. We can now relate the previous result t o the operator e-lDB as followa.
!CHEOREM 9.3. The condktims
are necessuy and sufica'ent that
where
(2) I l V(Y) I 1, < Jf= If (2) holds, the integral (1) converges absolutely for all x since (1)
ia9.2(1)witht= 1. Hencef(x)€A, --a < x < o o . Also
6 101 WEIERSTRASIS-STIBLTJES TRANSFORMS 197
by Theorem 7.3. Now apply Theorem 9.2 to the integral (3) (replacing t by 1 - t ) to obtain condition 2.
Conversely, condition 1 guarantees, by Theorem 3.4, that
belongs to H in --GO < z < CO, 0 < t < 1. Condition 2 enables us to apply Theorem 9.2 to the function (4), so that
00
u(x9 t ) = j" /(x - y, ~)c(Y)
where q(y) satisfies (2). As we have seen, this integral converges absolutely for all t > 0, so by continuity
00
1 - 4 = 1- =k(X -- y7 ~ A ( Y ) &I-
On the other hand, we showed in § 6.3 that whenf(x) € A, --a < x < a, then the function (4) + j ( x ) as t + 1-. Hence (1) holds, and the proof is complete.
10. WEIERSTRASS-STIELTJES TRANSFORMS
10.1. Theorems 9.2 and 9.3 &re no longer true if p = I throughout. To show this for Theorem 9.2 take u(x, t ) = k(x , t), the source solution itself. Then
so that conditions 1 end 2 are satisfied in -XI < x < cu, 0 < t < m. But the equation
00
(1) 4 x 9 t) = J" 2 ( x - y, t)q(y) du, with p(y) E L, is impossible. For, by Corollary 7.2b, we ahould have
lim k(x, t) = 0 = q(x) t 4 0 +
for almost all x # 0. That is, the integral (1) would be identically zero, contradicting equation (1). On the other hand (1) ia true if p(y) cly is replaced by da(y) where a ( y ) is constant except for a single jump at y = 0. That is, k(x, 1) may be a Weierstrass~Stieltjea transform.
10.2. The oorrect conclusion when p = 1 is the following. T I I E O B ~ 10.2. The conditions of Theorem 9.2 with p = 1 are
necessary and mficient thut
(1 -co<z<oo, O < t < c ,
where
198 THI WBIERSTBASS TRANSFORM [CH. VIII
Under ~lssurzlption (2) the integral (1) converges absolutely in the half-plane t > 0. Indeed
Hence by Theorem 3.3 21 E H there. Condition 2 is also satisfied since
Thus both conditions me necessary. To prove the converse we agein introduce the, function u,(x, t ) es in
the proof of Theorem 9.2. It is again bounded since
The rest of the proof is the same as for Theorem 9.2 except that the Helly and Helly-Bray theorems are used instead of Theorems 17a or 17b, D. V. Widder [1946; 29-33]. The procedure is standard (compare D. V. Widder [1946; 3071) and is omitted here. The conclusion is (1) (2), aa desired.
10.3, The previous theorem leads to the following representation theorem for the Weieratraas-Stieltjes trmsf'orm.
THEOREM 10.3. The conditions of Theorem 9.3 moth p = 1 are neces8a~y and su@ient that
QO
(1 I(.) = j - w z - y> 1 ) da(!l),
where
(2)
We saw in the previous section (t = 1 ) that the integral (1) converges absolutely when (2) holds. Hence f ( x ) € A in - oo < x < GO. Moreover,
The change in the order of integration needed to prove this is valid if
This is true if 0 < t < 1. Now i n e q d t y 10.2 (3), with t replaoed by 1 - t completes the proof of the necessity.
Bar the converse, the proof of Theorem 9.3 may be used, appealing to Theorem 10.2 in~tead of 9.2 and replacing p(y) dy by da(y).
3 111 POBITI VE TEMPERATURE F UNUTIONS
11. POSITIVE TEMPEUTURE FUNCTIONS
11.1. I n 4 5.3 we noted that u(x, t) = k(x, t )x / t is a temperature function not identically zero even though u(x, 0+) = 0 for all x. How- ever, this function takes on both positive and negative values in the half-plane t > 0. We wish to @how now that any temperature function which is known to be non-negative for t > 0 and to vanish for t = 0 must be identically zero (D. V. Widder [1944; 851).
11.2. The integral
(1)
generally gives the temperature of an infinite bar at time t + d when its temperature at time t = S is u(x, 8). This is not always the caee as the example of 5 11.1, with 8 = 0, showa. We shall ahow that it is so for non-negative functions u(x, t), but we must first show that the integral (1) always converges for such functions. To do so we need the following preliminary result,
LEMMA 11.2, If A
and f(x) is the function ofTheorem 6.3, then
It is of course assumed that ~ ( y ) E L in (-A, A). The inequality
is trivial. setting r = ylt/z therein, we have
Henm when 1 x 1 > A A ( F(Y) I 1
IY(XJ)IS [-A -Z jdy$ I_ZI -A
This proves the result. THRIORIDM 11.2. If u(x, t ) 2 0 and E H in the ship 0 < t - c, then
the integral (l), with 0 < t3 < oTonuerges in the strip 0 < t < o - B a d is 5 - u(x, t f 8 ) there.
We prove later that in this ambiguous conclusion (<) - only the equdty can hold.
Consider the function
$00 THB WEIERSTRASS TRANSFORM [Ca. VIII
for arbitrary positive constants A, 8, the latter < c. By use of Theorem 6.2, we show that v (x , t ) - 2 0 in the strip 0 < t <'c - 8. By Theorem 7.6
lim v ( x , t ) = u(xo , 8) 2 o IxoI ) A t T 4
= O 1x0 1 A- However, the integrd (2) generally approaches no limit &@ (3, t ) + ( A , 0) or (-A, 0) in the two-dimensional way. But by Corollary 7.6 it h a s rr,
limit superior (u(A, - 8 ) or u(-A, d), respectively. That is,
Hence v(x, b) satisfies hypothesis 2 of Theorem 6.2 at all point8 (x,, 0) of the x-&xh.
Now t o produoe a contradiction assume Chat v(x, , t o ) = -1 < 0 at some point (xo, t o ) , 0 < to < c - 8. By Lemma 11.2 we can d e t e d e R so large that V ( X , t ) 2 -112 on the segments x = f R, 0 < t < c - 8 because u ( x , t) 2 0 az the integral (2) tends uniformly to zero for 0 < t < co ITl--f m. That is, u(z, t) muat have a minimum inside a rectmguhr region D of thei type described in Theorem 5.2. This contradicts that theorem, so that the assumption v < 0 must have been false. The contrary assumption gives
in the strip 0 < t < c - 8 . Since thk integral inoreases with A the convergence of (1) is guturnteed and the theorem is proved.
11.3. By use of the foregoing result we oan now prove the required uniqueness theorem.
TEEOBEM 11.3. If
3. u(x, 0 ) = 0 then,
Observe that we are a~su~ning u(x, t ) E H on the lower bounbry of the atrip. It is for thia reason that we can give the boundary condition 3 without the use of limits. Set
v ( x , t ) = r ' U ( X , Y ) ~ Y *
By 1 and 3 we have
0 113 POSITIVE TEMPERATURE BUNUTIONS 201
so tht v € H there. It satisfies all the hypotheses of the theorem but h w the additional properties of being convex in x and non-decreasing in t. Moreover, if v(x, t) vaniehes identically, the same is true of u(x, t). Hence there is no loss in generality if we include these additioml properties as hypotheses on u(s, t).
Let 6 be an arbitrary positive number <c and set x = 0, t = to < o - 8 in the integral 11.2 (1). Then
by Theorem 11.2. Since u(x , t) is non-negative and non-decreasing in t the function f (x) of Tychonoff 's theorem is
By the convexity off(+) we have for x > 0
(the area under a convex curve 2 - the asea under a tangent). Hence
Since the latter integral is known to converge we have
I f x < 0 we have
and (1) is also valid as x 4 -00. Hence we may apply Tgchonoff's theorem and conclude that u(x , t) ia zero in the strip 0 ( t 5 - 8 . Shoe d was arbitrary the proof is complefe.
COROLLARY 11.3. If conditions 1 and 2 of the theorem hold, then
For, by Theorem 11.2 the funation
is 2 0 in the strip 0 < t < o - 8. Indeed it clearly satisfies all conditions ofrheorem 11.3 (with c replaced by c - d) cmd is consequently identically zero.
TEA? WEIERSTRAXS TRRNBFORM
12. WEIERSTRASS-STIELTJES TRANSFORMS OF INCREASING FUNGTIONS
12.1. In fj 10 we discussed Weierstrass-Stieltjes transforms of functions of bounded variation. One subclass consists of those trans- form for which the function to be transformed is non-deore~ing. The latter class is also a subclass of trmforms of unbounded non-decreasing functions. Illustrative examples are provided from the table of 8 2.6 by the pairs 1, 3 (p = yq f = x2 + 2), 5, 6, 7, 8, 9 and by the function a(y) given imrnedia,tely after Definition 2.1~. Note that the increseing fmction
of the pair 6, for example, is not of bounded variation on (- W, a). It is important to include fundions like this one, if the class of their trans- 'forms is to be neatly characterized by use of the operator e-'*'. The situation is analogous to one in Laplace transform theory. According to Bernstein's theorem the class of functions f (x) '%ompletely monotonic" on 0 < x < oo is equivalent to the clam
with a(y) non-decreasing. Here also the variation of a(y) may be infinite, as for example when a ( y ) = y and f ( x ) = l/z.
123. Aa in 8 8 6,9,10 we need a pre- result about temperature functions. The theorem will be the analogue of a familiar one by A. Herglotz [1911; 5011 concerning positive hermonic functions.
THH~OREM 12.2. A w s a y and eu&ient ctmditim that
where a (y ) M non-dec~easdng and the i&gral mvergea in the atrip 0 < t < c is that u(z, t ) € a, U(Z, t ) 2 0 there.
The neaes~iky of the condition follows by inspection of the integral (1). To prove the auf6(3iency set
8 121 TRANSPORMS OP INCREASING FUNCTIONS 203
Hence
By virtue of (3) we may now apply the Helly and Helly-Bray theorem, D. V. Widder [1941; 26-32], in the standad way to obtain
where B(y) is non-decreasing end bounded. We thus obtain (1) where
This function is non-decreasing, so that the theorem is proved. 12.3. Ae a consequence of the previous theorem we cm strengthen
Theorem 11.3 by dropping the demand that u(x, t ) E El a the x - a d and by replaaing condition 3 by u(x, O+ ) = 0.
T ~ E O R E M 12.3. I,',
For, by hypotheses 1 and 2 we have the integral repremnfation 12.2 ( 1 for ( x t). Let yo be any real number and let 6 be any positive number. Then
(1) YO - Y, t ) ~[N(Y) - a(yo)l
804 THE WEIRSTBASS TRANSFORM [CH. VUI
We have here integrated by parts and expressed the derivative of k in terms of E. Set
Then p(8) 2 0 by Theorem 12.2. Moreover,
d l d t > '3 (d)S yzb(y, 1) dy. = 2 0
Now let t + 0+, using hypothesis 3 and pair 3 of 8 2.6 to obtain
so that p(d) = 0. Allowing 8 to approach zero we have
That is, the lower derivate on the right for u(y ) is zero at yo. If the range of integration in the integral ( 1 ) is changed to (yo - 8, yo) we easily see that ( 2 ) still holds when y + 0- . By (2) it is cleaz tha% a@,+ ) = a(y , ) . Similarly a(yo-) = a(yo), and a (y ) is continuous. But a continuous function with any derivate constantly zero is constant (see, for example, C. J. de la Vall6e Poussin [1914; 991). But then u(x, t) is identically zero by equation 12.2 (l), and the theorem is proved.
12.4. We come fhally to the representation of functions as Weierstrms-Stieltjes tramforme of non-decreasing funations.
!THEOREM 12.4. The conditions
are mw8a y and suwent thub
where a(y) k non-decrtwting and the integral mvergm in a < x < b. As we b e seen, it i s no restriction t o msume that the origin lies in
(a, b). We prove the necessity, assuming (1). The absolute convergence of the integral ( 1 ) in (a, b) imures condition 1. By definition
3 121 TRANBFORMH OF INCBEAISINQ! FUNCTIONS
By Fubini's theorem and Theorem 2.5
Since a ( y ) f it is dear that condition 2 is satisfied throughout the whole infinite strip 0 < t < 1 .
For the suffioiency define
By conditions 1 and 2 it is well defined and 2 0 in the strip 0 < t < 1. By Theorem 3.4 it belongs to H there. ~ e n G we may apply Theorem 12.2 to obtain
a0
(2) U ( X , t ) = j" - Y, t ) d.(y).
where a(y) E and the integral convergerr throughout the strip 0 < t < 1. Since f (x ) E A in a < m < b we have by Cauchy's integral theorem, applied to the integral 3.1 (3), that
where M is some constant, a < A < B < b, and 6 is an arbitrary positive constant less than 1. Since the dominant integral converges and is independent of t we have by Lebesgue's limit theorem t h t
We show next that the integral (2) converges when t = 1 and a. ( x ( b . For everyx, t, R (0 < t < 1, R > 0)
The integral is dominated by
206 THE WEIEMTRASS TRANBPORM [CH, VIII
By allowing t to approach 1 in inequality (4) we obtain
and since a(y) € f' the convergence of (2) is immediate.
PinaJly, by Theorem 2.1,
and the proof is completed by equation (3).
13. TRANSmRMS OF FUNCTIONS WITH PRESCRIBED ORDER CONDITIONS
13.1. The behaviour of ~ ( y ) for large 1 y 1 affects the width of the strip in which the integral
converges. For example, if q ( y ) is bounded (1) converges absolutely in the half-plane t > 0. More generally p(y) = o(@), I y I + co, implies that (1) converges for 0 < t < I/(&). Let us now obtain con- ditions on a temperature function that it can be equal to an integral (1) with q ( y ) satisfging order conditions of the above type. We can then apply the result t o the Weierstraga transform.
133. THEOREM 13.2. The conda'tim
for s m a > 0 and some 6 - I/(&), are necessay and m$icient that
the i&qrul converging absolutely in - < x < oo, 0 < t < 1/(4a) and
The proof of the necessity is made by use of Theorem 3.3 and the pair 8 of fj 2.6.
3 131 BUNCTlONS WITH PRESCRIBED ORDER CONDITIONS 207
Conversely, condition 2 is equivalent to
where
Hence Mu - u and Mv + r are funotione which satisfy the conditions of Theorem 12.2 in the strip 0 < t < 8. Henae
for a, suitable function a(y). By the theorem used, the functions M@(y) & a(y) are both non-decreasing, so that
From these inequalities it is clear that a(y)'is absolutely continuous in any finite interval. Hence by rt familiar theorem, E. C. Titchmarsh [1939; 3641, a ( y ) is an integral of some function ~ ( y ) ,
where a' (9) = ~ ( y ) almost everywhere. By (1)
except perhaps at a set of meamre zero. Redefine q(y), if necessary, so that (2) is valid for all y.
in the strip 0 < t < 8 . This oompletes the proof. 13.3. THEOREM 13.3. T b conditions
for some a < 1/4 are necessa y and duf~~ient thut roo
where the integral cmve~ge8 for aU x and
If (1) (2) hold with a < 114, then the integral (1) converges absolutely in -a < x < m, andf(x) € A there. By Theorem 7.3
Heme by the necessary part of Theorem 13.2 with 8 = 1
Conversely, aet - -(I-t)DS u(x, t ) - e f (4
By Theorem 3.4 it belongs t o H in 0 < d < 1, and
by hgpothesis 2. By the sufficient part of Theorem 13.2 with d = 1
where y(y) satisfies (2). But in the presence of (2) the integral (3) con- verges absolutely in 0 < t < I/(&) and hence on the line t = 1. By Theorem 2.1
But we saw in 5 12.4 that
lim e -'IDs f (x) = f(x) 1-1 -
in my intarvd in whiohf(x) € A. Henae we have for all x
and the theorem is proved.
SUMMARY
14. SUMMARY
14.1. The principal results of the present chapter are the following. A. If E(e) is the special function e-" of class E, then as for other
functions of this class its reciprocal e* is the bilateral Laplace transform of a frequency function k(x, 1 ) = (4n)-112s-x'14 s
B. If this frequency function is taken as the kernel of a convolution transform, the latter, known as the Weierstrtrass transform,
is inverted by eeD' in the following sense:
C. Necessary and sufficient conditions, couched in terme of e -OD' f (x ) , are available to guarantee a representation (1) with ~ ( y ) in a prescribed class.
In conclusion we emphagize again that the basic ideas of this chapter are the same as the guiding ones for the rest o f the book. Any apparent difference is essentially due to the contrast of a discrete parameter with a continuous one.
CHAPTER I X
Complex Inversion Theory
1, INTRODUCTION
1.1. In thie chapter we shall obtain a complex inversion theory for a suitably restricted alms o f kernels, those of the form
(1)
where
the a, being real and such that
lim k/a, = Q b o o
In order to see what to expect let us consider the example a, = (2k -- I)/% Here E(8) = cos ire and Q(t) = (11%) sech ft. If in the equation
we set f (z) = F(eE)di2, ~ ( t ) = d ( e t ) e t P equation (3) becomes (sec 8 5.1 of I) the Stieltjee transform
F,,) = [a % at. 0 Q t - t
It is well known, see D. V. Widder 11941 ; 3261, that F(a) is an analy-tio function in the aeotor I mg z I < T. Correspondingly f(z) i s an analytic function in the strip I Im z 1 < n. The following formulas may be established by direct computation:
z 00
cos rre edz ds;
(5) COB 778 = - eaa d ~ .
8 11 INTRODUCTION 21 1
Here C is a rectifiable closed curve containing the points -k and i?r in its interior and proceeding counterclockwise. From Chapter I we have the symbolic inversion formula
Replacing 8 by D in equation (5) we fhd that
We have made use here of the familiar operational fornula ezDf(t) = f (t + 2). The formula (6) is only formally true since f (t + z ) is not in general defined for 2: on 0. A simple way of avoiding this difficulty is to replace the formula (6) by the formula
where C, (0 < p < 1) is a rectifiable olosed curve containing -in and in in its interior, lying in the strip I Im z 1 < +rip, and proceeding counter- clockwise. Employing the caloulus of residues in (7) we obtain
After a logarithmic change of variables t b becomea
(8') i r p e i ~ ~ 2 + ~ ( ~ - i * ~ ) ~ - i w / g l = lim g [ F ( k )
-1 - which is substantially the same as
1 #(t) = lim - [F(-t - i r ) - P(-t + ic)],
r-4-1.
the claesical complex inversion formula, for (3) due to Stieltjea; see D. V. Widder [1946; 3381. This example suggests preoedure for the general cam. We shall
show that if G is defined by equations (1) and (2) then
is analytic for I Im z I < nQ. Let
212 COMPLEX INVERBION THlCORY [CH. IX
We shall prove that K(z) ia andytio and single valued in the z-plane slit along the imaginary axis from -iQv to in*, and further that
where C is a closed rectifiable curve going counterclockwise around the segment [-inr, ~ Q T ] . Iff(z) is defmed by (9) then we have the symbolic inversion formula,
v(t) = mD)f(t)* Just aa before we obtain
Again this is not in general meaningful and it mugt be replaced by
y(t) = lim -.
where C, is a closed rectsable curve going counterclockwise around the segment [-in*, iQw] and lying in the strip I Im z 1 < mZl /p . The present chapter ie devoted to establishing the validity of this inversion formula.
2. TRANSFORMS IN THE COMPLEX DOMAIN
2.1. The preaent section ia concerned with the kernah G(t) and the oorresponding oonvolution tramforme in the complex plane. We suppose h t
where
We define
In order t o study U(z) we must first investigate l ( s ) . The following reault shows that E(e) behaves very much like cos irh.
b 21 TRANSFORMS IN THE COMPLEX DOMAIN 21 3
LEMMA 2.1. If Z(8) i8 defined by equations (1) and (2) then
lim r -' log I E (refl) I = ?m I sin 0 I r+oO
uniformly for 0 in any closed interval not containing an integral multiple of 7T.
Let N ( a ) be the function which counts the ak's. By equafion (2) N ( a ) = IRa + €(a) - a where €(a) = o(1) as a+ m. We have
log I E(s) 1 =Rl J log (1 - s2a-8) dN(a) . 0
Integrating by parts
N(a) -- %a log I E(s) 1 = R1 h,
= 11(4 + IZ(4* If s = reie then
I,@) = &r 1 sin t9 I If 8 lies in an interval Op - < 0 < el which does not include an integral multiple of w, t,hen there exists a constant A > 0 independent of 0 such that l a f . s l > ~ ( a + r ) . - Thus
( ) < 2A-2 jm I ' (a )1 do, I 1 - 0 (a + r)2
Since
our lemma, follows.
Lemma 2.1 has long been known, see V. Bernstein [I933 ; 2711. 2.2. The following theorem represents an extension to the complex
plane of the results of Chapter V. THEORBW 2.2a. If
1. E(s) is defined by equation8 2.1 (1) and 2.1 (2),
2. p = [multiplicity of e - a, as a zero of E(s)] ,
214 COMPLEX INVERSION THEORY [CH. IX
then : A. G(z) ia an analytic fundion in the atr ip I y I < nbl, 2 = x $- iy;
B. #(z) = p ( ~ ) e - ~ l ~ + R+(z),
Q(z) = p ( - ~ ) ~ " l ~ + R-(z),
where p(z) ia a polynomial of degree lc - 1 and where
for some e > 0, ufiijomZy in every proper &strip 1 y 1 < a(n - q) of the strip 1 y I < A.
By Lemma 2.2a if 7 > 0 then
uniformly for a in any finite interval. It follows that the integral 2.1 (3) defining G(z) converges unifody in the strip 1 y 1 < ~ (n ' - 27) and define8 an analytic function there. Since q is arbit& conclusion A f0110ws. TO establish conclusion B let us choose E > 0 so small that no zeros of E(s) lie in the interval -4 - s a < -al. Integrating about the rectangular contour with veirtioea-at j iT, -a, - E & iT and letting T increase without limit we obtain, as in tj 2 of Chapter V,
Using equation (1) it is easily seen that
uniformly for 1 y 12 r(a - 27). The second part of oonclusion B is established similaz&.
1. E(e) i s dejned by equatim 2.1 (1) and 2.1 (2),
8 21 TRANSFORMS IN THE COMPLEX DOMAIN 215
a d if the t~ansform 3 umve~geefor any v d w of z & thestrip (-a < z < 00;
I y I < ?rQ), it converges for all such z, uniformly in any m p t a&, so thut f ( z ) isanalyticfor (-03 < z < a ; l y l < ~ R ) .
Suppose that the transform 3 oonverges for z = z,, I Im z, 1 < whl. Let R be any compact subset of the strip I Im z 1 < no; we muet how that
(3)
lim B
uniformly for z in R. By Theorem 2.2a
uniformly for z in R. If L(t) = 1 i#(r,, - t ) do@) then
We have
Using equations (4) and (5) we see that equation (3) holds uniformly for z in R. We may similarly establish (3').
2.3. Let us consider some examples. The nth iterate of the Stieltjes kernel &(t) is defined by the formult~
We shall compute SJt) explicitly. We have
MaJzing the change of varieble een = y we obtain
$16 COMPLEX INVBRSION THEORY [CH. IX
Since
(2) 1 + Y)-~ CEY = r(s)r(v - g ) / r ( v ) o c R 1 8 < ~ ,
it follows t b t
In particular 1 t
dl = sech - 2'
t t 8, = - cosech - .
2srra 2
IJdaking use of the relation r(l + s) = sr(s) we obtain the recurrence formula
&(t) = [(n - 1) (n - 2)1"[(n - + +ttln)7fJn-e(t)
from which it follows that
t t 1 n-1
&(t) = - cosech - 2772
l-r [ ( t l ~ ) ~ + (2VI 2 (2% - l)! ~ - 1
These form& are due to Barrucand [1950]; see also D. V. Widder [ I W ; 259-2651. S,(t) and S,(t) have been previously evaluated in 5 9 of 111.
The function seoh 212 h e 8 simple poles at z = &(2r - l ) k , r = 1, 2, while the product
vanishes for z = &(2r - l)k, r = 1'2, , n. It follows that B,,,(z) is analytic in the strip 1 Im z 1 < (2n + 1 ) ~ . Similarly cosech 212 has the simple poles at z = &2&, r = 0 , l , . The product
vanishes for z = &2ri?r, r = 0,1, - n - 1 from which it follows that #,,(a) is analytic in the strip I Im z 1 < 2 m . We have thus verified directly conclusion A of Theorem 2.2a for these caees.
5 31 B ~ H A V I O UR AT INFINITY
A second example is given by the kernels
We have previously shown, see 8 9.5 of 111, that
3. BEHAVIOUR AT INFINITY
3.1. The results of the present section are an extension to the complex domain of the results of 5 2 of Chapter W.
then
The proof is left to the reader.
THEOREM 3.1. If 1. E(s) M &fined: by equdiolls 2.1 (1) and 2.2 (2),
2. a(z) ia d e $ d by equation 2.1 (3),
x = s + iy,
Let P
and let H2,(t) and G,,(t) be the corresponding kernels. It follows from Theorem 8.la of VI tbt if
m
228 OOMPLEX INVERSION THEORY [OH. IX
then CO
(1) f(4 = 1- a~2,(. - t)A(t) -co < x < 00.
a0
Since f(r) and 1-c04,(z - t)A(t) at are d y t i a in the strip I Im r I < nhl '
it follows that
By Theorem 2.1 of Chapter W
If al < a < a,+l then Theorem 2.2a implies fhat
uniformly for 1 y I - < w(Q - q), q > 0. Applying Lemms 3.1 we obtain our desired result.
As en example of this result we find, afhr a logarithmio change of variables, that if
then P(z) = o(r )
uniformly in every sector I arg z I < ?r - e. -
4. AUXILIARY KERNELS
4.1. Let 0 < p < 1 and let a > 0. We defme
where j ( t ) = 0 for t < 0, =f for t = 0, and =1 for t > 0. It is easily v d e d that h(p, a, t) is a, distribution function with mean 0 and variance 2(1 - and that
the bilataral Lsplace t d o m converging absolutely for -a < Rl s < a.
8 41 AUXILIARY KBRNELS
1. E(a) is &fined epuatiolas 2.1 (1) and 2.1 (2),
then : A. Q(p, 1 ) is a frequemyfunction with mean, 0 and variance
Q ( p , t)edt dt = B(p8)/B(8) tht? b i b ~ d t r a m f m
converging absolutely in the 8tra'p -a1 < Rl s < al ;
C. Q(P, z ) ia an. analytic function L the strip 1 y 1 < ?r(l - P ) S ~ ,
where p(p, z) is a polynomial of degre p - 1 and where
for erne e > 0, undfonnly in every proper substrip 1 y 1 < ~ ( 1 - p ) (a - 7) of t h 8trip Iy I < ~ ( 1 - p ) n .
By the convolution theorem if
Ll,(p, t ) = h ( q , p, r ) # h(a,, p, t ) # # h(a,, p, t ) then ilJp, t) is a distribution function with characteristic function
220 COMPLEX INVERBION THEORY [CH. IX
uniformly for 8 in any compact set of the 8-plane punctured a t &a,, &aa, . By Corollary 2.3 of 111 E ( p ) / E ( s ) is the characteristic function of a distribution function H ( p , t),
00
S- * e-" dH(p, t ) = E(pi.rl/E(h).
Moreover, by Theorem 6.2 of III,
Since by Lemma 2.1
(3) log i ( ~ i . r ) / E ( i . r ) - -v(1 - P)Q 1 r I 7+&%
d it follom that H ( P , t ) is infinitely differentiable. If a(;, t ) = - H(p , t ) at then Q ( p , t) is a frequency function, and
Conclusion C follow8 from (4). To demonstrate conolusion D the line of integration in (4) must be deformed to Rl 8 = &(a, $ s) as in 8 2. From
converges absolutely for I Rl8 I < % and defines in this strip an analytic function. Since
we have demonstrated conclusion B; that is, for I Rl s I < 4
(5) e4Q(p, t ) dt = E(pe)/E(a).
DiEentithtiug equation (5) with respect to 8, and setting 8 = 0 we obtain
Thw oonclusion A has been established. 43. Since the fumtionrr G ( p , t) are not vsriation diminishing the
changes of sign of their derivatives cannot be studied by the methods of Chapter IV. However some important information may be obtained by an elementary argument. See A. Wintner [1938].
8 41 AUXILIARY KEBNELS 221
THEOIL~U 4.2. V G(p, t ) ia dejened os in T h e m 4.1 then
A distribution function h(t) is said to be convex if
and if for r > 0
We will show that if hl(t) and hg(t) are convex distribution functions then h = h1 # h, is again a convex distribution function. There is no dif?iculty in showing that h(t) satisfies equation ( 1 ) . To establish equation (2) we note that
00 =I* hl(t - U ) d[hg(u + I ) - h,(u - r)]. M*g use of the relation h,(u) + A,(-%) = 1 we see that
The function hl(u - r) - h,(u + r ) is non-decreasing for 0 2 - v < a. For t 2 - 0 and 0 < - ec < oo the funotion h,(t f u ) - hl(t - r) is non- negative and non-increasing aa t increases. It follows that for t 2 0 h(t + r ) - h(t - r ) is non-increasing. The case t 2 - 0 may be dealt with similarly.
Let k( t ) , {kl(t)}zm be normalized dietribution tunctions and let lim k,(t) = k(t) at ell points of continuity of k(t) . Then if the functions ** k,(t) are convex ao is k(t) .
The functions b(a,, p, t ) are convex by inspection. It follows that for each HnFP, t) is convex and finilly that H ( p , t ) is convex. Our theorem is an immediate consequence of this fact.
222 UOMPLEX INVEBSION THEORY [CH. IX
COROWY 4.2. If Q(p,t) is defined aa in Theorem 6.1 then lin U(p, t ) = 0 for t # 0.
-1-
We may suppose t h t t > 0. We have
from whioh our assertion follows. The fanctions G(p, t) have in the oomplex inversion theory a role
malogow to the functions UJt) in the real invemion theory. The oontinuoua parameter p corresponds to the htegra-1 pameter la, and p 1 correeponda to n+ a. As n-c co the variance of 4 ( t ) decremes to 0; simihrly as p -c 1- the varirtnae of Q(p, t ) decreasee to 0, etc.
As an example let us compute explicitly the one parameter family of kernels associated with the Stieltjes transform. If
Making the change of variable earn = y
Since
we obtain
Bffer a few simplifications this becomes
Tp. t 1
00s - coah - 2 2
"(" t, = aos irp + cosh t
By inspotion h ( p , z) is adyt io in the strip 1 y 1 < (1 - p ) a thus verifying oonhion C of Theorem 4.1.
$a THE INVERSION PUNC!l'ION $23
5. THE INVERSION FUNCTION
5.1. In this seetion we shall determine the pxoprfies of K (2).
THEOREM 5.1. If B(8) ~&$d by epatim 2.1 (1) and 2.1 (2) art& if
QO
(1) K(z) = 2 mk)(0)rb l k-0
thm K(z) ia adytic and single v a Z d in the z p n e except m the 88e9&
[-krQ, kQ]. Moreover
whre 0 is a closed redi$ubZe curve going around [-ha, ha] in the positive direction.
The inequa&y I B(8) I 2 - E(i 1 8 1 ) together - with Lemma, 2.1 ~ h o w8 that E(s) is of order 1 type wQ. Hence lim I E(k)(0) I I l k < - wQ. Indeed if
-00
E > 0 then there exists A(E) such that I E(s) 1 < - A(.) exp [v(Q + a) 1 a I] for aJ18. We have
Choosing r = k[w(Q + €)]-I we h d that
Thus
It follows fhat the series (1) converges for I z I > dl. QO
Using term by term integr&tion of the series 4 ( s ) = 2 E("(Q)aX/kl 0
one may verify 'that for arg s = 8
the integral converging because of Lemme 2.1 in the hdf plane Rl eaz > lm 1 sin 0 I. That X(z) is analfiic exaept pomibly on the segment [-hQ, kSZ] is now evident. Equation (2) may be demon~trated
824 COMPLEX INVERSION THBORY [CH. IX
by &st taking C as a circle I z 1 = vQ(1 +- E ) and applying the calculus of residues to the integral (2). Cauchy's theorem then shows that C may be deformed to the more general curve described above.
Theorem 5.1 is well known, see V. Bemstein 11933; 294-31 11. E ~ m 1. E,(8) = (008 ~ 8 ) ~ . We have
Consider the function yp(x) which is e - 6 ~ - ' for x 2 0 and 0 for x < 0 - axld let *
(This formula ia valid for R1 p > 0.) ~f ~1 p > + then
thus, by Parsevd's equality, if R1 p > f, R1 cr > .f, we
Appealing to the principle of analytic continuation we see that this
x formula holds for R1 p + R1 o > 1. Let t = tan - p + + = v,
2' ( p - a)/2 = 8; we obtain
this result being valid for R1 v > 1 and d (complex) 8. Since
6 51 THE INVERSION BUNCTION
we have
The caae v = 1 is included under Example 1. We have
To obtain a formula valid for 0 < v -< 1, we note that
It follows that
Thus
Integration by parts gives
Thus h a y
t S' [-(aI~-l~;(~ - + :8.(;)] (Z - it) at (0 <V < 1). -a
This formula is of course valid for v 2 - 1; however if v - 2 1 it may be
286 COMPLEX INVERBION THEORY [CH. IX
reduced to the simpler formalas given above. Thia example is due to D. B. Sumner [1949].
6. APPLICATION OF THE INVERSION OPERATOR
6.1. We consider here the application of the complex inversion operafor.
THEOREM 6.la. Let
6. Cp be a h e d redflable curve going around the s e g m d
[-ha, i?lra] in the pm'tive dSrectim and lying in the a t ~ p I Im z I < A l p , trJlenfor-a<u<oo
The inner integral convergw uniformly for z on Op by Theorem 2.2b, and we may therefore interchange the order of the integrations to obtain
1 1 eCt da(t) 7 B(u + p - t)K(z) dz.
2m op We have wen in Theorem 2.2a that the integral
converges uniformly for z on Cp. h3ertin.g this integral and again invert- ing the order of integration we find t h t
as desired.
8 a APPLIUATION OF THE INVERSION OPERATOR , 427
C o r n m y 6.1. Under the assumptions of Theorem 6.la the
converges uniformly for u in any finite interval. If A and B are sats in the z plane and a and b are (complex) numbers
then by aA + bB we mean the set of all z of the form z = az, + h, with 2, E A, z, E B.
Let I be any finite interval - o ~ < u , < x ~ u , ~ c o . - - We aet J = I -+ pop. It is easily seen that J is a oompsct s e r h the z phne lying in the strip I Im y I < ?ram Given c > 0 we can find T so large that 1 t 1, 1 t* 1 2 - T, sgn t , = sgn t,, implies that
Let M = 1.u.b. I K(z) I for z on C p and let L be the length of C p . For t , and tp aa above we have
This establishes our assertion. THEOREM 0.1 b. Ulzder the as8umptiw of IPhemem 6. la. :
A. -a, < c < a, implies that
00 =La a(p, t)rctcr(z, -- t ) at - j':.~(~, t)e-ota(z, - t ) a;
B. c = a,, --a1, implie8 that
C. c > a, implies t b t oc(+co) exists and
228 COMPLEX INVERSION THEORY
D. c < -a1 implies that a(-oo) ezhb and
Let us first consider the case -a1 < c < a,. Lemma 2.10 of Chapter VI implies that a(t)U(p, -t)ect = o(1) as t + &co, fiom which it follows that
lim G(p, u - t)ebta(t) = 0 h f
uniformly for u in any finite internal. We have
QO -1, U(p , u - t)ect da(t) .
Equations ( 1 ) and (2) and Corollary 6.1 show that
f(u + P Z ) K ( Z ) az = - d u(t) - [G(p, u - t ) e-o(U-t)] dt,
dt
the integral on the left converging uniformly for zl in any finite interval. This may be rewritten as
1 QO d e-cu -. j f(u + p z ) ~ ( z ) dz = + J'_ 00 a( t ) - [a(p, u - t)e-c(u-t)] at.
2m c, du
Because of the d o r m convergence we may integrate under the integral sign to obtajn
(3) r e d . du 4 1 j(s + p z ) ~ ( z ) dr 2rrr 0, 00
-e(za-t) - G(p, xl - t)e-dal-t)]a(t) d t .
We have T
- t)e-c(*t)rr(t) dt - G(p, t)e-cta(x - t ) dt
=- x - t)e4(*')rr(t) dl + x - t ) ~ ~ ( * ~ ) a(1) dt.
Making use of equation ( 1 ) we see that P JIp~(p, x - t)e-c(etjoc(t) tit - t)e-ctoc(z - t ) = (T + m).
8 61 APPLICATION OF THE INVBRSION OPBIRATOR 229
Combining this result with equation (3) we obtain
J-a0
Using Theorem 4.1 we may show that if c is not equal to &a, then
[O(p, t)e-ct]/[G(p, t)e-ct]l' = O(1) t + &a
We have shown in the course of our proof that the integral
a(x - t ) [G(p, t)e-ct]' dt
is convergent. Employing the relatione (6) as in the demonstration of Theorem 2.2b we find that the integral
i~ also convergent. Thus
We have established conclusion A. If c > a, almost exactly the same argument shows that.
We have
230 COMPLEX INVERSION THBIORY
Using Theorem 3.1 we find that
Letting x, incream without limit in equation (6) and using the above result we obtain
We have established conclusion C. The same mgumenta suEioe to establish conoluaions B and D.
7. THE INVERSION THEOREMS '
7.1. We are now in position t o prove our principal theorems. It is convenient to estabhh them both at the same time.
THEORHIM 7.la. If
1. Q(z) b IEe$ned by quutelon2.1 (3),
C. G < -a, implies that a ( - a ) & and
8 a THB IN VERSION THEOREMS
3. K(z) ie &$d as k Theorem 5.1,
4. C,, h wined as in Theorem &la,
$7 +o, then
f(x + pz)K(z) = cp(x).
In view of Theorems 6,la and 6.lb it is suflicient to show that if the
is convergent, and if either
holds, then
Q(p, t)cp(t) dt = 0.
Let
md let
for k = 0,1, , p. If we define
then evidently 4 ( p , t ) = H,(t) * Q,(p, t ) -
232 COMPLEX INVERSION THflORY [CH. IX
It follom from this that t) is a frequency function with mean 0 and with variance 2kai2 + 2(1 - pa) 2ac2. We have
k > ~ r
the bilateral Laplace franaform converging absolutely in the strip
-a, (Rls < a, k = l ; - , p ,
- ap+1< Rl < a,,, it; = 0.
Furtherifk= 1, = , p, then
where p,(p f ) is a real polynomial of degree k - 1 and where 6 > 0; and
where p(p, t) is a real polynomial and where E> 0. From these mymptotio expansions we may h t verify that
and from this deduce that the integrals
are convergent. If the constants u,(p, t ) are defined by the equation
5 71 THE INVERSION THEORBINS
then we have U
' ( P , t , = Z u k ( p ) @ k ( p s t ) m k-0
Note that
Because the integrals (4) are convergent we obtain
By arguments like those given in fj 8 of Chapter VI one may easily show thah if
then OD
/-:G~(P, t ) ~ ( t ) at = / -a~o(p , t)B,(t) dt (k= 1, , p).
By Theorem 2.1 of Chapter VII
and thus if 1 < - k 5 - p then here exists a constant O(1) such that IB,(t) I O(l)cosha,t (-m <t < 00). Now
I-: %(p , t ) c0sl.1 at = i[Bo(p, -al) + E,(p, a,)] = O(1) p + I-.
f ik ing use of (6) we see that
a,(p, t)y(t) at = o (k = 1, . ,
It remains to show that if condition (2) or condition (2') is satisfied then
We first r t~sert if S > 0 then
(8) lim Ji, G,(p, t)yl(t) dt = 0. -1- 2 8
LBt Y (f) = f y~(u) du. Lemma 2.1 c of Chapter YI and the convergence
of the integral (1) imply that \Y(t) = o(e41cl) t + &-a. We have
234 COMPLEX INVBIRSION THEORY [CH. IX
By Corollary 4.2 lim [Qo(p, t ) ~ (t)ld_d = 0. -1 -
For 1 t 12 - d we have I Y . ( ~ ) 1 < - O(1) . Since by ~ h d r e m 4.2
Integrating by parts we find that
since lim B&p, c) = 1 for -a,+ 1 < a < Combining our re~dts , we -1-
have established equation (8).
Let condition (2) hold. Given E > 0 we choose 6 > 0 so small that I y(-t) + y(t) 12 6 for 0 < t l 6 ; - then
using equation (8) we obtain
/ {-:a,(p, t ) ~ ( t ) a 12 - E e
-1 - 6inae c is arbitrary, equation (7) follows.
Let condition (2') hold. Given 6 > 0 we choose 6 > 0 so small that
if y(f) = ~ ( u ) then I Y(t) 1 5 r 1 t 1 for / t 1 2 6 . Employing - !L+heorem 4.2 we .find that
8 81 A GENERAL REPRESENTATION THEOREM 236
R e c W g Corollary 4.2 we see that
Using equation (8) we obtain
Since e is tlrbitrq equation (7) followa. As an example we see that if
then
q(t) = lim - 1
(v > 11,
v P
p(t) = lim [I-* f(t + i p y ~ ( oos ).dy ptl -
+ 5 v 1' -= jt(t + ipy) (ooa $)'-' sin dr] (O < v < 1.1.
8. A GENERAL REPRESENTATION THEOREM
8.1. In the remaining sections of this ohaphr we shall construct a -.
representation theory oorresponding to our oomplex inversion operator. The following result is &n&logous to Theorem 3.1 of VII.
THEOBHIM 8.1. If 1. a( t ) hfid by v t h 2.1 (31, 2. K ( z ) defined as i~ T h e m 6.1, 3. flz) is andytic for I Im z 1 < nS1/8 0<8<1, 4. f (z ) = o(e.1"1"I)x+ &GO u n i f d y in every arrbst~p
I Im z I < wale', 8' > 8, then
raCl(z - t ) dt(l/%) f(t + z)R(r) dz = f(z) k where Cg is a recti$able &8& curve going wound [-kQ, hill in the positive direction, and lying C the strip I Im z 1 < ?rR/B.
236 COMPLEX IN VERSION THEORY [CH. IX
By assumption 4 and Theorem 2.2a we see that
For 6 < p < 1 let Cp be a cloeed rectifiable curve going mound [-id& i d ] in the positive direction and lying in the strip I Im z I < &/p. By Cauchy'a theorem
Assumption 4 and Theorem 2.2a show that the iterated integral
is absolutely convergent and so may be inverted to give
(112.74 K(Z) az f(t + p z ) ~ ( ~ - t ) at. 6, S-" Using Theorem 2.2a and assumptions 3 and 4 one may, for z on Cp, verify thaf
by integrating around the rectangular contour Im t = 0, Im t = -1m pz, Rl t = AT, and allowing T to increase without limit. Inserting this we obtain
U ( x + pz - t ) f ( t ) dt.
By Theorem 7.lb
r
Our theorem follows.
9. DETERMINING FUNCTION NON-DECREASING 9.1. THEOREM 9.1. Let U(t) be clejined as in equation 2.1 (3) a d K(z) as in
a T h e m 5.1. Necesscwy and 82cficient d i t i m that f (z) - - j" - with a(t) € f are:
1. f(z) ia analytic in the st+ I Im z I < A, 2. f(z) = o(&lxl) for x -+ &a, u n i f d y in every groper substrip
I Y IS""(Q - 7)'
The necessity of conditions 1 and 2 follows from Theorem 2.2b and Theorem 3.1. The necessity of condition 3 ia a oonsequence of Theorem 6.la which impliea that for 0 5 - p < 1, -co < x < ao
Let 0 < 9 < 1. It is evident from conditiom 1 and 2 t h t f (Ox) satisfies
the aesumptions of Theorem 8.1. T h e f (Ox) = J G(z - t )yo(t ) B - CO
where yo(t) = (1/2&) (8t + Bz)K(z) dz. By condition 3, yd(t) - 2 0.
By Theorem 5.18 of Chapter VI we have
where #,(t) = (1 - D a a ; 2 ) ~ ( t ) 2 - 0 for -00 < t < oo. It followa
that
Allowing 8 to approach 1 we obtain
Equation (1) and condition 2 for z rml together with Theorem 6.1 of Chapter VII now imply our desired result.
As an example we see that necessary and sufficient conditione that
1 "O 1 f(4 =, 1, sech - 2 (z - , t ) da(t)
with a(t) E f me:
1. f (2) analytic in the strip I Im z I < v,
9. f (2) = o(el"l12) x + fa, uniformly in every subatrip ~ I m z ~ < w - ~ ,
3. f ( x + i p n ) + f ( x - i p ) > O - 0 1 p < 1 ? - - a , < x < c a .
838 UOMP&EX INVERSION THEORY [CH. ][X
Aftex a logarithmic ohange of variable we find that necessazy and sufscient conditions that
win A(t) E f are: 1. P(z) d y t i c in the sector I arg z 1 < n, 2 . F(z) = o(1) for I z I -+ co, F(z) = o( 1 z 1-1) for I z I + 0, d o r d y
ineveryseotor I w g ~ I < a - ~ , - q > 0 , 3. e@l2~(mcd) + e-@l'~(re-~') - 2 0 0 < r < m , [ 6 l < n .
This result is evidently related to the theorem of Herglotz, conoerning functions with positive imaginq part. gee J. A. Shohat and J. D. Trtmarrkin [1943].
10. DETERMINING FUNCTION IN L*
10.1. We oonolude by estabhhhg one other typical representation theorem. Further multa may be obtained by essentially the same
as in T h e m 6.1. Nea88ay a d 8u&ie& wnditiOrCB that f ( x ) =
JI:(z - t)q(r) dt with ~ ( t ) E L.(-m, a), 1 < p < m, are:
1. f(x)~ana&bi?&the8t*ip Ihz.l<&; 2. f(z) = o(e41") for x -+ f 00, unifmnly in evey &strip,
I I m z J l a ( n - q), 1, > o ; C
39 ( ( ( 1 / % ) ] f(x 4- pr)K(z) hl(,s M, O S p - < 1 for s m CD
&nt M independkk of p. Here Cp ia d e j d ae ila T b e m 6.la. The necessity of conditions 1 and 2 follows from Theorem 2.2b and
Theorem 3.1. Besuming that f ( x ) hm the desired representation we have by Theorem 6.la
By Holder's inequa3ity if ~1 + q-1 = 1, then
I J;aa(P, 1; - t)lp(t) dl
- t ) le(t) lv at [Iw ~ ( p , - - O D
t ) a] Pk, from which it follows that
This establishes the necessity of condition 3.
8 101 DETERMINING FUNCTION IN Lp 239
To establish the sufEioienoy of our oonditions let 0 < 9 < 1. Using Theorem 8.1 we see that
f(e4 = j':'a(z - t ) l , ( t ) dt
where yro(t) = (11%) f(Bt + Bz)K(z) dr. Condition 3 implies that Qe
1 1 y e ( 1 1 M By Theorem 4.1 of VII there exists a sequence of values { o . } ~ - ~ , 8, + 1 - , and a function p(t) , 1 1 ~ ( t ) 1 l p < - M, W C ~ that if x(t) E LQ (-03, m) then
Since for each a, Q(x - t ) E La (--a, co) we obtain in the limit
11. SUMMARY
11.1. In this ohapter we have eeen how to sesociate a complex inversion formula with each convolution trantlform whose kernel is of the form
@(t) =l 2& f a -ice, [O(l --$)]-lC'da,
where O < 4 2 % 5 a 3 4 * ' * s
For a, = E - t thia formula reduces to the familiar complex inversion of the Stieltjes transform.
CHAPTER X
Miscellaneous Topics
1. INTRODUCTION
1.1. The present chapter consists of four short fopios whiah although connected with the remainder of the book me not related to each other. In eection 2 we consider the theory of generaked Bernatein polynomials, in aeotion 3 the behaviour of convolution trassforms at W t y , in section 4 the nlldytio character of kernels of c h s I1 and clam III, and in section 6 non-quai-mdytic olasses of functions.
2. BERNSTEIN POLYNOMIALS
2.1. A classical theorem of Weieretrass assertg that a funotion f(z) continuom for 0 5 x 1 can be uniformly approximated by polgnomkh. S. Bernsfein hae give& explicit method of effecting this approximation in terms of the fuatiom
the Bernstein polynomial of order n corresponding to f(x) being n
Bn[f(x)l = 2 f(~l%)l*,,(~). rn-0
Forf(x) continuou~ on 0 < - z I; - 1 Bernatein hae proved (see D. V. Widder [1946; 1521) that
lim B,[~(X)I = f(x)
u n i f o d y f o r O < z < - - 1. It is convenient to replace x(0 < - x < - 1) by et(-a, 2 - t < - 0) in t h e
formulae above. We h d thet if P(t) is continuous for -a < - t - 0 ( F ( - a ) exista) then
uniformly for -00 2 - t < - 0. The formula, defining Euler's constant y,
lim [ 1 L a g n ~=lj
8 21 BERNSTEIN POLYNOMIALS
shows t h t the quantities ?n 1
log- and - n d-m+lJ
are very nearly the same, at least when n and m are large. Thia implies that the following variant of Bernstein's result,
uniformly for -a, 2 t < 0, is true. Let us recall the following relation k r n 5 10 of c h a P t e ~ 1 1 ~
Expreseed in this form Bemutein's mult sug;g~~lts the following generalization: if
O = a , < c a l < a , - - * , where
""1 (3) 2 - = c o , l i m a , = c ~ ,
1 ak it-+ a and if
uniformly for -m < - t - 0. Note that the conditions (3) are obviously neceasary since otherwise no use would be made of value of P(t) in certain intervals. In the following sections we shall establish this conjecture. See I. I. Hirschman and D. V. Widder [1949], and aha A. 0. Gelfond [1960].
23. We msume as given a aequence of real numbers
O=a,<a,<a,< * * - .
We set
and
(2)
for (m = 0, 1, - , f i ; n = 0, 1, m).
242 MISOELLANE0 US TOPICS [W. X
! ~ E O R E M 2.2a. If D, , (e) and A,, are &Jned os above then:
Making use of the definition of An,,(t) we fmd that
We will establish this by induction on n. It is true if n = m. Suppo~e that it is true for a general n > m. We will prove that it is then true for n + 1. By our induction assumption equation (3) is valid. Multi- plying both sides of (3) by ( 1 - a,a;$,)/(l - aazl ) we obtain
8 21 BERNSTEIN POLYNOMIALS
Adding l / ( a - to both eidea of t h i ~ equation we have
2 p , l l k ( ~ m ) / ~ n + ~ # k ( ~ ) = 1 / (8 - a,)
as desired.
COROLLARY 2.2b. Annl(t) 2 - 1 ( - - m I t $ 0 ) . This is an immediate consequence of Corollary 2.2a and oonoluaion
A of Theorem 2.2b.
If 0 5 - a < - awl then we set
The function \,(a, t) is defined exactly as A,,(t) except that a is used in place of a,. As a consequence of this remark we have
T E I E O ~ M 2.20. If h,,(a, t ) M deF32ed as dove t b n :
2.3. We ahaU need the following result LEBUU 2.3. If 0s - x, 2 l ( i = I., 2, *, n) then
By Taylor's theorem we have
so that our lemma, is true for n = 1. I t is evident tbt
O< - e-(%+"'+Qm) - (1 - 2,) ( 1 - 2,).
To show that
we proceed by induction. We know it ia true for n = 1; assuming that it is true for n we will prove it for n + 1. If
1 = ' ' +"*+I) - (1 - x,) - (1 - xn+J then I = I , + I , where
- (1-23 * ( I --x.)lS f(4 + ' +Zf.), b m which it follows that
aa desired. We set
The generalized Bernstein polynomial of order n corresponding to a function P(t) (-00 2 - t - < 0) is
PZ
2. F(t) istxwatin,wfw -03 - 2 t < - 0, then
lh B,[F(t)J = P(t)
uniformly for -a < - t < - 0.
We first note that if I F(t) I < M for -00 2 t < - 0 then I B,[F(t)J I < M in the ~ a m e internal. Thisfollows from Corolhy 2.2a and Theorem z2b. We assert that it is sdficient to prove om theorem for the special fonotions f ( t ) = ef: j = 0, 1, . Indeed, let F(t) be an arbitrary fanation continuous for -a < - t < - 0. By the Weierstraes approxi-
?n mation theorem given c > 0 there exists a polynomial P(t) = zp,dt suah that 3-0
1 IP(t) - P(t) 1 la s € 0
By the above remark
I p%[P(t)l - B,CF(t)III, $ E (n = 0,1, - ). Sinoe we aswmed our theorem valid for dt j = 0,1, - B,[P(t)] tends d o d y to P(t) aa n + 03. That is, there exists an integer N such that
I I Bn[P(t) - P(t)ll , S E (a 2 N)* Thus combining our results
I I Bn[p(t)l - B(t) 1 1 , - 3~ (n 2 N). Since E is arbitrary our theorem follows.
We will now show that for 0 < a < co B,[eat] tends uniformly to eat as n + co, which will oompleG our proof. This is immediate for a = 0. Since B,[l] = 1, n = 0 , 1 , . We suppose a > 0. By definition
ta
Choose rn so that a,, > a; by conclueion B of Theorem 2 . 2 ~ n
eat = P,,,(a)k,,(a, t ) + P,,,(a) Aqr(t). k - m f 1
It follow^ that Bn[eatI eat = Jl( t ) $ J2(t) + J3(t)
where m
U~ing Corollary 2.2a and Theorem 2.2b we see that
Using Theorem 2.20 and Lemma 2.3 we find that
It follows that lim J,(t) = lim J,(t) = 0 uniformly for (-00 2 - t 2 - 0 ) . -00 %+a,
If c and n are given let r(n) be largest integer such that e-Runo..b 5 r for b < - r(n); r(n) is defined for all sufficiently large n and r(n) Yoo as n+m. Weset
Js(t) = J*(t) + J&t) where
- .--, JAt) = A,, ,(t) [e-mnvk - P,,, (a)] ,
b=-m+l
Using Lemma 2.3 and oonolusion A of Theorem 2.2b we see that
For r(n) + 1 < - E 2 - n we find, using Lemma 2.3, that n
If R(n) = (a,(,,,)-I then evidently n 1 2 ar2 (awl)-1~n8k R(n) - log (i) *
~ = E + I a
By conclusion A of Theorem 2.2b
from whioh it follows that lim J,(t) = 0 uniformly for --a, < t < 0. - - H a 0
Combining our results we have shown that
- - -
Since r is arbitrary our theorem is proved. 2.4. Instead of the points on, we may use any sufficiently dose set
of points t~&. We set u
COROLLARY 2 . h . If 1. {a,}: ssatiafies conditions 2.1 (3),
2. F(t) is continuous -00 < t < 0, - - 3. emu- - e-~:#m + 0 o n n m uniformly in m,
then
We have
If p(t ) is conthuons for --a, - t < - 0 and if oondition 3 is satisfied then
3. BEHAVIOUR AT INFINITY
aD
(1) E(a) = eba 1 - - e81ak k -1
where b, {ak}: me red and , b)
00
!I 31 BEHAVIO UR AT IN4INlTY
and let
(2)
(3)
We ah4 consider those kernels a(€) for which
In the following sections we shall ehow that if f(x) is defined by (3) then f ( x ) cannot approach zero too rapidly as x approaches infinity without being identically zero. Let us illustrate this by means of the Laplace and Stieltjes transforms. It will follow from our results that if
F(x) = dA(t) ( X ) 0,)
and if F(x) = 0(ewra) ( x + +a, for every r > 0)
then 6 ( x ) = 0.
Similarly if
F(x) = 6; ( x + t ) - I dA(1)
and if F(x) = ~ ( e - ~ ~ " ) (x+ $00, for r > 0)
or F(x) = ~ ( e - ~ ' " ) ( x + 0+, for every r > 0 )
then F(x) 1 0.
See I. I. Rirachman, Jr. [1951]. 3.2. We first consider the case when G(t) € class 11. LEMMA 3.2. If
1. a ( t ) € class 11,
then ~ ( t ) = 0 almost everywhere (p < t < a).
p*(s, = *d'v(t)c+l dt.
This bilaterd Laplace transform converges ab~olutely for ( G - 0 < CO).
We know that 00
(2) llE(8) = J- au(t)e4t &,
the integrd converging absolutely for (-a < cr < a2). Sinoe U( t )ce t /E(c) hea i ( s + c)/E(c) a8 its bilaterd h p h c e transform, and since B(8 $- c)/E(c) f a, it follows h r n Theorem 5.1 of that @(t)e-Ot is bell-&aped. Thus if x is sufficiently large and negative
I h(z) 1 - 2 eoa[rna. o(r - t)e-"*otl J Q I tp(t) I dt P S ~ Z ; ~ P
= O[@(x - plI (x+ -a).
From thb and from assumption 4 it fob that
for some positive constant A. This implies that the bilateral hplace
converges absolutely for -00 < o < a,. Since the integrals (1) and (2) b v e a common strip of absolute con-
vergence (c < cr < a,), the oonvolution theorem for the bilateral Laphoe transform, D. V. Widder C1946; 2571, tells us that in this strip
Equation (5) provides a, continuation of p*(s) into the hrwlf-plane o c so that v*(a) is an entire function.
By a change of varieble in equation (1) we obtain
It is clear that e@tp*(e) is bounded in the half plane B 2 c . In particular eHq*(a) is bounded on the line (a = o, --a < r < co)- Using inequality (3) we have
I h*(8) 1 s. A[E(4e?J-l (-m < 0 < a,), from which we obtain
(6) 1 * 1 01) IB(8)/E(4 I --a < u < a*. It follows that eMtp*(e) is bounded on the half line (-a, < u c, T = 0). -
8 $1 BEHA VIO UR AT INFINITY 249
We further amert that
that is, eMp*(s) ia at most of order two minimal type in the half plane Rl s < - c. It follows from (6) that
where t a k = a k C.
We have
Therefore given E > 0 we can choose n so large that
Making use of the elementary inequality (1 + x) - em for x 2 - 0 we 6nd that
From this it follows that - lim 1 8 1-2 log I ep8tp* (8) 1 5 - 6 RI. s < - 0.
la1400
Sinoe E is arbitrary our assertion is proved.
We now know that eP8d*(8) is bounded o n the linea (o = c, -00 < T < CO) and (-00 < s< c, r = 0) and is at most of order two minimal tspe for o < c. ~ ~ ~ 1 % ~ the Phrapen-LindelSf principle, see Titahmamh [193K 1781, to eaoh of the two quadrant8 of thia confipmtion we h d that e 9 * ( 8 ) is bounded for a c, and thus in the entire plane. By Lioudlle's theorem eNtp*(s), Ging entire and bounded, is a constant. See Titchmazah 11939; 853. Since
00
eiptp*(c + ir) = 6 p(t + p)~4r dt
it follows that p*(s) E 0. This implies that p(t) = 0 almost everywhere ( p 2 t < co) ; q.e.d.
1. U ( t ) € cl&8s II) Po0
2. f (x) = G(x - t ) da(t) ha8 ab8cis8a of mvergence. yo, J-a
3- f ( x ) = O[Q(x - p) l ( 5 - j +4, then a(t) is constand for p < t < co.
Let pa + 1 be the multipliaity of s - a, ae a zero I#($), end let
so that a = H , * H .
By Theorem 8. lb of Chapter VI
where
- it follows aa in 4 6.2 of Chapter VI that if e > 0 then
#(t - p)/H(t - p - r ) = O(1) ( t+ +a)= TO 8~ this let L(t) and Ll(t) be defined by the relations
00 r
03
t = b + z + Pa + 1 - (Pa + 1)Ll k - ~ a k ( a k + A,) a2 aa(aa + l;,)'
By T'heorern 3.4 of V (dl&) 1% act - p1l .W - p - €1
= -L(t - p $ ~ ( l ) ) $ Ll(t - p - E + 0(1)). The relation defUllng Ll may be written in the form
L-l[L1(t)] = t - Pa f 1 (pa + l )Ll a2
+ and thus
aa(aa + L1)'
Consequently (dl&) log U(t - p ) l W - P - 0
= - L ( t - p + o(1)) + L( t - p - 6 + o(1)). Since L ( t ) ia increasing we have, if t is sufficiently large,
or equivdently a(t - p) /H(t - p - E) E $ . Our aseertion now follows. Thus assumption 3 implies that
Chooae xo > yo. We have
If z > X , then, by Theorem 3.1 of IV, H ( x - t ) / R ( z o - t ) E f . By the mean value theorem [ ~ T H ( Z - t ) A ( t ) dt
The number 5: dependa upon x. However there is a constant dI such that
for all t, so that
(8) r 'd H(x - t ) A ( t ) dt = O[H(x - p - e)] x-+ +a.
Theorem 2.1 of Chapter VII implies that A ( t ) = o(saat) as t + + co. Let pa be the smalleat pole of ( 1 - 8~4-')h+'/E(e), and ohoose o, a, < o < p,. Then if p(t) = eWctA(t) we have
and (7) and (8) combined give
It follows from Lemma 3.2 that p(t) = 0 for p + E < t <. a. From this it ia eaaily deduced that a( t ) is conatant for p + e < t < a, and since E ia arbitrary for p < t < co.
mmonm~ 3.2b. If 1. a ( t ) e class 11,
a 2. = [' - t ) d.(t) ab8ckso of urnverge- yc,
3- f ( x ) = o[a(z - p)1 ( x + +GO, for every p), t h f ( x ) = 0.
This is an i m m a t e aonsequence of Theorem 3.2s. Applied to the Laplace tramform Theorem 3.2b gives the result mentioned in 8 3.1.
3.3. Let B(t) E olms I be deked as in 3.1(2) and 3.1 (4). We ctssochte with G(t) a, cltw I1 kernel 8 ( t ) as followa. W e set
(2) s( t ) = (2ni)-l J [&(a) ]-lest ds. -3 00
we 8haJ.l show that if
(3) f(x) = O[$(X - p)l (x+ $00, for every p), or (3') f(4 = OI@(p - 41 (x+ -00, forevery p), then
f(x) - 0. In order to illustrate the essential point of the mgument which follows let us consider s very special case. Let
where q(t) f L'(-co, m). We denote by S the Fourier transformetion
S t y = l 2 i m fT y(u)eitU du,
and by F-1 its inveree
$C;-'y = (2rr)-lP i p ( u ) ~ - ~ ~ ~ a%. 2-00
We ass& h t i f f ( x ) iB given by equation (4) and if
(5) cy(t) = ~ r ~ ~ ( - i u ) b ( - i u ) - ~ P ~ ~ ( t ) then
(6)
The txansfonnaton ~ ( t ) + y(t) defined by (5) ia unitary; that is I I p, 1 1, = I I cy I I,. This follows from the relation
and Plancherel's theorem. Transformations of thia type are called Watson transformations, see Titohmarsh [1948; 212 et eeq.]. To establish (6) we note that as a conaequenoe of (4) we have
0 31 BEHA VIO UR AT INFINIITY
Equation (6) is equivalent to
F* f = S(-iu)SJJ = S ( - i u ) E ( - i ~ ) B ( - i u ) - ~ S ~ q = E(-iu)Fuv,
which we have just seen to be true. Thus f(x) is represented aa a trams- form with a class 11 kernel g ( t ) . If (3) holds then Theorem 3.2b shows that f(x) must be identically zero. (A similar argument shows that if (3') holds then f ( x ) is again identically zero.) THEOREM 3.3. If
1. a ( t ) is d e f i d by 3.1 (2) and 3.1 (4) a d bebngs to ch i3 1, 00
2. f (x) = 1' a a(. - t ) da(t) m e r p , '
3- f(z) = O [ ~ ( X - p) ] as (z+ +a, for every p) , or 3'. = O [ S ( p - x ) ] MI ( x + - a, for every p ) ,
t h f(x) = 0. Using Theorem $.la of Chapter VI and Theorem 2.1 of Chapter M
and mguing as in the proof of Theorem 3.2a, it is easy to see that it is no restriction to Etssume that
where
pl and p2 being such that
Suppose that 3 holds. Let o - 2 2a,
and let A,= 1ak1+6,
g1(t) is a class I1 kernel olosely related to B(t). We will show that f(z) can be expressed in the form
This is snalogous to the formula (6) and serves the same purpose. We note that if 7 > 0 then
%(t) = o(e"l+q%) t - t $-a (9) = o(e(%-vy t+-00.
(Actually much more ia true.) It is evident that
(a) G?~(S)/E (a) is andytic for a, < Rl s < a,. Because of our choice of c
decreaaa as 1 T 1 inoreases if a, < cr < a,. It followe that if a1 < al < u, < or, then there existe a constant A euch that - -
A aimple application of a theorem of Hamburger, see Widder [1Q46; 2651, now shows that if
than
the bila,terd Laplace tranaform converging ab~olutely for a, < R18 < a$. Because of the order relation (b) we may deform the line of integration of the integrd defining J(t) to R1 a = a, - r] or to Rl s = al + 7. We obtain
and theae together with (b) imply that
FinaJly the aonvolution theorem for the bilateral Laplace transform, D. V. Widder [1948; 2671, gives
Thus 00
g l ( x - u)J(u -- t ) du.
Because of the aonditions (8), (B), and (10) this iterated integral is absolutely convergent. Consequently the order of the integrations may be inverted fo give
00
( 1 1 ) f(x) = j" a 4 ( x - ~)r(u) du,
where
I(%) = j':,"b -- t)e)(l) a* We next aaaert that if
then
for any c > 0. To prove this we introduce
Proceeding ae in 8 6.2 of Chapter VI one may show that
We have, see 8 10.2 of 111,
from which it follows that
thua proving our assertion. Assumption 3 now implies that
The relations (11) and (12) and Theorem 3.2b show that f (x) must vapish identically.
3.4. The following result may be proved exactly as Theorem 3.2a was proved, .The analogue of Lemma 3.2 whioh is needed here reduces to a special, w e of a well known result, see Titohmarsh [1948; 322-3271,
266 MISOELLANEO US TOPICB
! C H E O ~ 3.4. If 1. a(t ) &fined, by 3.1 (2) bebnqs to class 111, 2. a(t) is of bounded variation for T < tl 5 - t < - t2 < a,
thn a(t) is colaatant for ( p < t < a).
4. THE ANALYTIC CHARACTER OF KERNELS OF CLASSES I AND I1
4.1. The best known example of a class I1 kernel is
which is an entire h t i o n . Thia suggeats that conceivably all class 11 kernels may be entire functions. We dull prove that this is true.
then Q(t) M the ratridioll to the rsat m ' 8 of a function U(w) ad3(tic in the entie plane.
We m y assume without loss of generality that b is zero. It is easily verified that if a - ) 0, then
It follows that for s in these seotora we h v e
From inequality (1) we see that for each m there exists a constant B > 0 8uch that
5 41 THE ANALYTIC7 CHARACTER OF KI3RNELS 367
Let C be the contour below where n/4 < - 8,< - n/2.
Inequality (2) immediately enables ua to deduce by e standard argument t h t r
Again using inequality (1) we see that for each n, there exists a constant B' > 0, such that
It follows that the integral ((3, with t replaced by w, convergea uniformly for w in any compact subset of the open sector shaded in the figure below.
FIG. IV n
Since n may be chosen arbitraxily large and since lim Z arl = co the wco 1
integral (3) (with t replaced by w) converges uniformly for w in any compact set, and our theorem is proved.
4.2. In section 9.9 of Chapter I11 it waa ahown that if
then G(t) = @i(f, t) for oo < t < 0. Thus Q(t) is, for -00 < t < 0, the restriction to the r-1 ax& of a function Q(w) analytic for R1 w 2 - 0. !Chis is a special case of the following result.
Q)
then #(t) ie, fm -00 < t < B + Z a ~ l , the r a t r k t h to th real d 'I
" 1 of a function G(w) analytic in th hal f -pbe Rl w < b + 2 -.
1 ak
It is no restriction to aaswae that b is equal to zero. It is ert~ily verified that given 0 > 0, there exists a constant A such that
It follows that for s in these seotors we b v e
From inequality (1) we see that for each n there exists a constant B > 0 such that
Let C be the contour below.
PIG. v We have
Inequality (2) enables us to deduce by a atandard argument that
Born inequality (1) we see that for each n there exists a constant B' such tbt
It follows that the integral (3) with t replaced by to converges uniformly for w in any compact subset of the open sector shaded below.
Since 8 may be taken arbitrmily small and since
we see that the integral (3) converges for w in any compaot subset of the "O 1
half plane Rl w < 2 - . k - 1 ak
It can be further shown that under the assumptions of Theorem 4.2 OD
the line Rl w = b + 2 a;' ia a cut for G(w). Since the demonstration 1
of this makes use of results from the theory of general Dirichlet series we omit it here.
5. QUASI-ANALYTICITY
5.1. Let M,, n = 0, 1, - , be a sequence of positive oonstaxlts, normalized by the condition Mo = 1. The sequence (111,): is said to be non-quasi-analytic if there exiats a function f(x) + 0 such that
(here A and k are constants which may depend upon f(z)) and suoh that for some x,
If ' n ' ( ~ o ) I =: 0 (n = 0,1, ).
Using our theory of convolution transforms we can prove the following r e d t which is of fundamental importanoe in the theory of quasi- analybicity. See alao I. I. Hirschman and D. V. Widder [1952].
THEOBEM 5.1. ~j {M,}: M a m i z e d sspuewe of positive mwttznta and if
then M,, is m-quad-anatytic.
260 MISOELLANEO US TOPICS [Ck. X
Let a, = Mk/M,, k- l ,2 , 9 and let us define
U(t) ie a kernel belonging to 0 1 ~ 8 111. We assert that ifp(t) E La (-03, a) and if
then
We know h m Theorem 4.2a of Chapter VII that if {A,, l , A,) is any selection from {a,, a,, ) then
(3)
Since
Dn/al a, oan be expressed as a sum of 2" term of the form
Thus IDnj( . ) Ig2na~o * a n l l ~ I l a ,
If y(t) = 0 for t 2 T then f ( x ) = 0 for t 2 T, which implies that f("(z,) = 0 for T - zo < a. It ia obvioas thet f cannot be identically zero unless q(t) = 0 almost everywhere.
For an extensive treetment of the theory of quai-analytic functions see S. Wmdelbrojt [IW].
ASSEN, M., E D ~ I , A., SCHOENBJERQ, I. J., and WEXTNEY, A. 1961. On the generating f~nation8 of totally positive sequenctee. Pro&- of
$he Ncctkmat Aea&emy of S h e s , vol. 37, pp. 303-307. AISSEN, M., S C H O E N B ~ Q , I. J., and W ~ Y , A. 1963. On the generating funotions of totally positive sequencee. J m l d;'Andyae
Matirkrrscctiqw, vol, 2, pp. 93-103. A~UTOWXCZ, E. J. 1948. The third iterate of the Laplace transform. Duke Mahmatical J-,
vol. 16, pp. 1093-1132. BANAOH, S. 1932. Thdmk des opdrathrrs Ziradai~ea. Warsaw. BARRUOMD, P. 1960. M6rJization de la transformation de Stieltjes it8de: tl;snaformation d'ordre quelconque. Comlptss Reradw d@ 8hncss du Z'Aoadthis th Sce'mea, vol. 231,
pp. 748-760.
BOAS, R. P., JR. 1937. Asymptotio relations for derivatives. Duke M c s t ~ ~ J w W , vol. 3,
pp. 637-646. 1942. Inversion of s generalized L a p h integral. Procedirag8 of the N m
Academy of Sdmca, vol. 28, pp. 21-24,
BOAS, R. P., Jn., end WIDDEB, I). V. 1939. The iterated Stieltjes tramform. Tmnsuc-tiona of th AmsP&c&n M&htmatiwl
Sodcty, vol. 46, pp. 1-72.
B~CHER, M. 191 7. L e v Bzrr 148 d thod&8 de S t m . Paris. BOOHNEIR, S., and C F I A N D R A S E X E ~ , K. 1949. Po21der tmr~g fm. Princeton. BOOIINHIR, s. , and WIDDBR, D. V. 1948. A homogeneous differential system of inhito order with non-vanishing
solution. Buttatin 01 the Amdcan Mathmuticat Bociety, vol. 54, pp. 409416.
EDREI, A. 1953. On the generating functions of totally positive eocponoes. JozM.rrcsl cl'duaatyss
Ildatllematiqw, vol. 8, pp. 104-109. See also Arssm, M. Q A N T ~ ~ A X ~ R , I?., and KRSIN, M. 1937. Sur lee matrices cornpl6toment non-negatives et osoillahires. Comp&o
Mc&emtka, vol. 4, pp. 446-476.
GABDEB, A. 0. 1964. Topios in tho theory of convolution transforme. Thesis, Washington
Univensity .
G m f g ~ m , A. 0. 1960. On the generalized polynomials of 8. N. Bermtein (in R d a n ) . Izu. A M .
Naak B.S.S.R., 8er. meth., vol. 14, pp. 413420. H~RQZOTZ, A. 1911. Ober Potenmihen Mit posit-ivem mlen Teil im Einheibkmis. B&e
itbw die Vwhuncilungm &I kmigZi& iraciraischra Ges&ch@ &er Wk8enso- zu Leipig, mthemat&?ch-p?by86eche Kk.886, vol. 63 [6-93, pp. 601-511.
HILml, E. 1948. Functional analpie and aemi~groups, Amwkan Mctbhmticd Society
CoZloqudum PubZicatim, vol. 31. IEESaxm, I. I., JB. 1951. The behaviow at city of certain convolution t rdornns. Tmnaadorw
of the Ammidata Mahm&xal Society, vol. 70, pp. 1-14. 1953. Systems of partid differentid equations which generalize the heat equation,
Carwcdk9~ JozMnal of M a h m d c a , vol. 5, pp. 118-128. Hnmomzm, I. I., JB., and W I D D ~ , D. V. 1948a. Generalized inversion formdm for convolution tranefonhs. Duke Mathe-
mathaZ J W , vol. 16, pp. 659-696. 1948b. An invereion and representetion theory fop convolution transforms with
totally positive kernels. P&blag8 of the N a W Academy of Scimse, vol. 34, pp. 152-156.
1949a. mneralized Fhm~teh p o l y n o m . Duke ddcrthendad J d , vol. 16,
1949 %Po . 433-438* T h e inversion of e general olaee of convolution transforms. T K I C ~ ~ of the American Mathemdm1 Sode#y, vol. 66, pp. 135-201.
1949c. A repmaentation theory for a general o h of oonvolution traaeform9. T m ~ c t G m s of #he A m e h n MatlsePrwrtU So&&, vol. 67, pp. 69-97.
1950~. Generalized invereion formdm for aonvolution, t d o r m s , 11. Duke MdAsmatkaZ J d , vol. 17, pp. 391402.
1960b. A minhtum theory in illUBtmtion of the oonvolution traaeform. d m ' w n Mat- M o n t Q , vol. 67, pp. 667-674.
196la. Convolution t r d o r m a with complel kernels. PmYc Journal of Math- mdim, vol. 1, pp. 211-225.
1961b. On the produots of funotions repreerented ae oonvolution t r d o r x u . Pmesdkqa of the Amrimn Mot?mmthZ Socie#y, vol. 2, pp. 97-90.
1962. A note on quasi-malytio funations. PubZicccW cle Z lmti&ut Matipkddqzce de Z'Acudhd Berbs b Scimceu, vol. 4, pp. 57-60.
Hoseow, E. W. 1926. The theory of f d w of a real vaddte, vol. 2, ed. 2, Cambridge, k m , M. Sw U-, F. b a w m t m , E. 1882. Sur lee fonotions du genre d r o et du germ un. Cmpttw R d w dee Shmss
d6 Z'Acaddmk &a Scignccs, vol. 98, pp. 828-831. L&m, P. 1926. O W (IC8 p W l i t $ s . Pmh. LoREmz, Q. Q. 1963. Bemrrtain PoZywmiale. Toronta, Maams, W. , and O~E-GER, F, 1948. Pwnaeln und E W fih. die q h e k Fuddbam &r snatligtrwhchera Phyebk:.
Berlin.
~ E A J ~ B I I I B E O J T , 5. 1942. Analytio functione and o w of idnitely differentiable funotiona. Rice
Iwthte Pamphlet, vol. 29, pp. 1-142. M:-, T. 1936. B&tmge zur T M & linewen Unglaiciucngm. J e d e m . OB-QEB, F. Bee M a ~ m e , W.
POLLARD, H. 194th. The reppeeentation of 6 - 3 ~ E I a Laplaoe integral. B d l i n of Ua American
Mat- SoCiStg(, V O ~ . 52, pp. 908-910. 194613. Integrd t ~ f o ~ . Dub M J m d , vol. 13, p 307-330. 1 9 4 , ~ ~ The inversion of the trmsfo--eiterated Stieltja gmeh. Dubs
~ a t ~ a ' c c l l J d , vol. 14, pp. 129-142. 194713. The integral transforms with iterated Laplaue kernels. &ke, M-Z
JoUI*GQI, vol. 14, pp. 669-676. P~LYA, u. 1913. 'Uber Anniihe-g d w h Polynome mit lsuter reelen Wurzeln. Rendiconti
deZ Circolo M w dsl Palmno, vol. 36, pp. 279-296. P~LYA, G., and SZEG6, a. 1926. Aztfgabm urad Luhr8dtzc aw deT Analgeis, vol, 1. S a ~ o m m s ~ a , I. J. 1930. Ober variationsvernaindernde lineare Transformationen. Mat--
Zeitachrift, vol. 33, pp. 321-328. 1947. On totally positive fumtions, Laplace integrals and entire f~~t i0XlS of the
Laguem-P6lya-Sohur type. Promdin* of th National Aaademy of 8-, vol. 33, pp. 11-17.
1948a. Some mdyticral mpeota of the blem of smoothing. Stdim a;& E s w a r P r ~ t w d so R. Uourant an his 60 B&&y, January 8, 1948, pp. 351-870. Interscience Publishers, Ina:, New York.
1948b. On variation dhiniahmg intagrd operatore of the oonvolution type. P d i ? b Q 8 of thO N&*0d A c a k ~ of SChC88, ~01. 34, pp. 164-1.69.
1960. On Pblya frequenoy funotionsl. 11. Variation-diminishing hhgre;l operators of the convolution type. A& S-m Mathma&amm Smged, vol. 12, Leopoldo Fejbr et Frederioo Rima LXX -08 natis dedioatus, Pam B, pp. 97-106.
1961. On Pblyrr, frequenoy functions I. The totally positive funotion~ cmd their Laplace transforme. J m d d9Arcdy8s M a t h h d q w , vol. 1, pp. 33 1-374.
$oao~mmw, I. J., and WHITNEY, A. 1949. Sur la positivite dee determinant0 de translatiam de funotiom de fdquence
de Pblya, aveo une appliosltion & une problbme d'intarpol&tion, 00~lly3ta R d w dm SBamw de Z'Acudhie d68 S d e w , vol. a28, pp. 1990-1998.
1961. A theorem on polygons ka n dimensions with appkoatione to variation diminbhing and oyclio variation d imehing linear t r ~ o ~ t i o n e . Cony~~= eitio Mdhsma;ticcc, vol. 9, pp. 141-160.
86s &Q &SSEN, M. SHOIRAT, J . A., and TAIMARKXN, J. C. 1943. Th p b l s m 04 ?nmmts. New Yoxk. smmu*, W. 1920. Sur lee fonotione oonvexee rn~urables. FundQmta M c c t ~ o a e , vol. 1,
pp. 125-120. S ~ B , D. B. 1949. An invemion formula for the Btieltjes trsnrafonn. Buthin of the A d c a n
Mathm&icaZ Sooicly, vol. 66, pp. 174-183.
Sma6, G. See P~LYA, Q. TAMARKIN, J. D. See SXOEAT, J . A. TXTOHWH, E. C. 1939. The theory o j f ~ n c t i m , 2 ed. Oxford. 1948. ItLtrodu&on to thc tJCFJOrY of F&r id@, 2 ed. Oxford. 1961. Xb theory of th Ritrrsann z & a . f ~ ~ . Oxford. TYOHONOB~, A. 1936. Thborbmee d'unicit4 pour 1'8quation de la ohelem. Mdenadwbi i S b h k ,
vol. 42, pp. 189-218. DE LA. VKLLEIO ' J P o u ~ ~ x N , C. J. 19 14. Cow8 d'awlyae ~ j e n ~ ~ . Paris.
Symbols and Notations
S p b o l Page
4 4 17 54
180 25 25 28 28 38 5
48 48 64 42
5 172 171 17 181 172
a 173 28 193 152 18 20 20 19
Index
AISSEN, 261 U m o w ~ c z , 261 Aqmptotic behaviour of k e m l , 108;
&(% 117 AUXili&ZIY kernel, 21 8
BAIUACH, 153, 261 B-ucm, 216,261 Beheviour a t inffnity, 147,217, 246 Bell-shaped, 92, 248 B ~ E N s ~ , S., 240 B ~ s m m , V., 213,224, 261 Bernstein polpomial, 240 Bernstein's kheorem, 202 Beesel funotion, rnodilbd, 74 Beta-funotion, 70 BOAS, 73, 76, 148, 281 BBcmm, 33,261 Boomma, 26, 193,261 B ~ Y , 203
Cauohy value, 28 & I I ~ ] C of gr8df y, 19 Centre1 limit theorem of prob~bility, 113 C ~ B T D R A B E I K ~ ~ A N , 193, 261 C h v of sign of Q("'(t), 91 Changes of trend, 95 chamoterietio funotion, 20 Cbrwferietio funotiom, multiplioation
of, 23 Cless A, 180 h D,, 48 C b s D,, 48 C h E, 42 Claw 8 , 1 8 1 Clam LP, 193 Claw I, 120; kernel, 14 Class II, 120; kernel, 14 Claaa III, 120; kernel, 14 C l m of D,, 48 Column d a t e matrix, 99 Complex inversion theory, 210 Conjugate index, 162 Continuity oondition, 138 Continuity theorem, 80 Continuous parameter, 173 Convergenoe, 123 Convexity, 106
Degree d, polynomial of, 30 Degree of kernel, 68 Derived Stieltjes t d o r m , 78 Determining funation in I;*, 162, 238 Defermining h o t i o n non-deoreasing,
168, 236 Defermining funotion of bounded btd
variation, 166 D r m ' s symbolic h o t i o n , 7, 173 DiriWt kernel, 30 Diricblet series, general, 269 Dhrete pmanmter, 173 Distribution function, 17, 27; olosure
Laurent aeries, 3 Law of the mean, second, 61 LEBE~SGUE, 40,41,60,66, 187, 194, 205 Lebesgue eet, 130 bw, 39, 50, 262 Limit operation, 38, 64
Liouville's theorem, 249 Logarithmio convexity, 85 Logid symbol, 17 L O R E ~ , 241,262
MAGNUS, 71,74, 262 MAXOBLBROJT, 260, 262 Maee distribution, 19 Matrix t~8nsformation, 97 Mean, 23, 56; of distribution funotion,
18; of frequency funotion, 20 Median, 142 MEIJEB, 38 Meijer transform, 73, 78; general, 76 Nillinkowski's inequrtlity, 193 Minor definite, 98 Mode, 140 Moment of inertia, 19 MOTZKXN, 262 Multiplioation of characterbtio funotiona,
23
Non-finite kernel, 37; properti- of, 56 Norm, 193 Normal frequency funotion, 20 N o d i z e d distribution fmotion, 18
T-, 238,263 Tabberim theorem, 148 Temperature funation, positive, 199 Theorem of Herglotz, 238 Theta-hotion, 77 T ~ ~ ~ ~ o I T u , 78, 110 TITO-H, 263 Totally positive fiwquenoy h o t i o n , 103 Totally poaitive funotions, generation
of, 0s Trmsfom in the complex domain,
212 , TYOHOWO~, 183,201, 263
Tychonoffs uniqumem theorem, 183, 1 84
Uniqueness of, 41 Uniqueness theorem for Fourier-Stieltjes
lx-anafom, 41
Uniqueness theorem of Tychonoff, 183, 184
de la V- Poussm, 204, 263 Vmiance, 23, 66; of distribution funo-
tion, 20; of frequenoy function, 20 Variation dimininhing convolution, 12,
