Universitext
Komaravol u Chand rasekharan
Classical Fou rier Transforms
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Komaravolu Chandrasekharan Professor of Mathematics Eidgenossische Technische Hochschule ZOrich CH-8092 ZOrich
Mathematics Subject Classification (1980): 42-XX, 10-XX, 60-XX
ISBN-13: 978-3-540-50248-7 e-ISBN-13: 978-3-642-74029-9 DOl: 10.1007/978-3-642-74029-9
Library of Congress Cataloging-in-Publication Data Chandrasekharan. K. (Komaravolu), 1920-Classical Fourier transforms / Komaravolu Chandrasekharan. p. cm.-(Universitext). Bibliography: p.
1. Fourier transformations. I. Title. QA403.5.C48 198988-38192 515.7'23-dc 19 CIP
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© Springer-Verlag Berlin Heidelberg 1989
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Preface
In gratefuZ remerribrance of Marston Morse and John von Neumann
This text formed the basis of an optional course of lectures I gave
in German at the Swiss Federal Institute of Technology (ETH), Zlirich,
during the Wintersemester of 1986-87, to undergraduates whose interests
were rather mixed, and who were supposed, in general, to be acquainted
with only the rudiments of real and complex analysis. The choice of
material and the treatment were linked to that supposition. The idea
of publishing this originated with Dr. Joachim Heinze of Springer
Verlag. I have, in response, checked the text once more, and added some
notes and references. My warm thanks go to Professor Raghavan Narasimhan
and to Dr. Albert Stadler, for their helpful and careful scrutiny of the
manuscript, which resulted in the removal of some obscurities, and to
Springer-Verlag for their courtesy and cooperation. I have to thank
Dr. Stadler also for his assistance with the diagrams and with the
proof-reading.
Zlirich, September, 1987 K.C.
Contents
Chapter I. Fourier transforms on L1 (-oo,oo)
§1. §2. §3. §4. §5. §6. §7. §8. §9. §10. §11. §12. §13. §14. §15.
Basic properties and examples •.•••••..••..•.•...•.•..•....• The L 1-algebra ••.......••••..••.•..••..••..•...••....••.•.. Differentiabili ty properties ...•••.•.•••••••....••••.•...•. Localization, Mellin transforms ......•.•......•......•..•.. Fourier series and Poisson's summation formula .......••.••.. The uniqueness theorem .......•...........•....•............ Pointwise summabili ty .••.•••......•••.••..••.••.•.••.••.•.. The inversion formula ......••.•........•••...•.........•... Summabili ty in the L1-norm ••.••....•.•••....•..•....•..•.•. The central limit theorem .••...•.•...........•..•.....••.•• Analytic functions of Fourier transforms •.•.•...•.•...••..• The closure of translations •.•....•..........••••..••.•.••. A general tauberian theorem ..•....•••..••.......•.••...•... Two differential equations ..•...•..••.......•.•..••.••••... Several variables .•..............•...................•.•...
Chapter II. Fourier transforms on L2 (-oo,oo)
§1. §2. §3. §4. §5. §6. §7. §8. §9. §10. § 11.
In troduction .............•..•..•.••.•..•.••......•........• Plancherel's theorem ..•..•........•.••.••..•..........•.... Convergence and summability ••.......••.•..•.....••..•...... The closure of translations ....••.....•........•..•..•..•.• Heisenberg's inequality ..•..•......•...•••..•......•.••.•.. Hardy's theorem •••...•....•..•....••.••....•..•..••.•..•.•• The theorem of Paley and Wiener .••.....••......•....•..••.. Fourier series in L2(a,b) ..••••.......•.....••....•.....•.. Hardy's interpolation formula ...•......•......••..••.....•• Two inequalities of S. Bernstein ......••••••••.....••..•... Several variables ......•••....••.......•...•.••.....•.....•
Chapter III. Fourier-Stieltjes transforms (one variable)
1 16 18 25 32 36 38 44 51 56 60 68 73 77 83
91 92
100 103 105 112 116 122 128 131 134
§ 1. Basic properties ...•.......•...•.•.•..••....•..•..•••••.•.• 137 §2. Distribution functions, and characteristic functions .•.••.. 140 §3. Positive-definite functions ......••....•••........•.•••...• 144 §4. A uniqueness theorem .............•..••.....••.•.••......••. 154
Notes ...•......•••......•.......•.•..•............•••...•..•.... 160
References .....•..•••...•.•..••.•.......••....•......•.......•.. 169
Chapter I. Fourier transforms on L1(-00, 00)
§1. Basic properties and examples
We assume as known Lebesgue's theory of integration.
If P is any real number, with p!:.. 1, we denote by Lp(-oo,oo) the vector
space of all complex-valued functions f(x) of the real variable x,
-00 < x < 00, such that f is Lebesgue measurable, and
II flip ( (I f (x) I p dx r / p < co.
-00
We call the number I I f I I p the Lp -norm of f.
If f,g E L (-00,00), we say that f is equivalent to g, and write f::: g, p
if and only if f = g except for a set of Lebesgue measure zero. The
relation ':::' is reflexive, symmetric, and transitive, and partitions
Lp(-oo,oo) into equivalence classes, and Lp(-oo,oo) is a Banach space if
it is looked upon as the set of all such equivalence classes, the
norm of an equivalence class being defined as the Lp-norm of any of
its members.
We shall use the same symbol L (-00,00) to denote the Banach space of p
equivalence classes, as well as the vector space of all functions
belonging to them, and make the distinction clear when necessary.
If f(x) ELl (-00 < x < 00), and a. any real number, we define, for
-00 < ex. < 00,
( 1. 1 ) f(a.)
and say that f is the Fourier transform of fELl (-00,00). We shall also
2 I. FOURIER TRANSFORMS ON £/-00,00)
use the notation
( 1 .2) F[f] 1(~TI) f, or F[f](a) = ~ f(a).
(Cf. (8.15) of Ch.I, and §2 of Ch.II.)
In the special case when f is even, f(-x)
of x, (1.1) takes the form
( 1 .3) f(a) 2 J f(x) cos ax dx. o
f(x) for all real values
If f is odd, f(-x)
form
-f(x) for all real values of x, (1.1) takes the
A
( 1 .4) -i f(a) 00
2 J f(x) sin ax dx. o
If f is defined in (0,00) only, and f E L1 (0,00), then the integrals on
the right-hand side of (1.3) and (1.4) define respectively the cosine
transform, and the sine transform, of f.
We list below some basic properties of Fourier transforms of functions
in L1 (-00,00) •
A
(1.5) If f(x) EL1 (-oo<X<00), then f is bounded on (-00,00), since for
all reai a, we have
00
If(a) 1.2. J If(x) Idx = II f 111 < 00,
where II f 111 denotes the L 1-norm of f, so that
sup If (a) I .2. -oo<a.<oo
II f 111 < 00.
( 1 .6) If f(x) EL1 (-00< x < 00), then f(a) is continuous for -00< a < 00.
For if h is a real number, h * 0, then
If(a+h) - f(a)
§1. Basic properties and examples
where
and
I f (k) I I e ihx - 1 I ... 0, as h ... 0 ,
for almost all x E (_00,00). It follows from Lebesgue's theorem on
dominated convergence that
as h'" 0,
A
and hence f is continuous at the point CI., where -00 < CI. < 00.
A
(1. 7) The operator f ... f is linear in the sense that
where c 1 , c 2 are complex constants, and f 1 , f2 E L1 (-00,00) •
(1.8) Let h be a fixed real number, and f(x) E L1 (-00< x < 00). Then
3
the Fourier transform of f(x+h), the translation of f(x) by h, equals
f(CI.)e-iCl.h, since
_00
(1.9) Let t be a fixed real number, and f (x) E L1 (-00 < x < 00). Then itx A
the Fourier transform of f(x)e equals f(CI.+t), since
f=f(X)ei(t+CI.)Xdx f (CI.+t) _00
A
the last being a translation of f(CI.) by t. It follows that the trans-
lation of a Fourier transform is again a Fourier transform.
( 1.10) Let A be a fixed real number, 1.*0, and f E L1 (-00,00). Then
the Fourier transform of f(AX) equals
_1_ f(~) II.I A
since
4 I. FOURIER TRANSFORMS ON L/-=,oo)
__ 1 __ f=f(y)ei(aY)/Ady.
IAI -=
It follows that f is an odd or even function, according as f is odd
or even.
(1.11) If f denotes the complex conjugate of f, and fELl (-=,=) ,
then the Fourier transform of f(x) equals
f(-a)
since the complex conjugate of f=f(X)eiaxdx equals f(-a). -=
(1.12) If fEL 1 (-=,=), and f n EL1 (-=,=) for n 1 ,2, ... , and
II fn - f 11 1 .... °, as n .... =, then we have
A
f(a)
uniformly for -=<a<=, since by (1.5) and (1.7), we have
sup Ifn(a) - f(a) I < II fn - fill -00<0..<00
(1.13) If fl ,f2 ELl (-=,=), then we have the composition rule
til (y)f 2 (y)dy = f fl (Y)f 2 (y)dy
-= -=
since
= f fl (y) f2 (y) dy -=
f=f 2 (Y) (f=fl (x)eiYXdx)dy -00 -00
f= f=f 2 (Y)f 1 (x)eiyxdx dy -00 -00
by Fubini's theorem, and
= = f f If 2 (y)llf 1 (x)ldxdy f If 1 (x) Idx f If 2 (y) Idy<=.
-= _00 -00
The integral
f f=f 2 (Y)f 1 (x)eiyxdx dy -00 -00
§1. Basic properties and exampZes
is symmetric in f1 and f2' so that we can interchange f2 and f1' and assertion (1.13) follows.
5
A fundamental property of Fourier transforms of functions in L1 (-=,=) is contained in the following
Theorem 1 (Riemann-Lebesgue). If f(x) €L 1 (-=<X<=), and f denotes the Fourier transform of f, then
feed + 0, as lal + =.
Proof. Consider the special function X defined by
{1 ,
x(x) = 0,
for -=< a~x~b < 0:>,
for x < a, and x > b,
referred to as the characteristic function, or indicator function, of the finite interval [a,b]. Its Fourier transform is
for a real, a * 0, so that
I A 2 x(a) 1<--+0,
-Ial as lal +0:>.
By linearity, this property holds also for any step-function (such a function being a finite linear combination, with complex coefficients, of characteristic (or indicator) functions of finite intervals). And step-functions form a dense subset of L1 (-0:>,0:». That is to say, given
e: > 0, and f € L1 (-0:>,0:», there exists a step-function fe:' such that
II f-fe:111< e:. Since
f(a) fe:(a) A
- fe:(a», + (f(a)
we have
If(a) I ~ I fe: (a) I + If(a) - f (a) I < If (a) I + e:, e: e:
because of property (1.5) and the choice of f . Hence e:
6 I. FOURIER TRANSFORMS ON L/-<»,oo)
lim sup If(a) I < lim sup If (a) I + £ = £, lal+co - lal+co £
since we have just seen that the theorem holds for the step-function
f£ •
Cor-oZZar-y. If f E L1 (-co,co), then we have
co f f(t)cos(at)dt+O, as lal + co, -co
and
co f f(t)sin(at)dt+O, as lal +co. -co
Remarks. 1. The Riemann-Lebesgue theorem is related to a property of
the Lp-moduZus of aontinuity Tf of the function fELp(-CO,co), 1~p<co,
which is defined by
(1.14) ( CO )1/P f I f (x) - f (x+y) I p dx , -co
-co < Y < co.
Clearly we have
(1.15)
and
(1.16 )
since
(1.17) If fELp(-co,co), 1~p<co, then Tf(h) is continuous in h; in
particular, Tf(h) +0, as h+O.
To prove this we note that given £ > 0, there exists a continuous
function ~(x) which vanishes outside a finite interval, such that
§1. Basic properties and examples 7
( CO )1/P f If(x) - lP(x) IPdx < E. -co
If we set
then ~(h) is continuous in h, and
(CO )1/P
Tf(h) ~ f If(x+h) - lP(x+h) IPdx -co
( CO )1/P (CO )1/P + f IIP(x+h) - lP(x) I Pdx + f IIP(x) - f (x) I Pdx , -00 -00
so that
and similarly also
so that ~ + Tf uniformly in h as E + 0. Hence Tf(h) is continuous in h,
and tends to zero with h.
Reverting to the proof of Theorem 1, if f E L1 (-co,co), then
f(a) = fcof(x)eiaxdx, -f(a) = fcof(X)ei(x+n/a)adX -00 -00
co n ia f feY-ale Ydy, -co
so that
12f(a) I < f If(y) - f(y-1!) Idy, a -co A
and, because of (1.17), f(a) +0, as lal + co.
A
2. If f E L1 (-co,co), the Fourier transform f(a) is a continuous function
of a, -co < a < co, which vanishes at infinity; that is to say,
lim f(a) = 0. Not every continuous function that vanishes at inlal+co
8 I. FOURIER TRANSFORMS ON £/-00,00)
finity is necessarily the Fourier transform of a function in L 1 (-=,=).
To construct an example, let fELl (-co,=), and let f be odd, that is
to say f(-x) -f(x), -oo<x<oo. Then obviously we have
00 f(a) 2i J f(x)sin ax dx.
o
Let e denote the exponential, and R> e. Then we have
R A
J f(a) da a e
R do. co 2i J J f(x)sin ax dx
e a 0
2i 7 f(x)(i sin ax da)dx o e a
= (Rx. . Sln a = 21 J f(x) J -a- da )dX,
o ex
the change in the order of integration being permitted by Fubini's
theorem, because of the assumption fELl (-=,00). Now
I J Si~ a da I < MO < 00 a
for all real a,b, where MO is independent of a and b, and hence
( 1.18)
If we define the function g by
{a/ e, 0 ~ a ~ e,
g(a) = 1/1og a, a> e;
g(a) -g(-a), a.:: 0;
then g is an odd, continuous function, which vanishes at infinity, for
which
R J g(a) da e a
R da J aloga e
loglog R ->- 00, as R ->- 00.
Clearly g cannot be the Fourier transform of a function in Ll (-00,00) ,
since it cannot take the place of f in (1.18).
§1. Basic properties and exampZes
Examples
1. If
then
" f(a)
f' 0, f(x)
for Ix I ~ 1,
for I x I > 1,
1 1 f eiaxdx = 2 f cos ax dx -1 0
Note that here f(a) ¢ L1 (-co,co) •
Similarly if
x-1 [ 2 Si~ ax] -
x=o
(, for I x-a I ~ R,
0, for Ix-al > R,
-co < a < co, R > 0,
then f(a)
2. If
then
since
f(a)
Note that
f(x)
2 e iaa sin aR a
__ {1- IX I , f(x)
0,
for Ixl ~ 1,
for I x I > 1,
( sin (a/2»)2 a/2 '
1 dx = 2 J (1-x) ~(sin aX)dX 2 f (1-x) cos ax
0 o dx a
1 sin J ~(sin2(axL21)dx sin2 (aL2) 2 f ax dx =
0 a o dx (a/2)2 (a/2)2
here f ( a) E L 1 (-co, co ).
Similarly if a> 0, and f (x) =
Ixl > a, then f(a) = sin2 (aa/2) a(!!)2
2
1=1, for Ixl ~a, and f(x) 0, for
3. If f(x) = x -e x e e, then f(a) r (1+ia) * 0, where r stands for
Euler's gamma-function, since
9
10 I. FOURIER TRANSFORMS ON L/-"'.oo)
f(a)
Similarly
4. If f(x)
00 x eXeiaxdx J
-e e -00
-x if f(x) -e -x e e
e- 1xl , then f(a)
J e-x+iaxdx o
00 -y ia
J r (1+ia) . e y dy = 0
then f(a) r (1-ia) .
2 For 1+0'.2 .
and the result follows from adding these two integrals. Here again we
have £ (a) E L1 (-00,00) •
If a> 0, and f(x) = e-a1xl , then £(0'.) 2a
a 2+a 2
5. Let <;; be a
and if n < 0,
Then
for, if n > 0,
-00
and, if n < 0,
6. If f (x)
complex number, <;; = f; + in, with n * o. If n > 0, let
f(x)
let
f(x)
~
f(a)
2
t, -ix<;; e ,
f O' l -ix<;; -e ,
i(a-<;;) ,
for
for
for
for
x> 0,
x < 0;
x < 0,
x> o.
J e-x(n-if;+ia)dx o
-J e-ix1;;+iaxdx o
-J e-x(-n+if;-ia)dx o
e - x then f (a)
i(a-<;;) ,
i(a-1;;)
§1. Basic properties and examples 11
To prove this, we note, first of all, that
0:> 2 0:> 2 211 2 f e-x dx f e-Y dy -0:>
2 2 f f e-(x +y )dx dy -(X) -00
J de f e-r r dr o 0
0:> 1 2 2 211 f - e- r d(r) 11,
o 2
so that
0:> 2 f e -x dx /iT.
~
Next we note that the derivative of f(a) is given by
~ d ~
( f (a) ) I :: da ( f (a) )
so that
(f (a) ) I
and hence
a (-2) f(a), or
i -2 f
(f (a) ) I
f(a) -a/2 ,
log f(a) 2 -a 2 14
-(a 14) + c 1 , or f(a) = c e ,
~
where c 1 , c, are constants. On setting a = 0, we see that f(O) = c = /iT,
0:> 2 because of the evaluation of f e- x dx first made.
-00
It follows that if f(x)
1 2 -2x
e then f (a)
7. Let ((In (x)
1 2 -2 x (d)n _x 2
(-1)n e - e dx where n is a positive integer. Then
;Pn(a)
for
12 I. FOURIER TRANSFORMS ON L/-OO,oo}
fOO<pn (x) eiaxdx -00
2 1 2 . 00 e-x (ddX)n(e2X +~Xa)dX (by f partial integration)
-00
e~a2 00 _X2( d )n( ~ (x+ia) 2) fe dx e dx -00
00 _x2( d )n( ~ (x+ia) 2) f e da e dx -00
1 2
= (_i)n e 2a (d~)n (1(2nf e-(2 ) = inl(2nf <pn(a) •
The functions (<pn (x» are known as the Hermite functions.
8. If o for Ix I ~ 1, f(x) = {' v_I
( 1_x2 ) 2, for 0 < I x I < 1, v > -~ ,
then f E L1 (-00,00), and
since
r(n+~)
1 v_I f(a) = 2 f (1-x2 ) 2cos(ax)dx
o
00
2 n=O
00
n=O
1 1 r(v+2 )r(n+2 )
r (v+n+1)
( 2n-1 ) (2n- 3 ) ••• 5. 3 • 1 liT 2n
(2n)!1iT
22n(n! )
§1. Basic properties and e~Zes 13
The BesseZ function J v of order v is defined by
(_1)n(lx)v+2n 2
00 , for v > -1 •
n=O n! r (v+n+1)
Hence
A
f(a.)
Incidentally we obtain integral representations for Jv(X) , namely
and, on setting t = cos a,
9. If f(x) = cosh 7fX' then f(x) EL 1 (-00<X<00), and f(a.) cosh(a./2) •
eixa. If we apply Cauchy's theorem to the integral ~ cosh 7fX dx taken along
the contour C, which is a rectangle with corners at -R, +R, R+i, and
-R+i (where R> 0), and note that the residue at the pole x = i/2
equals e-a./2/(7fi), and then let R+oo, we get
1 27fi
00 ei(x+i)a.dx ] - f cosh 7f(x+i) -00
so that
10. An integraZ of Ramanujan. . 2
A
, or f(a.) -1
[COSh (a./2)]
e-~7fX
If f(x) = cosh 7fX' then f(x) EL 1 (-00<x<oo), and
Ijl(a.) ;:: f -i7fx2 -ixa. e e dx
i 7f ia. 2 T 4iT e - ie
00
-00 cosh 7fX cosh(a./2)
Proof. It is immediate that
14 I. FOURIER TRANSFORMS ON L/-=,oo)
co ( i) 2 f
. 2 Since e-1.7TX f L1 (-co<X<co), the integral on the right-hand side is de-
fined as a Cauchy principal value (cf. §4). To evaluate it one can use
Cauchy's theorem, or use Example 6, which gives
(ii) f for \ > 0.
By analytic continuation, this holds good also for compZex \, with
Re \..:::.0, provided that \ *0. Thus, for \ = ± 7Ti, we get:
(iii) f f
On using the first formula in (i), we obtain
(iv)
We shall see, on the other hand, that
(v)
To prove this, we note that (ii) implies that
(vi) 2
/TI e-a /(4\)
15:
7Ti T
-i7T 2e-4-
-ixa e for Re \ > O.
Let E>O, and \ = 1j[4(E+i7T)]. Using the composition rule (1.13)
together with the Fourier transforms given in Example 9, and (vi), we
get
1 1
27T 2 (E+7Ti) '2 f -co cosh 7TX
dx f -\(x+a)2
e
cosh(x/2) dx,
\ 1/[4(E+7Ti)], Re \>0.
Lebesgue's theorem on dominated convergence permits us to let E + ° in
this relation, and we obtain
§1. Basic properties and exampZes
-1Ti e i (x+a)2/(41T) (jl(a)
-4-f dx = e 21T cosh(x/2)
-1Ti e i (21Ty+a)2/(41T) -4- 00
= e f dy, cosh(1TY)
-1Ti e i (-21Ty+a)2/(41T) -4-f dy = e
cosh 1Ty
ia 2 1Ti 2 4TI ""4 00 ei1TY -iay
= e f dy cosh 1Ty
This leads, as before, to the relation
x 21TY,
-1Ti 2e-4- f
2e- ia2 /(41T)
if we use the second formula in (iii), and this leads to (v).
On multiplying (iv) by e-a / 2 , and substituting from (v), we obtain
-1Ti (jl(a+i1T)(ea / 2 _ e-a / 2 ) = 2e--4- (1 _
and on replacing a by a-i1T, we get
(jl(a)
as claimed.
2e-~i [1 _ ei(a~;1T)2 - (a~i1T)] e(a-i1T)/2 _ e-(a-i1T)/2
. 2 l.a
4TI e
1Ti ia2 ""4 41T e - i e
cosh(a/2)
15
On equating the real and imaginary parts of both sides, and on setting
a = 21Tt, we get Ramanujan's formulae:
f cos 21TtX cos 1Tx 2dx o cosh 1TX
1 + 12 sin 1Tt2
212 cosh 1Tt
16
f cos 2ntx sin nx2dx o cosh nx
§2. The L 1-algebra
I. FOURIER TRANSFORMS ON L/-OO,oo)
-1 + /2 cos nt2
2/2 cosh nt
The Banach space L1 (-oo,oo) can be made into a Banach algebra by the
introduction of an operation of 'multiplication', which is defined by
the convolution of any two functions. To make this possible we need thE
following
(2.1) Lemma. Iff,9EL1 (-oo,oo), then the integral
f f(x-y)g(y)dy (= foo f(Y)9(X-Y)dY) -00
exists for almost all x E (-00,00), and is an integrable function of
X, -0:;) < X < 00.
Proof. The function f(x-y)g(y) is a measurable function in (x,y), and
the double integral
00 00
f f If(x-y)g(y) Idx dy 2 00. -00
By Fubini's theorem, this integral equals the repeated integral
fOO Ig(y) l(fOO If(x-y) IdX)dY -00 -00
fOO Ig(y) l(fOO If(x) IdX)dY < 00,
-00 -co
and hence we have also
so that If(x-y)g(y) 1 is integrable as a function of y for almost all
x E (-00,00), and
f If(x-y)g(y) Idy -00
is integrable as a function of x, and therefore also f f(x-y)g(y)dy. -00
17
Definition. Let f,g E L1 (-=,=), and let
h(x) = f f(x-y)g(y)dy,
when the integral exists. Then the function h is defined to be the
convolution of f and g, and is denoted by h f*g.
By Lemma 2.1 we note that hE L1 (-00,00). We also note that the convolution
is associative:
where f, g,h E L1 (-=,=). It is also commutative, that is to say
f*g g*f for f,g E L1 (-=,=) as can be seen by a change of variable in
the defining integral.
Theorem 2. If f,g E L1 (-=,=), and h
(2.2) fog,
where the dot denotes pointwise multiplication, and
(2.3) II h 111 = II f*g 111 .:: II f 11,11 g 111 .
Proof. Lemma 2.1 shows that hE L1 (-=,00). On using the definition of
h and of h, together with Fubini's theorem, we see that
-= -=
f=g(y)dy f=f(U)ei(u+y)udu -= -= -00
fooeiUYg(y)dy J=f(U)eiUUdu -00
On the other hand, we have
f Ih(x) Idx < f dx f If(x-y)g(y) Idy -00 -00
18 I. FOURIER TRANSFORMS ON L/-oo,oo)
f [g(y)[dy f [f(x-y)ldy I[ g I[,·I[ f 1[,.
Remarks
The properties of convolution just established show that the space
L,(-oo,oo) is a cOIllIllutative Banach algebra under ordinary addition,
with convolution as multiplication, and [I • [I, as norm. This Banach
algebra is also known as the L,-algebra on the real line.
§3. Properties of differentiability
We have seen that if a function f belongs to the Lebesgue class
L,(-oo,oo), then its Fourier transform f is a bounded, continuous
function on (_00,00). By imposing a simple additional condition on f,
one can secure the differentiability of f. If, on the other hand, the
function f E L, (_00,00) itself is assumed to be differentiable, and the
derivative also belongs to L, (_00,00), then the Fourier transform of
the derivative f' is related, in a simple way, to the transform of f
itself.
Theorem 3. A. If f(x) EL,(-oo<X<oo), and ix·f(x) EL,(-oo<X<oo), then
ita) exists, and
00 (f) , (a) f [ix·f(x) leiaxdx
B. If f(x) EL,(-oo<x<oo), f is continuously differentiable, and
f'(x) EL 1 (_00<X<00), then
so that
Proof. A. Let
00 f [f'(x)leiaxdx -ia·f(a),
f(a) 0(-'-)' as [at +00. [a [
( ihx
f(x) e h-')
§3. Properties of differentiability
for h real, and h * O. Then
~
( 3 • 1 ) f (o:+h) - f (0:) h
Now fh(x) +ix-f(x) pointwise, for almost every x, as h+O; and
1 ihx 11 I fh (x) 1 ::. I f (x) I- e h - ::. 1 x I -I f (x) I E L1 (-00 < x < 00) ,
by hypothesis (on using the first mean-value theorem). Hence
Ifh(x) - ix·f(x)1 ::. 2Ixl-lf(x)1 EL1 (-oo,oo).
By Lebesgue's theorem on dominated convergence, we conclude that
fh(X) + ix·f(x), in the L 1-norm,
as h + O. By property (1 .12) it follows that
(3.2)
uniformly, as h + O. The last integral is a bounded, continuous
function of 0: for -00 < a < co. From (3.1), however, we see that
hence
lim fh (a) h+O
(f) , (0:)
(f) , (0:)
B. Let a be real, 0: * O. For R> 0 we have
(3.3)
Now f(x)
so that
R . [eiaX ]X=R R io:x J f(x)elo:xdx = -. - f(x) - J e f' (x)dx. -R 10: x=-R -R ia
tends to a finite limit as x -+ ± co. For
f(x) - f(O)
f(± 00)
X
Jf'(t)dt, and f' EL1(-co,co), o
±co f(O) + J f' (t)dt.
o
19
20 I. FOURIER TRANSFORMS ON L/-=,oo)
But f(:I: 00) = O. Hence, on letting R+oo in (3.3), we have
00 e iax - J ia f' (x)dx, a * 0,
-00 -00
which implies that
(3.4)
This holds also for a = 0 by continuity (since the left-hand side is
zero, while the right-hand side is
00
f f' (x)dx f(oo) - f(-oo) 0) • -00
Since
f If'(x)ldx= Ilf' 11 1 <00, -00
and f (a) 0 (1), as I a I + 00, by Theorem 1, we conclude that
f(a) = O(1/Ial). Actually f(a) = o(1/Ial), as lal +00, because of
(3.4) and Theorem 1, since f' E L1 (-00,00) by assumption.
Remarks. Theorem 2(A) can be given a more general form. If m is a
positive integer, xm f(x) EL1 (-00<X<00), then f(a) is continuously
differentiable m times for -00 < a < 00, and we have
(f) (m) (a) 00
f(x)}eiaxdx, = f {(ix)m -00
so that
I (f) (m) (a) I 00
< f I (ix)m f(x) Idx. - -00
One has only to use Theorem 2(A) and induction on m.
Theorem 2(B) can also be similarly generalized. If f is continuously
differentiable m times, and f (r) (x) E L1 (-00 < x < 00), for 0::.. r::.. m, then
so that
§3. Properties of differentiability 21
Thus one can roughly say that the faster f decreases, the more often ~
is f differentiable (with bounded derivatives), and the more integrable
derivatives f has, the faster f decreases.
The spaces S and V of L. Schwartz
A complex-valued function f(x) of the real variable x is said to
belong to Schwartz's space S, if f is differentiable infinitely often,
and for any integers p,q,
f(q) denoting the qth derivative of f.
We note the following properties of S.
(3.5) If f E S, then x£f(m) (x) is bounded, and belongs to L1 (-00,00),
for any integers £ ,m ~ O. For
implies that
£ (m) I M Ix f (x) < --2 E L1 (-00,00). 1+x
In fact, f belongs to L (-00,00) for every p such that 1,::, P < 00. p
(3.6 ) If f E S, then (x£f(x» (m) is bounded, for any integers
£,m~ 0, and belongs to L1 (-00,00) .
This follows from (3.5), if we just use the rule for the differentiation
of a product.
~
(3.7) If f E S, then f E S.
First, if fES, then X£f(X~ EL1 (-00<X<oo), for every integer £~O, so
that (after Theorem 2(A» f is differentiable infinitely often.
Secondly, if £ and m are positive integers, then by Theorem 2(A), and
the remarks thereafter, we have
22 I. FOURIER TRANSFORMS ON £/-"",00)
so that by Theorem 2(B), and the remarks thereafter, we have
(3.71 )
Thus
I (f) (,1',) (a) I
for every m..:: 0, hence
I (f) (,Q,) (a) I
for every m..:: O.
(3.8) 2
We note that e-x ES, ~f S, e- 1xl fS, the last not being l+x
differentiable at the origin.
(3.9) A concept of convergence, and therefore of continuity, can be
introduced into the vector space S as follows. Given an infinite se
quence (f.) of functions, all belonging to S, we say that (f.) con
veY'ges inJs to zero as j->-oo, if for any integers ,Q"m>o'{X,Q,f~(m)(X)} - J
converges uniformly to zero on the line: -00 < x < 00.
(3.10) If {f. (x)} converges in S to zero as j ->- 00, then the sequence J ~
of Fourier transforms (f.(a)) converges in S to zero. J
For by (3.71) we have
and the right-hand side tends to zero as j ->- 00, since
tl {(ix),Q,f. (x)} (m) Idx -00 J
J IXI >R
R>O
I1 + I 2 , say,
where I2 -+ 0 as R ->- 00, uniformly in j, while I1 ->- 0 as j ->- 00 since the
integrand there is bounded and converges uniformly to zero.
§3. Properties of differentiabiZity 23
A continuous linear functional on S is known, after Schwartz, as a
tempered distribution. A study of the theory of Fourier transforms of
tempered distributions is outside the scope of the present text.
The space V
The space S contains the vector subspace V of infinitely differentiable
functions on (-00,00) with bounded supports. The support of a function f
defined on (-00,00) is the closure of the set of points x at which
f(x)*O.
The function f defined by
, 0, for I x I ~ 1 , f(x) { -1/(1-x2)
e , for I x I < 1 ,
belongs to V.lts derivatives, of all orders, vanish at x ± 1.
We can introduce into V a concept of convergence.
(3.11 ) We say that an infinite sequence (f.) of functions in V con-J
verges in V to a function f of V as j ->- =, if the supports of (f j ) are
all contained in the same bounded set (independently of j), and the
sequence of derivatives (f. (m», of any given order m, converges uni
formly, as j ->- co, to the de~i vati ve f (m) of f.
If (f j ) converges to zero in V, then clearly (f j ) conver~es to zero in
S as well.
A continuous linear functional on the space V is known, after Schwartz,
as a distribution. One can study the Fourier transforms of particular
classes of distributions, such as those with bounded supports, or
those with point supports, though such a study is outside the scope of
the present text.
We have already noted, and used, the fact that step-functions are dense
in L (-00,00) for 1~p<co (cf. §1). P
Given the function X b defined by a,
-co < a ~ x ~ b < +00, X b(x) a, forx<a,x>b,
24 I. FOURIER TRANSFORMS ON L/-OO,oo)
we can approximate to it by a function W b € 1), such that for any a, given <5 > 0,
Let
(3.12)
while
where
II X b - W b II < <5. a, a, p
W(x)
w(x) c 1
c 2
a c 1 = f
a-E
x
{1 , for a < x < b, 0, for x < a-E , 0, forx>b+E,
1
f e (y-a) (y-a+E) dy, a-E
b+E 1
f (y-b) (y-b-E) dy, e x
c = 2 (y-a) (y-a+E) e dy,
E > 0,
for a-E..::x..::a,
for b..::x..::b+E,
b+E 1 f e ly-b) (y-b-E) dy. b
Then w(x) = 1, for x = a and x = b, while w(x) = ° for x = a-E, and x b+E; and the derivatives of w, of all orders, vanish at x a-E, a, b, b+E. If E is chosen sufficiently small, the graph of
the function w is as shown in the figure:
--------~--~--r_------------_+----_r~~--~-------x o-€ a o+€ b-€ b b+€
Clearly wE L (-00,00), for every p such that 1..:: p < 00, and p
II X -w" < (2E) liP. a,b p
§3. 'PPoperties of differentiability 25
It follows that V is a dense subspace of L (-co,co), 1.::. p < co, and since p S ~ V, S is also a dense subspace of Lp (-co,co) •
Since w E V c: S, we have OJ E S •
We can similarly construct an infinitely differentiable function
w = W b' such that a,
W (x) (' 0,
for a+e:.::. x'::' b-e:,
for x.::. a, x 2. b,
so that
w(x) <X b(x) <w(x), a,
and
co f {w(x) - w(x) }dx < 4e:. -co
These auxiliary functions will corne in useful in the proof of the Central Limit Theorem in §10.
§4. Localization, Mellin transforms
If f(x) E L1 (-co < x < co), it does not follow that the Fourier transform
f (cd belongs to L1 (-co < a < co), as in Example 1 of § 1. We can, however,
find a simple condition, such that at a given point x, we can "invert"
the Fourier transform; that is to say, obtain the relation
f(x) 1 co -iax 2n J f(a)e da,
-co
the integral being defined as (a Cauchy principal value)
RA -iax lim f f ( a ) e da , R > 0. R-+oo -R
Let f E L1 (-co,co), and
26
( 4 • 1 )
Then we have
R
211 f -R
I. FOURIER TRANSFORMS ON L/-oo,oo)
-ixa~ e f(a)da, 0< R < =.
and since the repeated integral is absolutely convergent, the order
of integration can be interchanged, so that
Hence we have
(4.2) SR (x)
or
SR(x)
since, for R > 0,
(4.3)
211 f
= 11 f f (x-t)
sin Rt t dt.
11 7 {f(X+t) 0
- f(x) 2 11
f sin Rt dt o t
f 0
+ f(X-t)} sin Rt
t
{f (x+t) + f (x- t) 2
11
2" '
dt,
- f (x) } sin Rt dt, t
the integral being convergent though not absolutely. Thus we obtain
the following
(4.4)
( 4 • 1 )
and
(4.5)
then
Lemma. If f E L1 (-=,=), and
R
211 f -R
~ -iax f(a)e da, o < R < =,
~ [f(x+t) + f(x-t) - 2f(x)],
§4. LoaaZization. MeZZin transforms 27
2 ~ sin Rt 'IT J gx{t} t dt.
o
This lemma will enable us to prove that the convergence of SR {x}, as
R+~, to f{x} at a given point x, depends only on the behaviour of the
function in a neighbourhood of that point. This is usually referred
to as Riemann's ZoaaZization theorem, and is contained in the following
two theorems.
Theorem 4. If fEL1{-~'co}, xE {-~,~}, and gx{t} is defined as in
(4.5). and there exis ts a 0 > O. suah that
{4.6} dt < co,
then we have
lim SR{x} R+co
f {x} ,
where SR{x} is defined as in (4.1).
Proof. For any fixed 0 > 0, we have
2 [0 CO] g {t} S {x} - f{x} = - J + J _x ____ sin Rt dt = I1 + I 2 , say.
R 'IT 0 0 t
Since gx{t}/t is absolutely integrable in {O,o} by hypothesis, we have
I1 = e:{ o} + 0, as 0 + 0;
while Theorem 1 implies that for fixed 0 > 0,
co J f{x+t} sin Rt dt + 0, o t
and j f{x-t} sin Rt dt+O, o t
as R + co. The integral
co J f {x} sint Rt dt o
00
J sin T f{x} --T-dT+O, oR
as R + co
{cf. {4.3}}. Hence I 2 +O, as R+co, which proves the theorem.
Remark. Condition {4.6} is satisfied at any point x at which the
function f has a finite derivative, or satisfies just a Lipschitz
condition of order a, namely: If{x+h} - f{x}1 =O{lhl a }, 0<a<1.
28 I. FOURIER TRANSFORMS ON L 1 (-00,00)
The next theorem gives a sufficient condition for the convergence of
the integral SR (x) as R ->- 00, not only at the point x, but in an inter
val containing the pOint.
Theorem 5. If f E L1 (-00,00), and f is of bounded variation in a neigh
bourhood of the point x, then we have
~[f(X+O) + f(x-O)],
where SR(x) is defined as in (4.1).
Proof. If for a given 0, 0> 0, the function g is of bounded variation
in the interval [0,0], then we shall prove that
(4.7) lim ~ f g(t) sint Rt dt = g(O+) . R->-oo 0
Since g can be expressed as the difference of two bounded, monotone
increasing functions, it suffices to prove (4.7) on the assumption that
g is a bounded, monotone increasing function. We may further assume
that g(O+) = 0, for if g(O+) '* 0, we set G(t) = g(t) - g(O+), so that
G(O+) = 0, and if (4.7) holds with G in place of g, then we have
o lim f G(t) sint Rt dt R->-oo 0
G(O+) 0,
which implies, in turn, that
2 0 f g(t) 'IT 0
sin Rt dt t
2 0 Sl' n Rt 2 cS 'Rt f ( d f ( ) Sln dt G t) t t + g 0+ t :n: 0 'IT 0
->- 0 + g (0+), as R ->- 00,
because of (4.3), which proves (4.7).
We assume therefore that g(O+) 0, and that g is a bounded, monotone
increasing function. Given E > 0, therefore, we can find n, such that
o < n < 0, and I g (t) I 2. E for 0 < t 2. n. We now apply the second mean-value
theorem, which states that if f is integrable on the finite interval
(a,b) and ~ bounded and monotone in (a,b), then we have
b f f(x)~(x)dx a
I; b ~(a+O) f f(x)dx + ~(b-O) f f(x)dx,
a I;
§4. LoaaUzation, Mel-Un i;ransforms
for some l; E [a,b]. On setting (j) = g, f(t)
there exists l; E [O,n], such that
29
sin Rt t ' we see that
n f g(t) o
sin Rt dt t
g(n-O) f sin Rt dt l; t
nR sin t g(n-O) f -t- dt.
l;R
Hence
I nf sin Rt I I I g(t) t dt ~ g(n-O) ·M ~ E M < <», o
where E is an arbitrary positive number, and M is independent of l;,n,
and R (cf. (1.18». And we have
I j g(t) sint Rt dtl ~ E M + I j g(t) sint Rt dtl. o n
Since g is bounded and measurable on [0,0], the function g(t)/t is
integrable on [n, 0], so that Theorem 1 gives
and hence
or
lim j g~t) sin Rt dt 0, R+<» n
lim sup R+<»
o If g(t) o
sin Rt t dt\ < EM,
o lim f g(t) sin Rt dt = 0 = g(O+) • R+<» 0 t
We shall now use (4.7) to prove the theorem.
Given x, we choose 0, such that 0> 0, and 0 is so small that f is of
bounded variation in [x-o, x+o]. By (4.2) we have
Since {f(x+t) + f(x-t)}/t is integrable, as a function of t, on the
interval o~t<<», Theorem 1 implies that 1 2 +0, as R+<». By (4.7), how
ever, we have
30 I. FOURIER TRANSFORMS ON L1 (-OO,"')
I 1 + ~ [f (x+O) + f (x-O) ], as R + 00,
and Theo~em 5 follows.
Remarks. The criteria for
given by Theorems 4 and 5
by f(x) = x sin(1/x), for
x > 1 ~ sa tis fies condi tion
the convergence of SR (x) to f (x), as R + 00,
are not comparable. The function f defined
o<x<l~ f(x) =0, forx_<O~ f(x) =0, for -1f
1f (4.6) of Theorem 4
of bounded variation in any neighbourhood of
at x = 0, but it is not
the origin.
The function f defined by f(x) = 1/log(1/x) for
for x> 1, f(x) = f(-x), is of bounded variation -e
o < x < ~ , f ( x) = 1 / (ex) 2
in a neighbourhood of
the origin (since it is bounded and monotone), but does not satisfy
condition (4.6) of Theorem 4 at x = O. Both the functions belong, of
course, to L1 (-00,00) •
Mellin transforms
Theorem 5 leads us, by a change of variable, to MeZZin transforms and
MeZZin inversion.
Theorem 5'. Let ya-1 f (y) EL 1 (0<y<=), for a reaZ, and Zet fey) be of
bounded variation in a neighbourhood of the point y = x. If
00 (4.8) F(s) s-1 f f (y) y dy, s = a + it, i r-T,
o
then
(4.9) a+iT
2ni lim f F(s)x-sdx = ~ [f(x+O) + f(x-O)]. T+= a-iT
Proof. The hypothesis on f ensures the existence of the integral in
(4.8). As hitherto f is a complex-valued function of the real variable
x~ on the other hand, s is complex with real part a and imaginary
part t. If we make the substitution x = e Y in (4.8), we get
00 F(a+it) = f f(eY)e ay eitYdy,
-00
so that F(a+it), considered as a function of t, -00< t < 00, may be looked
upon as the Fourier transform of f(eY)eaY EL1 (-00<y<00). By Theorem 5 we
get
§4. Localization, Mellin transforms
1 R . lim -- f F(a+it)e-ltYdt R+oo 2'lT -R
31
~ [g(y+O) + g(y-O)],
where g(y) = f(e Y) e ay , provided that g is of bounded variation in a
neighbourhood of the point y, or
R lim ~ f F(a+it) e-(a+it)Ydt R+oo 2'lT -R
On setting x = e Y , we get
e- ay ~ [g(y+O) + g(y-O)].
a+iR f -s lim ~ F(s)x ds
R+oo 'lTl a-iR ~ [f(x+o) + f(x-O)],
which is (4.9).
Similarly we have also the following
Theorem 5". Let F(a+iu) E L1 (-00< u < 00), and let F be of bounded
variation, as a function of u, in a neighbourhood of the point u t.
If
then
a+ioo fix) 2'JTi f
a-ico
-s F(s)x ds, s = a+it,
R . 1 lim f f(x)xa +lt- dx R+oo 1/R
~ [F{a+i(t+O)} + F{a+i(t-O)}].
The function F in (4.8) is usually referred to as the Mellin transform
of f; and (4.9) is referred to as the Mellin inversion formula. We
note that the Mellin transform is just another version oj the Fourier
transform obtained by a change of variable.
Examples
1. The integral representation for the gamma-function given by
r (s) f -x s-1 e x dx, a = Re s> 0, o
shows that in Theorem 5' if fix) = e-x then F(s) = r(s), for a>O.
Thus r(s), for a>O, is the Bellin transform of e-x , O<X<oo. And we
have
-x e a+ioo
2'JTi f a-ioo
-s r(s)x ds, a>O, X>O.
32 I. FOURIER TRANSFORMS ON L/-<»,oo)
2. The series L n=1
-s n , s = a+i t, converges for a > 1, and the surn-
function ~(s) is known as the Riemann zeta-funation. If in Theorem 5'
f(x) = 1/(ex-1), then F(s) = r(s)~(s), for a> 1.
To see this we note that for any integer n, n ~ 1, we have
and since
00
r(s)n- s = f e-nx x s - 1dx, for a> 0, o
; j Ixs - 1e-nx ldx n=1 0
00 00
L f e-nx xa - 1dx n=1 0
for a> 1, we have
00
I: n=1
r(s)~(s)
00 00
I: J x s - 1 e-nxdx 00 00
f xs~1 I: e-nxdx n=1 0 o n=1
-a r(a)n <00,
00 s-1 f x dx, o e X_1
for a > 1. Thus r (s) ~ (s), for a > 1, is the Mellin transform of
1/ (ex-1), 0 < x < 00, and we deduce that
a+ioo 21Ti f r(s)~(s)x-sds, a>1, x>O.
a-ioo
3. Let L(s) denote one of Dirichlet's L-functions, defined by the
series
L(s) .... , for a> 0;
then r (s)L(s), for a> 0, is the Mellin transform of
and we deduce that
a+ioo
x -x e +e , 0 < x < 00,
x -x e +e 21Ti f
-s r(s)L(s)x ds, a> 6, x> O. a-ioo
§5. Fourier series and Poisson's summation formula
If f(x) EL1 (0..::.x..::.21T), and f(x+21T)
series of f is defined to be
f (x), for -00 < x < 00, the Fourier
(5.1) 0..::. x"::' 21T,
§5. Fourier series and Poisson's summation formuZa
where the Fourier coefficient Cv is given by
(5.2) c v
21T
21T J o -ivx
f (x) e dx.
33
If gx(t) = ~ [f(x+t) + f(x-t) - 2f(x)], and there exists a 0> 0, such
o that J Ig (t) It- 1dt < 00, then the Fourier series of f at the point x
o x
converges to sum f(x). This is the well-known criterion of convergence
due to Dini, of which Theorem 4 is the analogue for Fourier transforms.
If on the other hand, f is of bounded variation in (O,21T), then at
every point xo the Fourier series converges to i [f(xO+O) + f(xO-O)].
In particular, the series converges to f(x) at every point of con
tinuity of f. If further f is continuous at every point of a closed
interval, then the series converges uniformZy in that interval. This is
the well-known criterion of convergence due to Dirichlet and Jordan,
of which Theorem 5 is the analogue for Fourier transforms.
The following lemma establishes a simple connexion between Fourier
transforms of functions in L1 (-00,00) and the Fourier series of related
periodic functions, of period 21T, belonging to L1 (O,21T).
00
(5.3) Lemma. If f(x) EL 1 (-00<X<00), then the series L f(x + 2k1T) k=-oo
converges absoZuteZy for aZmost aZZ x in (O,21T) and its sum F(x) be
Zongs to L 1 (O,21T), with F(x+21T) = F(x) for aZZ reaZ x. If Cv denotes
the Fourier coefficient of F, then
(5.4)
Proof.
A 21T
- 21T J o
-ivx J F(x)e dx = 21T - 21T f (-\!) • -00
We have
21T N 21T L J If (x+2k1T) I dx - lim L J If (x+2k1T) I dx
k=-oo 0 N->-c:o k=-N 0
N (2k+2)1T lim
C:-N J I f(y) Idy)
N->-oo 2k1T
(2N+2)1T 00
lim J If(y) Idy J If(y) Idy < 00
N->-oo -2N1T
34 I. FOURIER TRANSFORMS ON L/-=,oo)
It follows by Lebesgue's theorem on monotone convergence that
2 'IT 00 2 'IT f L If(x+2k'IT) Idx L f If (x+2kTI) I dx < 00,
o k=-oo k=-oo 0
hence L f(x+2k'IT) converges absolutely for almost all x in (0,2'IT), k=-OO
N L f(x+2k'IT), then lim FN(X) = F(x), where
k=-N N->-oo
FE L1 (0,2'IT),
given by
and F(x+2'IT) = F(x). The vth Fourier coefficient of F is
since
2 'IT J -ivx
2 'IT lim FN(x)e dx o N->-OO
l ' 1 1m 2'IT
N+oo
(2N+2)'IT -ivx J f (x) e dx -2N'IT
IFN(X)I < L If(x+2k'IT)I EL1 (0,2'IT). k=-oo
JOO -ivx 2 'IT f(x)e dx,
_00
Theorem 6. Let fEL 1 (_00,00), and be of bounded variation on (_00,00),
and let fix) = ~ [f(x+O) + fix-Oj 1 foY' all x in (_00,00). Then we have
(5.5) N
lim L N->-oo V=-N
) .
Proof. Let vk denote the total variation of f in the interval 00
Ik = (2k 'IT, (2k+ 2) 'IT), k = 0, ± 1, ± 2, The series L f(x+2k'IT) con-k=-OO
verges absolutely at some pOint Xo in 10 , and
L If(x+2k'IT) I < L If(xo+2k'IT) I + L If(x+2k'IT) - f(xo+2k'IT) I, Ik I"::'N Ik I"::'N Ik I"::'N
where
n Since L vk = lim L vk < 00, by hypothesis, the series L f (x+2kll)
k=-oo n->-oo k=-n k=-oo
converges absolutely, and uniformly, in 10 to sum F(x), say, which is
of bounded variation, and such that F(x) = ~ [F(x+O) + F(x-O) l, F(x+2'IT) = F(x). By the Dirichlet-Jordan test mentioned above, the
§5. Fourier series and Poisson's summation formula
Fourier series of F, say +ivx Cv e ,converges to F(x), so that \)=-00
L f(x+2k~) = F(x) k=-=
L C v v=-oo
ivx e ,
and at the point x 0, we have, by Lemma (5.3),
= L f(2kn)
k=-=
35
Remarks. Formula (5.5) is referred to as Poisson's summation formula.
The conditions for its validity can, of course, be relaxed. A more
symmetric form can be obtained by modifying the definition of Fourier
transform. If we write f (x) = g(~~), where a> 0, and ab = 27f, and
define
(5.6) v F [f] (a) J
ffn -=
-iax f(x)e dx,
then (5.5) takes the form
v (5.7) ra L g(ak) Ib L F[g](bv),
k=-oo V=-oo
where g ELl (-=,=) .
Examj2les 2 1 . If we take g(x) = -x (7ft) 1/ 2, t > 0, e a =
and use Example 6 of § 1 , we obtain from (5.7)
(5.8) 2
1 e-~k It It L
k=-= k=-=
(=
ab
b
f (-a) ) I21T
27f, a> 0,
= (2/TI)IIf, as we
the theta-relation
t> o.
may,
2. If we take g(x) -Ixl e , and use Example 4 of § 1 , we obtain from
(5.7) the formula
(5 .9) ra L e- Ikla =j(ii) L
1 ab 2~, a> O. 1+n2b 2
, k=-oo n=-oo
36 I. FOURIER TRANSFORMS ON L/-=,oo)
§6. The uniqueness theorem
If the Fourier transform f of a function f E L1 (-eo,oo) vanishes every
where, then the function itself must vanish almost everywhere. This
can be proved in many different ways, as we shall see later. We can
prove it, at this stage, by using the infinitely differentiable
function w which vanishes outside a finite interval, introduced in
(3.12), and applying Theorem 4.
~
Theorem 7. If f(x) E L1 (-00 < x < 00), and f denotes the Fourier transform
of f, and f(a) = 0 for every a such that -00 < 0.<00, then f(x) = 0 for
almost all x, -00 < x < 00.
Proof. Given real numbers c > 0 and e: > 0, let
for x < -c-e:, and x> c+e:, w (x) C,e: for-c<x<c,
and let w (x) be infinitely differentiable for -00 < x < 00. Its deri-c,e: vatives, of all orders greater than zero, vanish at x = -c-e:, -c, +c,
c+e:. Such a function exists; we have only to take a = -c, b = +c in
(3.12). Obviously we have
;;;c,e:(a) J
and on integrating this by parts sufficiently often, we see that ~ -k we e:(a) = 0(10.[ ), as 10.[ +00, for any integer k.:: 1, and hence, in
pa~ticular, w (a) E L1 (-00 < a < (0). Because w (x) has a finite c, e: c, e: derivative at every point x, -oo<x<oo, assumption (4.6) of Theorem 4 is
satisfied everywhere, and we can conclude that
00 A -ixo; w (x) = 2n J w (a)e do., c, e: -00 c, e:
-00 < x < 00,
where the integral converges absolutely, since;;; E L1 (-00,00). By the C,e: composition rule (1.13), and (1.9), we obtain
00 1 00/\
J f(y) w (x-y)dy = ~ J f(a) -00 C I £ 7f -00
~ -iax (w (a) e ) do.. C,e:
~
Since f(a) 0, for every a, we obtain
§6. The uniqueness theorem
(6.1) f f(y)w (x-y)dy = 0, c,t:
which holds for every c > O. From the definition of w , we have c,t:
where
(X-C
f f(y)w (x-y)dy = . f _= c,t: 'x-c-£
x+c x+c+£ + f + f )f(y)wc (x-y)dy,
x-c x+c ' £
/x-c / x-c f f(y)w c £(x-y)dy < f if(y)i'1 x-c-£' x-c-£
dy + 0, as £ -} 0,
and similarly also
while
Ix+c+£ f f(y)w (X-Y)dY / +0, x+c C, £
x+c f x-c
x+c f(y)w (x-y)dy = f
C,£ x-c
as £ -} 0,
f(y)dy.
By (6.1) it follows that for arbitrary x,
x+c f f(y)dy 0, x-c
(3
37
for every c>O; that is to say, f f(y)dy = 0, for arbitrary a and (3, a
which implies that f(x) = 0 for almost all x, -=<X<=.
Remark. The above proof makes use of the infinitely differentiable
function w together with the validity of "Fourier inversion", namely
w(x) 21T f ~ -iax w(a)e da.
We shall presently see that if both f and f belong to L1 (-=,=) such an
inversion holds almost everywhere, from which Theorem 6 would follow
at once.
(6.2) Corollary. If f1 ,f2 E L1 (-=,=), and f1
f1 = f2 almost everywhere.
~
f2 everywhere, then
38 I. FOURIER TRANSFORMS ON L/-=,oo)
§7. Pointwise summability
Examples given in §1 show that if fELl (-co,co), it does not necessarily
follow that the Fourier transform f of f also belongs to Ll (-co,co), so
that the integral - referred to sometimes as a Fourier integral -
co f f(a)e-iaxda
21T
may not exist as a Lebesgue integral, or even as a Cauchy principal
value. We can, however, introduce into the integrand a function K(a),
called a kernel, or a convergence factor, or a summability factor,
and formulate general conditions on K, and on its Fourier transform,
to secure the relation
R A a -iax lim f f(a)K(R) e da R->- co-R
f (x) ,
for almost every x.
Theorem 8. If K E Ll (-co,co), K is even, and K := H, and R> 0, then we have,
for every fELl (-co,co), the formula
(7.1) f f
R H(Rx).
If we assume further that 21T f H(t)dt = 1, then we have the formula
(7.2)
where
(7.3)
f gx(t) RH(Rt)dt, 1T 0
1 gx(t) ="2 [f(x+t) + f(x-t) - 2f(x) l, 0 < t < co.
Proof. If fELl (-co,co), the composition rule (1.13) gives
(7.4) f . co
A -~xa f f(a)KR(a)e da = f(x+t)HR(t)dt f f(y)HR(y-x)dy. -co
If we assume, in addition, that K is even, then H is even (cf. (1.10))
and the last integral equals the convblution (f*HR) (x), giving (7.1).
§7. Pointwise summability 39
If we assume further that 211 J H(t)dt = 1, then
00 i1l(f*HR) (x) - f(x) = i1l J [f(x+t) - f(x) J RH(Rt)dt
11
giving (7.2).
J g (t) RH(Rt)dt, o x
Formulas (7.1) and (7.2) can be used to formulate conditions under which
as R -+ 00.
Theorem 9. Let f E L1 (_00,00), and for each x E (_00,00) let
1 gx(t) ="2 [f(x+t) + f(x-t) - 2f(x) J, 0 < t < 00.
Let K E L1 (-00,00), Keven, K =: HE L1 (-00,00), and 211 J H(t)dt 1. Le t
H (t) be mono tone decreasing for 0 < t < 00. Then
(7.5) 211 (f*H R) (x) -+ f (x), as R -+ =, RH(Rx), R>O)
at every point x at which
(7.6) h
lim 1 J gx(t)dt = O. h-+O h 0
In particular, (7.5) holds at every point x at which f is continuous;
and uniformly over every closed interval of points of continuity of
f. In general, (7.5) holds for almost all x.
Proof. We note that the conditions imposed on H imply that
H(t).:.O for 0':-' t< 00, and that
(7.7) tH (t) -+ 0, as t -+ +00, or t + 0,
since
1 t "2 tH(t) < J H(x)dx -+ 0, as t-++oo , or t+ 0,
t/2
40 I. FOURIER TRANSFORMS ON L/--.oo)
so that there exists a constant C, such that tH(t) ~ C, for 0 < t < 00.
If we define
(7.8) G(t) t J gx(y)dy, t~O, o
then G is absolutely continuous. Because of assumption (7.6), given
e: > 0, we can choose 11 > 0, such that I G(t) 1 < e:t, for 0 ~ t ~ 11. Having
chosen such an 11, we keep it fixed.
By Theorem 8 we have
(7.9) 00
2~(f*HR) (x) - f(x) = n b gx(t)RH(Rt)dt
say. Then, for R> 0, we have, by (7.8),
1 11 1 11 11 = - J RH(Rt)dG(t) = - RH(R11)G(11) + - J G(t) R d{-H(Rt)},
1T0 1T 1T0
by partial integration of the Stieltjes integral, where the integrator
G is continuous and of bounded variation in [0,11], and the integrand
H is of bounded variation. By the choice of 11, we have
(7.10) 1111 ~ ~[11RH(R11) + R l t d{-H(Rt)}] = ~[r H(t)dt] < e:,
because of the particular normalization of H that has been assumed.
As for 12 in (7.9) we have
(7.11) 11 00 1 1121 = 21T J [f(x+t) + f(x-t) - 2f(x) ]RH(Rt)dt
11
as R+oo, for a fixed 11>0. From (7.9), (7.10), and (7.11), we obtain
(7.5). In any closed interval of points of continuity, f is uniformZy continuous, so that the choice of 11 = 11(e:) in (7.9) can be made inde
pendently of x, and (7.5) then holds uniformly in that interval.
Finally we note that condition (7.6) is equivalent to the condition
lim h+O
1 h 2h J f(x+t)dt = f(x),
-h
§? Pointwise swrrrnahiZii;y 41
which is satisfied for almost all x, since f E L1 (-=,=), because of
Lebesgue's theorem that the indefinite integral of f is absolutely
continuous and has a finite derivative, which equals f almost every
where.
Remarks 2 1. We can take for K(a) the Gauss kernel e-a , or the Abel kernel
e- 1ai , and conclude that if fEL 1 (-=,00), then at every point of
continuity x of f, and for almost all x, we have
(7.12) 2n f
as R -+ =. An equivalent statement is the following
(7.13) Corollary. If f E L1 (-=,00), then the Fourier integral
f f(a) e-iaxda 2n
is Gauss summable, and Abel summable, at every point of continuity x
of f, and for almost all x, to sum f(x).
2 2. If K(a) = e- a , then K(x) _ H (x) and since HR(X)
= RH(Rx), R> 0, we have
If we set t 1/R2 > 0, and
(7 • 14) W(x,t) 2
e- x /(4t) t>O,
the last integral becomes
(7.15 ) f f(~) W(x-~,t)d~ U(f; x,t),
say. This is referred to as the Gauss-Weierstrass integral of f, and
because of (7.1) and (7.12) we have the following
42 I. FOURIER TRANSFORMS ON L/-=,oo)
(7.16) Corollary. If fE:L 1 (-=,=), then the Gauss-Weierstrass integral
U(f; x,t) of f, given by (7.15), converges to fix) as t+o+, for
almost all x.
3. If K(a) = e- 1al , then K(x) _ H(x)
t = 1/R > 0, and
2 1+x2 ' and on setting
(7.17) p(x,t) = 1T
we obtain the following
t
t 2+x2 '
(7.18) Corollary. The Cauchy-Poisson integral of f, namely
= V(f; x,t) = J f(~) P(x-~,t)d~, t>o,
converges to fix) as t + 0+, for almost all x.
Theorem 10. Let f E: L1 (-=,00), and for each x E: (-=,=) let
1 "2 [f(x+t) + f(x-t) - 2f(x)l, 0.2. t <oo.
A 1 = Let K E: L1 (-=,00), Keven, K H, and 21T J H(t)dt = 1. Suppose that
there exists a function HO' such that
and HO is monotone decreasing in [o,~). Then
(7.5)
at every point x at which
(7.20) lim h+O
1 h
h Jig (t) I dt = 0; o x
RH (Rx) )
in particular, at every point x at which f is continuous; and uni
formly over any closed interval of points of continuity of f; (7.5)
holds for almost all x, in general.
Proof. As in the proof of Theorem 9 , given s > 0, we choose n such
that IG(t) 1< st for 0.2. t.2. n, where
§7. Pointwise swnm::rhiZity
G(t)
and write
t f Ig (u) Idu , o x
43
l(J + i)g (t)RH(Rt)dt 1T,0 n x
11 + 1 2 , say.
We then have
n 1111 ~ f Igx(t) IRHo(Rt)dt .... 0, as R .... oo,
o
as in the proof of Theorem 9, while
1121 ~ Ii -t-[f(X+t) n 1T
+ f(x-t) - 2f(X)]RH(Rt)dt\
Rn Ho(Rn) I f(x) I 00
< - 211 f 111 + f HO(t)dt - 21T n 1T nR
.... 0, as R .... 00,
for a fixed n > 0, as before. By a theorem of Lebesgue, condition
(7.20) holds almost everywhere for any function f E L1 (-00,00) •
Remarks. We may take for K(a) in Theorem 10 the Cesaro kernel given
by K(a) = 1 - lal for lal ~ 1, and K(a) = 0 for lal > 1. Its Fourier
transform R(t) = H(t) = (si~I~/2)2 is the Fejer Kernel (cf. Example 2,
§1), which is not monotone decreasing in [0,00). But we may take
HO (t) = ~ , with a suitable constant c > 0, so that IH (t) I ~ 1+t
c 1+t2 = HO(t), with HO(t) EL1(0~t<00), and monotone decreasing in
[0,00). Thus we obtain the following analogue of Fejer's classical
theorem on (trigonometric) Fourier series.
(7.21) Corollary. If fEL 1 (-00,00). then
R lim ~ f f(a) (1 - 1~I)e-iaxda f(x) R .... oo 21T -R
at every point of aontinuity of f, and uniformly over any alosed interval of points of aontinuity of f, and for almost all x, in
44 I. FOURIER TRANSFORMS ON L/-OO,oo)
general. The Fourier integral
00
21T f
is, in other words, (C,1) summable (Cesaro summable of order 1) to
sum f(x) at every point of continuity of f, and uniformly in every
closed interval of points of continuity, and for almost all x in
generaL
§8. The inversion formula
The theorems on pointwise summability can be used to "invert" the
Fourier transform almost everywhere.
Theorem 11. If f E L1 (-00,00), and f E L1 (-00,00), then we have
for almost all x E (-00,00).
Proof. By Theorem 9, Corollary (7.13), we have, for almost all x,
lim 21T f f(a) e- 1ai / R e-iaxda = f(x). R-+oo -00
~
If f E L1 (-00,00), then the left-hand side equals
by Lebesgue's theorem on dominated convergence.
Remarks. If fE L1 (-00,00) , the integral
defines a continuous function of x, so that the function f E L1 (-00,00)
that we started with in Theorem 11 is continuous almost everywhere.
Hence we obtain
§8. The inversion formuLa 45
Theorem 11'. If fEL 1 (-co,co}, and fEL1 (-co,co}, and f is continuous in
(-co, co} , then
f(x} t f(a.} e-ia.xda. -co
for every x E (-co,co) •
Examples
1. We have already (§1, Ex. 2) seen that if
{ 1- IX I , Ixl~1, K(x}
0, Ixl>1,
A (Sin a./ 2)2 A then K(a.) a./2 Here both K and K belong to L1 (-co,co}. Hence
we have, by Theorems 11 and 11',
(S .1) ~ fCO(sin a./2)2 -ia.xd 21f a./2 e a. -co
f CO(Sin a./2)2 eia.xda. 21f a./2 -co
f-'X" Ixl~1, 0, Ixl>1.
For x 0, we get the formula
(S.2) 1f.
2. If K(x} = e-a1xl , with a> 0, then K(a.}
By Theorems 11 and 11' we have
2a
a 2+a.2 (Example 4, §1).
21f fco 22a 2 e-ia.xda. = 2a j cos a.x da. = e-a1xl, -co a +a. 1f a a 2+a. 2
hence
(S.3) 1f -a I x I 2a e , a> O.
3. If a> 0, b ~ 0, we have the formula
(S.4)
For
j e-a2x-b2/x x- 1/ 2dx = ~ e-2ab • a
46
f()1+1)
so that
hence
I. FOURIER TRANSFORMS ON L/-oo,oo)
J e-x x)1dx, for)1 > -1,
° 2 2
J e-(x +a)y y)1dy, (x real, a>O, )1>-1)
° 00 2 2
1 J e- a y y)1dy J e-x y cos Sx dx f()1+1) ° °
2 if we use the expression for the Fourier transform of e-x (Example 6,
§1). On taking )1 = 0, S = 2b ~ 0, and using (8.3), we obtain (8.4).
The next theorem gives sufficient conditions for the Fourier transform
of fELl (-00,00) to belong also to L1 (-00,00) .
Theorem 12. If f(x) EL 1 (-00<X<00), and there exists h>O, such that
If(x)I~M<oo for -oo<-h~x~h<+oo, and f(a) >0 for every aE (-00,001,
then we have
00 J I f (a) I da
so that (by Theorem 11)
J f(x) = 2rr
for almost every xE ( -00,(0) •
Proof. Let K(a) = -Ial e , so
J -00
~
f(a)
that
~
f(a) da < 00,
-iax e da
K (x) := H(x) 2 --2 ' and l+x
irr Joo H(t)dt = 1. As in Theorem 8, (7.4), we have by the composition -00
rule, for any R> 0,
(8.5) J f(a) e- Ial / R e-iaxda 00
J f(x+t)RH(Rt)dt _00
J f(X+~)H(t)dt -00
§3. The inversion formula 47
For x = 0, we get
f f(~)H(t)dt , R> ° -<X> -<X>
[-hR hR <X>
f +f +f] -<X> -hR hR
say. We have
1121 ~ M f H(t)dt M·21T, since H(t) .::0. - <X>
Since tH(t) is bounded,
where Nl is a constant, and similarly
constant (since H is even). Hence
N2 I III ~ 11 II fill' where N2 is a
-00
A
where N is independent of R. Since f(a) .::0, we have, by Lebesgue's
theorem on monotone convergence,
lim f f(a) e-Ial/Rda A
f f(a)da~N<oo. R+o::> -00
(8.6) Corollary. If h > 0, f(x) ELl (-h~x~h), and If(x) I ~M< 00,
for I x I ~ h, and
h (j)( a) :: f
-h
iax f(x) e dx.::O,
then (j)(a) E Ll (-00< a < (0).
We have only to define f(x) ° for I x I > h, and use Theorem 12.
_1 I _a 2 Remark. Instead of the kernel e la , we could have used e or the
Cesaro kernel: K(a) = l-Ial, for lal ~ 1, and K(a) = 0 for lal > 1.
Theorem 12 implies also the following
A
(8.7) Corollary. If fEL1(-00,~), f(a).::O for -oo<a<oo, and f is
continuous at the origin, then fELl (-00,00), and
48 I. FOURIER TRANSFORMS ON L 1 ( -«>,00)
(8.8) co
f(x) - f Af(N) e-io.xdN - 27T ~ ~
for a~most every x E (-co,co). In particu~ar,
(8.9) co
f(O) = 217T f A
f(o.)do.. -co
We shall denote by L1 o n o L2 the class of functions f, such that
f E L1 (-CO,"") and f E L2 (-co,co) •
(8.10) Lemma. Let f,gEL1 o n o L2 . Then the function h defined by
(8.11) h(x) = f f(x+y)g(y)dy = f f(x-y)g(-Y)dy
is bounded, and continuous, and be~ongs to L1 (-"",co) •
Proof ° If we define G(y) = g(-y), then G E L1 (-"","")' and h is the
convolution f*G, hence hE L1 (-CO,""), by Theorem 2. Since
Ih(x)l..:. II fl1 2 o llg11 2 , where f,gEL2 (-co,co), h is bounded. Since
( "" 2 )1/2( co 2 )1/2 Ih(x+t) - h(x) I..:. £""If(x+t+y ) - f(x+y) I dy £""Ig(y) I dy
= T f (t) II g 112 + 0 as t + 0, (af. (1.17), § 1)
we see that h is continuous.
2 Theorem 13. IffEL,.n o L2 , then IIfl12
transform of f be~ongs to L2 (-co,,,,,).
27T II f II ~ , so that the Fourier-
Proof. Let F(x) = f(-x), so tha~ FE L1 o~. Let h = f*F E L1 (-"","").
Since the Fourier transform of f(x) is f(-o.), we see that A A 12 h(o.) = If(o.) ~O. Since h is bounded, and continuous, it follows from
Theorem 12 that h E L1 (-"",""), and that
h(x) = f - 1 f"" I Af(N) 12 e-io.xdN f(x+y) f(y)dy = 27T ~ ~
for every x E (-"",""). On setting x = 0, we get the theorem.
§8. The inversion formuLa 49
co
Theorem 14. If f,gEL 1-n-L2 , then f f(x)g(x)dx co
21T f f (a.)g(a.) da.. -co -co
Proof. By Theorem 13 we have II f 112 < co, II g 112 < co, so that II t g 111 < co
by Schwarz's inequality. If hex) is defined as in (8.11), h = f*G, where G(y) = g(-y), then hE L1 (-co,co), and h(a.) = f(a.) -g(a.), and since h is continuous, its Fourier transform can be inverted everywhere (Theorem 11'), so that
co
hex) = 21T f t(a.) g(a.) e-ia.xda., -co
and on setting x ° we get the required result.
Examples
1. If f(x) = 1, for Ixl ~ 1; and f(x) = 0, for Ixl > 1, then
t(a.) = 2(Si~ a.) (cf. Example 1, §1). Theorem 13 gives the formula
2. If a>O,
then t(a.)
formula
co sin2 a. f 2 da. = 1T. -co a.
and f(x)
sin2 (¥)
a(a./2)2
= 1 - I~I, for Ixl..::a; and f(x) = 0, for Ixl >a,
(cf. Example 2, §1), and Theorem 13 gives the
fco (sina. ba.)4 21T 3 da. = 3"" b, for any b.::O. -co
3. If a> 0, and f(x) = e-alxl , then t(a.) = 22a 2 (cf. Example 4, §1). I I a +a.
For b > 0, let g(x) e-b x • Then Theorem 14 gives the formula
1T 2ab(a+b)
4; If f(x) = 1, for Ixl~a; and f(x) = 0, for Ixl >a, where a>O, and g(x) = 1, for Ixl..::b; and g(x) = 0, for Ixl > b, where b > 0, then Theorem 14 gives the formula
co • f s~n ax-sin bx dx
° x2
1T "2 min (a,b).
Fourier transforms in S
We have considered in §3 Schwartz's space of infinitely differentiable
functions f which are "rapidly decreasing". We have noted in (3.7) that
50 I. FOURIER TRANSFORMS ON L/-oo,oo)
if f E S, then f E S, S being a dense subspace of L (-00,00) for every p, p
1 ..s. p < 00. I t is a trivial consequence of Theorem 11' that if f E S ,
then
(8.12) f(x)
for every x E (-00,00), so that the Fourier transform maps S onto itseZf.
Further we have for any two functions f, g E S,
(8.13)
"A For gES, and since g(x)
co A -iax 2n f g(a) e da, we obtain
_ 1 00 i g(x) = 2n f g(a) e aXda
A
2n g(x). By the composition rule (1.13),
however, we have
00 f f(x) g(x)dx
or
00 f f(a) ~(a)da,
-00
as claimed. On taking g = f, we get
00 2 1 00 A 2 f I f (x) I dx = 2n f I f (a) I da • -~ -co
These are special cases of Theorem 14, but simpler to prove directly.
Finally, if f,g E S, then £,g E S, hence also fog E S. But (hg) fog,
by Theorem 2. Hence
(8.14) 1 00
2n f -00
which means that f*g E S.
In the notation of (1.2) we can write (8.13) as
(8.15 ) f f(x) g(x) dx f F[f](a) F[g](a)da. -00 -00
The fact that S is a dense subset of L2 (-00,00) leads (in Ch.II) to
Plancherel's theorem.
§9. Summability in the L1-norm
§9. Summability in the L1-norm
We have seen in §7 that if fELl (-oo,oo), for special choices of the
kernel K, we have
lim 2~ f f(a) K(~) e-iaxda = f{x), R+oo -00
pointwise almost everywhere. It is somewhat simpler to consider
51
this limit in the L1-norm. Before doing so, we shall prove a general
result on approximating any fELl (-oo,oo) in the L1-norm.
Theorem 15. Let HELl (-oo,oo), with 21~ f H(a) da = 1, and for R> 0
let HR(a) = R H(Ra).
If fELl (-=,00), then we have
(9 • 1 )
Proof. By definition we have
2~ f f(x-y) RH(Ry)dy , -oo
and, by assumption, we have
2~ f HR(y)dy 2TI f RH (Ry) dy 2~ f H(a)da 1 .
Hence
2~ f [f(x-y) - f(x)] RH(Ry)dy ,
and
II 2~ (hHR) - fll12 2~ f dx f If(x-y) - f(x) I·RIH(Ry) Idy -oo
2~ f 'fry) R!H{Ry) Idy (cf. (1.14»
where 'fry) is the L1-modulus of continuity of f {see (1 .14», which
is bounded, even, non-negative, and tends to zero as y ~ O. Given
52 I. FOURIER TRANSFORMS ON L/--=,oo)
£ • 211 £ > 0, we can choose II > 0, such that 0.:::. 'f{y) < ---
IIH 111 for I y I .:::. ll. We
then write
11 + 1 2 , say,
where
II21.:::.c J IH{t) Idt+O, as R+cx>, Itl>llR
(c being a suitable constant), while
by the choice of ll, and the- theorem follows.
Remarks
= (sin X/2)2 If H{x) x/2 ' then HEL1 {-eo,eo), and 211 J H{a)da 1, {see
(8. 1 ) ), and
J H ( ) eiaxdx 211 x
Then
( - lal,
0,
K{a), say.
for I a I .:::. 1,
for I a I > 1 ;
where KR vanishes outside the interval [-R,Rj. The Fourier trans
form of (1/211) (f*HR), in Theorem 15, is f.K R, which vanishes there
fore outside [-R,Rj, and from Theorem 15 we can deduce the following
(9.2) Corollary. Every function fELl (-eo,eo) can be approximated
in the L l -norm by a function in Ll (-eo,eo) whose Fourier transform
§9. Summability in the L1-norm 53
vanishes outside a bounded interval.
Theorem 15 has also another interpretation. There is no unit element relative to multiplication in the L 1-algebra over (-=,00). That is to
say, there exists no function IEL1 (-00,00) such that I*f = f, for
every fEL 1 (-00,=). For if it did, we would have, in particular,
1*1 = I, which implies that I(a) = {I(a)}2, hence I(a) = 0, or 1, for
each given a. Since I(a) is continuous, we must have I(a) 0: 0 or
I(a) 0: 1. By the Riemann-Lebesgue theorem, however, I(a) +0 as lal +00.
Hence \le must have I (a) 0: 0 identically. By the uniqueness theorem for
the Fourier transform (Theorem 7), it follows that I(a) = 0 for al
most all a. If fEL 1 (-00,00) is such that it is non-zero almost every
where, the equation I*f = f will be contradicted. We have, however,
an approximate unit in L1 (-00,00), by which we mean that we can find a
sequence of functions (on)' such that 0n.?O, 0nEL1(-00,00), IIonl11 = 1,
for each n, and such that 0n*f + f in the L1-norm for every f E L1 (-00,00). (sin nx) 2
Theorem 15, and Corollary (9.2), show that Hn(x) 2 is 2 'IT n(x/2)2
such an approximate unit.
Theorem 16. Let KEL 1 (-00,00), Keven, Ko:HEL1 (-00,oo), 2'IT J H(a)da 1.
Let R> 0, and HR(a) = R H(Ra).
If fEL 1 (-00,oo), then
(9.3)
where (as in (7.1))
(9.4) 00
2'IT (f*HR) (x) = 2'IT J [(a) K(~) e-iaxda.
Proof. Since f,H E L1 (-00,00), we note that f*H R E L1 (-"",00), and the
integral equalling the convolution in (9.4) exists for every x, since
K E L1 (-00,00) and f is bounded. To prove the theorem we have only to
use Theorem 15.
Remarks
By taking K(a)
the following
2 e-a and making use of (7.14) and (7.15), we deduce
54 I. FOURIER TRANSFORMS ON L/-OO,oo)
(9.5) Corollary. If f E L1 (-00,00), then the Gauss-Weierstrass
integral
U(f;x,t) f f(s) W(x-s,t)ds, W(x,t) 21(1Tt)
of fconverges in the L 1-norm to f (x), as t + 0+.
_- e-iai, By taking K(a) and making use of (7.17), (7.19), we deduce
the following
(9.6 ) Corollary. If f EL1 (-00,00), then the Cauchy-Poisson integral
V(f;x,t) f f(s) P(x-s,t)ds , P(x,t) 1 t -2 2' t > 0,
1T t +x
of f converges in the L 1-norm to f (x), as t + 0+.
The principal deduction from Theorem 16, which results from taking
for K(a) the Abel, Gauss, and Cesaro kernels separately, as in (7.12)
and (7. 2 1 ), is
(9.7) Corollary. The 'Fourier integral'
1 f ~ -iax -- f(a) e da, 21T -00
is Abel, Gauss, and Cesaro (C,1) summable in the L1-norm to f(x).
This is just another way of expressing (9.3) and (9.4).
For instance, in the case of the Gauss kernel, we have:
21T f 2/ 2 .
f(a) e-a R e-laxda + fix), as R+oo, in the L1-norm.
By Weyl's formulation of the Riesz-Fischer theorem, there exists a
sequence +00 as k + 00, such that
2 2 f f(a) e-a /Rk e-iaxda + fix), as k+ oo ,
21T
for almost every x E (-00,00). If we assume, in addition, that
f E L1 (-00,00), then, by Lebesgue's theorem on dominated convergence,
we obtain Theorem 11 on Fourier inversion.
§9. Summability in the L1-norm
As another application of Theorem 16 we shall prove
Theorem 17. Let f,g E L1 (-00,00). If geed
f(x) - J g(y)dy x
-iaf(a), then we have
55
Proof. Case (i). Let us assume, in addition, that g,f E L1 (-00,00). Then,
by Theorem 11, we have
00 -iax f(x) 2n J f(a) e da
-00
and
00 -iax -iax g(x) 2n J g(a) e da 2n J (-ia)f(a) e da,
-00 -00
for almost aU x, so that
b ...l.- ( (e- iba e -iaa)
~
J g(x)dx - f(a)da, -00 < a < b < +00, 2n a -00
= feb) - f (a) ,
x hence g(x) = f' (x) for almost all x. (Note that if G(x) = J g(y)dy,
o where g E L1 (-00,00), then G' (x) = g(x) for almost all x. And if
G(b) - G(a) = feb) - f(a), for all a,b, such that -oo<a<b<+oo, then
G differs from f by a constant, so that f' = G' = g almost every
where. )
Case (ii). Let K(a)
For R> 0, define
00 (9.8) FR(X) 2n J
-00
and
GR(x) J = 2n -00 ~
e
2 -a e
-iax
e-iax
~
-00< a < 00, and let K(a) _ H(a)
f(a) a K(R)da,
00 -iax ~ a g(a) a J K(R)da 2n e (-iaf(a))K(R)da. -00
We note that f is bounded, and that FR is the Fourier transform of
a function in L1 (-00,00) , and therefore FR(X) +0 as Ixl +00.
By the composition rule (1.13), we have
56 I. FOURIER TRANSFORMS ON L/-oo,oo)
'" 2rr f f(x+t) RH(Rt)dt , -",
so that
(9.9) '" 1 '" f IFR(X)ldx ~ 2rr IIfll1 f RH(Rt)dt = IIfll1 <'" -00 -0)
because of the choice of K. Hence F R E L1 (-""",), and similarly also
GR E L1 (-'" ,"'), for each R> O.
Since
b> a,
we have, on letting b -+- "',
'" FR(X)
Since g E L1 (-"',ca), we have, by Theorem 16,
'" lim f R+'" -'"
which implies that
'" '" lim f g(y)dy • R-+-co -00
Hence, for every fixed x, we have
00
- f g(y)dy . x
On the other hand, by Theorem 9, FR(X) ... f(x), for almost all x, as
R'" "'. Hence
'" f(x) = - f g(y)dy ,
x
for almost all x E (-"',"') •
§10. The central limit theorem
As an illustration of the method of Fourier transforms, we shall
state and prove a theorem which corresponds to what is known as the
§10. The central limit theorem 57
central limit theorem in the theory of probability.
00 Theorem 18. Let fELl (-00,00), f(x) 2:. 0, f f(x)dx = 1, f xf(x)dx = 0,
-00 -00
and f x 2f(x)dx = 1, and let fn = f* ..• *f, the convolution of f with
itself n times. Then we have
blil lim f fn(x)dx n-+oo alil
Proof. Let
Xa,b(x)
so that
f
and
2 e-x /2
f 1(2Tf) -00
b 2
f e-x /2
dx, -00 < a < b < 00. 1(2Tf) a
f' for a 2. x 2. b,
0, otherwise,
2 b e-x /2 X b(x)dx f 1(2Tf) a, a
dx.
It is sufficient therefore to prove that
(10.1) lim f n-+oo
f -00
2 e-x /2 1(2Tf) Xa,b(x)dx.
In order to prove this, it is sufficient, in turn, to prove that
2 00 e-x /2
Iil fn(xlil) k(x)dx = f 1(2Tf) k(x)dx, (10.2) lim f n4-(X) -00
where k E S, where S denotes Schwartz's space of infinitely differen
tiable functions on (-00,00) which are rapidly decreasing, defined in
§3.
For, given Xa,b we can find two functions kl ,k 2 EVe S, where V denotes
the subspace of infinitely differentiable functions on (-00,00) with
bounded supports, such that for any s > 0, we have
(10.3) kl (x) < Xa,b (x) < k2 (x), and f [k 2 (x) - kl (x) jdx < 4s.
58
We have only to define (as in (3.12), (3.13})
{ 1, a+£ < x < b-£ k (x) = - -
1 0, x':'a, x'-:b {1' a < x < b
k 2 (X) = 0, x2 a-£, x> b+£
both k1 and k2 being infinitely differentiable everywhere.
If (10.2) is proved, then it holds with k = k1 and k = k 2 • Since
X b(x}dx a,
co
< f
X b(x)dx a,
< £ + 4£, for n.::nO'
by (10.3), we obtain
00
f -co
and by using k1 in place of k2' we see that the left-hand side is
greater than -5£ for n..: n', thus proving (10.1).
A
It remains to prove (10.2). If k E S, we have seen that k E S, and by
Theorem 11', inversion holds everywhere. Hence
fCO in fn(xin}k(x}dx = Joo in fn(xln){dn fco k(a)e-iaxda}dx _OQ _00 _00
ex> (t e-iaxdx)da 2n f k(a) In fn(xlil)
-co -co
co
(Jex> fn(y)e-iay/lndY)da 2n f k(a)
_ex> -co
(10.4)
A a -a where f(- 7n) denotes the Fourier transform of f at the point 7n Now
§10. The aentraZ Zimit theorem
( 10.5) 1 ,
and for every fixed CI., CI. * 0, -00 < CI. < 00, we have
( 10.6)
00 {' 2 2 = £00 f(x) 1 - ~/nx - Cl.2~ (l+r(n,X»}dX
where r(n,x) is bounded uniformly in x and n, and r(n,x) -+- 0, as
n -+- 00, for every fixed x, -00 < x < 00. (Note that if ITn 1 ~ 1, then 1 r(n,x) 1 < c, where c is a constant independent of x and n, while
1 ;n 1 > 1 implies that
11 + r(n,x) 1 = I (_e-iCl.x/1n + 1 - i;:)( ~n2)1 = 0(1), CI. x
the constant implied by the 0(1) being independent of x and n). It follows that
00 2 "" 2 { } lim J x r(n,x)f(x)dx = f x f(x) lim r(n,x) dx = 0(1), n-+oo -00 -00 n-+CX)
"" as n-+-eo, since f x 2f(x)dx 1, and f(x) ~O. By hypothesis, we also have
f f(x)dx = 1, f xf(x)dx 0, -00
so that (10.6) yields the relation
(10.7) A CI. f(- -) In
2 1 - ~n (1 + 0 (1) ), as h -+- 00,
A
which holds also for CI. = 0, since f(O) 1. Hence
59
( 10.8) {1 _ (CI./ ;7) 2 (1 + 0 ( 1 ) ) } n -+- e- (CI./ 12) 2, as n -+- 00,
since lim (1 - ~)n - e-x If we take (10.4), and use (10.5), then n-+-oo n-
by Lebesgue's theorem on dominated convergence, we obtain
lim f"" In fn(xln) k(x)dx·= in Joo k(CI.) lim {f(- In)}n dCl. n+oo -00 -00 n+oo
60
21T J
I. FOURIER TRANSFORMS ON L/-oo,oo)
~
k(a) 2 -a /2 e da, by (10.8)
k (a) I( 21T) 2 -a /2 e da,
if we use the composition rule (1.13) and the Fourier transform of
_a 2/2 e (see §1, Example 6). Thus (10.2) is proved, which, as we have
already shown, implies (10.1) and the theorem.
§11. Analytic functions of Fourier transforms
~
If f(x) 1 for all x, -= < x < =, then obviously (by Theorem 1) f
cannot be the Fourier transform of a function in L1 (-=,=). If, instead
of the interval (-=,=), we had only a bounded interval, say [a,b],
then ~ere exists a function f E ~1 (-=,=), such that its Fourier trans
form f (a) = 1 for a E [a,b], and f vanishes outside a larger interval.
We have constructed in (3.12) an infinitely differentiable function
w, which equals 1 in [a,b], and vanishes outside (a-E, b+€), where
E > O. Such a function belongs to Schwartz's space S, which has the
property that if f ES, then f E S {see (3.7)). Further, as a trivial
consequence of the inversion formula (Theorem 11 '), and the fact that
S eL1 (-=,=), we note that if we define ~ by the relation
~(x) 21T J w(t)e-itxdt, -=<X<=,
v then wE S eL1 (-=,=), and
Hence we can assert the following (by taking ~ 6) :
(11.1) Given two real numbers a,b with b > a, and a number E > 0,
there exists a function 6 E L1 (-=,=), such that its Fourier transform
6 has the property
§11. Anatytic functions of Fourier transforms
1, a':'O:2.b,
;S (0:) { 0, 0:':' a - E, 0: 2:. b+ E, E > 0,
infini tely differentiable in (-00 < 0: < 00) •
(11.2) There exists a function fEL 1 (-00,00), such that its Fourier
transform f has the property
f(o:) > 0, for 0: > 0,
and
f(o:)
For if F(x)
then FE L1 (-00,00), and
F(-x)
say, so that
A
0, for
-x r e ,
0,
0: < 0.
for x> 0,
for x.:.O,
f t e- t (1+ix) dt
°
F(x) 27T f(-x).
(1+ix) 2 27T f(x),
Since F,F E Ll (-00,00), and F is continuous, we have (by Theorem 11') ,
61
l°O A -itx F(t) = 27T f F(x)e dx = f f(-x)e-itxdx = f itx A
f(x)e dx = f(t), -00 -00
and the function
f(x)
satisfies (11.2).
(11.3) Given an interval (-oo,al, or [a,oo), where a is a real
number, there exists a function fELl (-00,00), such that its Fourier
transform vanishes on the given interval, and does not vanish outside
that interval.
A A
For if we c0nsider h(o:) defined by h(o:) g(o:-a), where g is defined
~y (11.2), then h is the Fourier transform of a function h E L 1 , and
h vanishes on the interval (-oo,al but not outside.
62 I. FOURIER TRANSFORMS ON L/-OO,oo)
~
Similarly the Fourier transform of h(-t) is h(-a) 0, for
a.::. a, and h(-a) *0, fora<a.
Theorem 19. Let f E L1 (-00,00), f(O) = 0, and E > 0. Then there exists a
function hE L1 (-00,00), such that (i) IIhl11 < E, (ii) h = f in a neigh
bourhood of the origin, and (iii) f(a) = ° implies that h(a) = 0.
~roof. By (11.1) there exists a function A E L1 (-00,00), such that
A(X) 1, for Ixl.:::1. If we set, for any fixed R>O, AR(X) = RA(Rx),
then
1, for I a I .::: R,
since
-00
~
Since f(O) = ° by hypothesis, we have f f(x)dx = 0, and
Hence
00
IIAR*£1I1<f dxf If(y)IIAR(x-y)-AR(x)ldy
00
f If(y) Idy f IAR(X-y) - AR(X) Idx (by Fubini's theorem)
f If(y) Idy f RIA(RX - Ry) - A(Rx) Idx -00
00
f If(y) Idy J I A (x-Ry ) - A (x) I dx .
Now -00
f I A ( x-Ry ) - A (x) I dx .::: 2 II A 111 -00
§11. Analytic functions of Fourier transforms
for every fixed y, and the (L1-modulus of continuity of A) integral
tends to zero as R ->- 0 for any fixed y (see (1.17». Now choose H so
large that
-M J If(y)1 211AI11 dY<E 1 , J I f (y) I 211 A 111 dy < E 2 '
M
for any two strictly positive numbers E1 ,E 2 , given in advance. If
lyl~M, then lyRI~MR, and yR->-O as R->-O. Given E3>0 and M>O, we
can therefore choose R sufficiently small to ensure that
M 00 J If(y) I dy J IA(X-Ry) - A(X) Idx< E3 · -M
Thus we have II AR*f 111 < E, for any arbitrary E > 0 given in advance,
63
by proper choice of R> o. With such an R we define h = AR*f, so that
h = ~R·f. By the definition of A, we have ~R(a) = 1 for lal ~R, hence
h(x) = f(x) for Ixl ~R. Obviously h(a) = 0 if f(a) = o.
Theorem 20. If R> 0, and tp (z) is holomorphic for I z I < R, with tp(O) 0,
and hE L1 (-00,00), with Ilh 111 < R, then there exists a function
g E L1 (-00,00), such that tp(h) = g.
Proof. Let f E L1 (-00,00). Then the Fourier transform of the convolution
of f with itself n times is the nth power of f (Theorem 4), and since
the functional f ->- f is linear, it follows that the function
n p(z) = L a k zk
k=1 (z, a k complex)
carries Fourier transforms into Fourier transforms.
By hypothesis we have tp (z) = L a zn, I z I < R, the series converging n=1 n
absolutely (ao = 0 since tp(O) = 0). Since IIhl11 < R by assumption, we
have Ih(x) I ~ IIhl11 < R, for all x, so that
00 L a (h(x»n, -oo<x<oo.
n=1 n
Set h1 = h, hk = h k _ 1*h, for k~:2. Then obviously Ilhkll.::.llhll~, for ~ ~ k
k~2, while hk(x) = (h(x» , by (2.3) and (2.2).
If we choose integers m,n, such that n~m~ 1, then
64 I. FOURIER TRANSFORMS ON L/-oo,oo)
n n n II 1: ak hk 111 ~ 1: I ak I II hk 111 ~ 1:
k=m k=m k=m 00
k IIhll1 converges, so that
as m,n .... 00.
Since the function space L1 (-00,00) is aomp'lete (if II fm-fnl11 .... 0 as
m,n .... oo , then there exists fEL1 (-00,00) , such that IIf-fnIl1 .... 0 as n .... oa ) ,
there exists a function g E L1 (-oa,oo), such that
Hence
n II 1: ak hk - gI11 .... 0, as n .... oo.
k=1
n 1: ak hk(X) .... g(x) , as n .... CD ,
k=1
uniformly in -CD < x < CD (see (1.12». Thus we have
~ k ~ 1: ak (h(x» = ql(h(x», for -CD < x < 00.
k=1
co
(11.4) Coro'l'lary. If ql is an entire funation, with ql(O) = 0, then ql aarries Fourier transforms (of funations in L 1 (-CD,CO») into Fourier transforms (,!f funations in L1 (-CD,CO)')'
Remarks
1. The condition ql(O) = 0 in Theorem 20 is necessary. For if n k
Q(z) = aO + P(z) = aO + 1: ~ z , with aO *' 0, then by Theorem 20, k=1
and the Riemann-Lebesgue theorem (Th.1), we have
~
lim P(f) (x) = 0, Ixl .... co
while Q(f) ~
aO + P(f), and
~ ~
lim Q(f) (x) Ixl .... CD
aO + lim P(f) (x) Ixl .... CD
§11. AnaLytie funetions of Fourier transforms
so that Q(f) cannot be the Fourier transform of a function in
L1 (-co,co) •
65
2. Given a finite interval [a,b], there exists a function 0 E L1 (-=,=),
such that ;S(a) = 1 for a<a<b (see (11.1». Thus, if fEL 1 (-=,co),
then
Q(f) (x) = a o 6 (x) + P(f) (x), fqr a < x < b.
Hence Q(f) coincides on a bounded, closed interval [a,b] with the
Fourier transform a 0 6 + P(f) of a function in L1 (-co,co). More
generally we have
Theorem 21. Let D be a domain (that is. an open. eonneeted set) in the
eomp~ex pLane. and ((J a funetion hoLomorphie in D. Let f E L1 (-co,=).
and f (x) ED for -= < a.::. X'::' b < co. Then there exists a funetion
gEL 1 (-co,=). sueh that ((J(f(x» = g(x). for a<x<b.
For the proof we shall use two lemmas.
(11. 4) Lemma. If fEL 1 (-co,=). f (0) O. and ((J is hoLomorphie at the
origin. and ((J(O) O. then there exists a funetion gEL 1 (-co, =) • sueh
that ((J(f) A
in neighbourhood the origin. = g a of
Proof. There exists a number £ > 0, such that ((J is holomorphic in
I z I < £. Hence, by Theorem 19, there exists a function hE L1 (-=,=) ,
such that IIhl11 < £, and
f(x) hex), XENO'
where NO is a neighbourhood of the origin. By Theorem 20 there exists
a function g E L1 (-=,=), such that ((J(h) = g in (-=,=); in particular,
((J(~(x» = g(x), for x E NO.
(11.5) Lemma. If f E L1 (-=,=). f(a) = S. and tp is hoLomorphie at S.
then there exixts a funetion g E L1 (-=,=). sueh that
((J(f(x»
where Na is a neighbourhood of a.
66 I. FOURIER TRANSFORMS ON L/-oo,oo)
Proof. If the lemma holds for a = 0, then it holds for arbitrary real a. For suppose that Il * 0, and set f, (t) = eiatf(t), -00< t < 00. Then
flex) = f(x+Il), therefore f,(O) = f(ll) = 8. If we assume the lemma
true for Il = 0, then there exists a function g, E L, (-00,00), such that
~(f,(x» = g,(x), for all x .in a neighbourhood of the origin. There
fore ~(f,(X-Il» = g,(X-Il), for all x in a neighbourhood of a, say
Nil. If we define g by the condition g(ll) = g,(X-Il), then we have
since flex-a) = f(x).
If the lemma holds for 8 = 0, then it holds also for 8 * 0. For suppose
that 8*0, and set flex) = f(x) - 88(x), where 8(x) =, for Ixl.::' (as in (".,», and let 1jJ(z) = ~(z+8). Then f,(O) = f(O) - 85(0) = 8-8 = 0, and 1jJ(z) is holomorphic at z = 0. By the assumption that the
lemma holds for 8 = 0,. there exists a function gEL,(-oo,oo) , such that
where NO is a neighbourhood of the origin. We may assume that
Noc[-',,]. Then we have, for'xENo '
which proves the lemma in the case 8 * 0.
If the lemma holds in the case ~(O) = 0, then it holds also in the
case ~(O) * 0. For suppose that ~(O) * 0, and set 1jJ (z) = ~(z) - ~(O), so that 1jJ(0) = 0. On the assumption that the lemma holds in the case
~(O) = 0, we can conclude that there exists a function g, E L, (-00,00) ,
such that 1jJ(f(x» = g,(x), XENO' where NO is a neighbourhood of the
origin. If we define g by the condition 9 = 9, + ~(0)5, (g exists
since g, + ~(O)g is the Fourier transform of a function in L,(-oo,oo»,
then we have, for x E NO'
g, (x) + ~(O) 5 (x) g(x) •
§11. Analytia funations of Fourier transforrru"
The proof of Lemma (11.5) is thus reduced to that of Lemma (11.4),
which is already proved.
67
Proof of Theorem 21. By Lemma (11.5), every pOint x E [a,b] is covered
by an open interval on which ~(f) coincides with the Fourier trans
form of a function in L1(-~'~). By the theorem of Heine-Borel, a
finite number of such intervals cover [a,b]. We may suppose that none
of those intervals is wholly contained in another. Let (a 1 ,8 1 ) and
(a 2 ,82 ) be two of those intervals. We may suppose that
We choose g1 ,g2 E L1 (-~,~), such that
and A
~(f(x»
It follows that g1(x) g2(x) for a 2 <x<8 1 ; in fact, for a 2 .::.x.::.8 1 ,
since the Fourier transform of a function in L1(-~'~) is continuous.
A (IlZ,1) (,8,.1 )
A W, Wz
)( I i~ I Il, IlZ {J, (Jz
As in (3.12), and (11.1), there exist infinitely differentiable
functions w1 ,w 2 ' such that
0, x..:: 81 1, for a 1 .::. x.::. a 2
(y-8 1 ) (y-a 2 ) e dy,
for a 2 ': x.: 8 1
68 I. FOURIER TRANSFORMS ON L 1 (-oo,00)
~ x ( y- Cl 2 ) (y- 13 1 ) f e dy,
c Cl 2
For Cl2.:::.x.:::.131, we have therefore (;;'1 (x) + (;;2(x) 1.
If we define ljJ
~(x) g 1 (x) 4J(f (x) ) ,
and if x E (13 1 ' 13 2)' then
~ (x)
If x E [Cl 2 ,13 1 ], then g1 (x) g2 (x), so that
ljJ (x) = [(;;1 (x) +(;;2 (x)] g1 (x) g1 (x)
(by choice of g1)' Hence
This argument can be repeated for a finite number of such intervals,
which together cover [a,b], and the theorem follows.
By choosing 4J(z) = 1, we get the following z
(11.6) CoroZZary (Wiener). Let fEL 1 (-oo,00), and "let f(Cl) *0, for
x E [a,b]. Then there exists a function g E L1 (-00,00), such that
g(Cl), for Cl E [a,b].
§12. The closure of translations
Let fELl (-00,00), and let Sf denote the set of finite linear combi
nations of translations of f, that is to say
§12. The closure of translations 69
where c k is real, a k complex, and m an integer, m.2:. 1. Clearly we
have SfcL1(-00,1X». Let Sf denote the closure, in the L 1-norm, of the
set Sf' Then g ESf if and only if there exists a sequence (gn) ,gn E Sf'
such that Ilg-gn l1 1 +0 as n+oo. Obv~ously sfcSf , and since L1 is
complete, we have S f cL1 . Clearly Sf = Sf' since Sf is closed.
The set Sf is linear, in the sense that if h1 ,h 2 E Sf' then
a 1h1 + a 2h2 E Sf' for any two complex numbers a 1 ,a2 . The set Sf is
translation invariant, in the sense that if h (.) E Sf' then
h(. + c) ESf' for any real c. Finally, if hESf' and we consider the
closure (of the set of finite linear combinations) of translations of
h, namely Sh' then Sh C Sf' so that we do not obtain any functions not
already contained in Sf' For let F E Sh' Then there exists a finite n
linear combination of translations of h, say ~ akh(x + c k ), such k=1
that for any given E > 0,
n II F - ~ akh (. + c k ) 111 < E.
k=1 n
If h ESe then ~h(. + c k ) ESf' hence ~ akh(. + c k ) ESf' so that n k=1 ~ akh(. + c k ) can be approximated in the L 1-norm by translations of
k=1 f, and hence also F, so that F ESf .
Theorem 2.2 (Wiener~ A
If f E L1 (_00,00) > and Sf
f (x) * 0, _00 < x < 00.
Proof. Suppose ~e theorem false. Then there exists a real number a,
say, such that f(a) = O. By (11.1) there exists a function
<5 E L1 (_00,00), such that ;\ (a) = 1. We will show that if /) E Sf L 1 ,
then ;\(a) = 0, which is a contradiction.
A
For if f E L1 (_00,00), and f(a) = 0, then g(a) = 0 for every g E Sf' To
prove this, take a~y hE Sf' Then h(x) = ~ akf(x + ck ), and A A -lCkX A A
h(x) = ~ akf(x) e . If f(a) = 0, then h(a) = O. If now g E Sf'
then there exists a sequence (gn),gnESf' such that Ilg-gn I1 1 +0, as
n + 00, Taking h = gn' we see that gn (a) = 0 for every n. Therefore
g(a) = lim g (a) = O. (Note that if Ilg -g111 +0, then, by (1.12), n n n+oo
70 I. FOURIER TRANSFORMS ON L/-OO,"')
gn (a) -+- g(a), uniformZy in a).
If Sf L1 , then c5 E Sf' and by what we have just proved, g (a) 0,
whereas 6(a) = 1, which is a contradiction.
Theorem 23 (Wiener}. If fEL 1 (-eo,eo), and if f(a) *0, -eo<a<eo, then
Sf = L1 (-eo,eo) •
In the arrangement of the proof, we follow Bochner, and prove three
lemmas first.
(12.1)
Proof. We may assume that neither 9 nor h is almost everywhere zero.
(Note that 9 E L1 (-eo,eo), since Sf c: L1 ) •
Let H = g*h, so that
eo H(x) J g(x-t)h(t)dt,
where the integral exists for almost all x, and belongs to L1(-~'eo).
(see Lemma (2.1».
Given E > 0, choose N so large that
(12.2)
Let
so that
and
J I h ( x) I dx < ---CE"---_
Ixl~N 211g111
N J g(x-t)h(t)dt, -N
H(x) - HN(X)
eo < J
J g(x-t)h(t)dt, Itl>N
dx J Ig(x-t) I Ih(t) Idt Itl~N
§12. The eZosure of transZations
( 12.3)
by (12.2).
00
f I h ( t) I d t fig ( x- t) I dx It I2:.N -00
f Ih(t) Idt Ilglll Itl2:.N
£ <-2
Given E: > 0, there exists 0 > 0, such that
(12.4) f Ig(x-y) - g(x) Idx < £ , for Iyl ~o - 211hl11 -00
(see (1.17». Choose a finite sequence t 1 , •.. ,tn such that
and such that tk
disposal. Then
If we now define
t k - 1 ~ 0, for 1 ~k~n, where n remains at our
n tk HN(X) L f g(x-t)h(t)dt.
k=l t k _ 1
n tk hN(x) = L g(x-tk ) f h(t)dt,
k=l t k _ 1
then clearly hN E Sg' and we have
n tk L f [g(x-t) - g(x-tk ) ]h(t)dt ,
k=l t k - 1
hence
n tk L f Ih(t) Idt f Ig(x-t)-g(x-tk ) Idx.
k=l t k _ 1 -00
00
fig (x-t) - g (x-t ) I < ---,E'=--_ k - 211hlll
so that
71
72 I. FOURIER TRANSFORMS ON L/-<»,oo)
(since h $ 0), and
by (12.3) and (12.5). Since hN E S , it follows that HE S • By hypo-g g thesis g E Sf' hence Sg ~Sf (as noted before the enunciation of the
theorem), therefore H E Sf'
(12.6 ) Lemma. Let f E L1 (-co,co), and Zet
_ (sin(Rt/ 2»)2 HR(t) - R (Rt/2) , R> 0, -co < t < co.
Proof. Take any function hE L1 (-co,co). If HR E Sf for R
by the preceding Lemma (12.1), we have HR*h E Sf for R
Theorem 16, (9.3), we know that
1,2, ••• , then
1 ,2, •... By
Hence hE Sf Sf' for every hE L1 (-co,co), which implies that
L1 (-0>,0» cSf • But SfCL1(-co,co), and the lemma follows.
(12.7) Lemma. If fEL 1 (-co,co), and f(x) *0, -co<x<co, then
Proof. Let co>R>O, and f(x) *0 for -R~x~R. By corollary (11.6) of
Theorem 21, there exists a function g E L1 (-co,co), such that
f(x) as the Remarks following Theorem 15, §9), then KR (x) = ° for I x I ~ R,
and
hence
§13. A general- tauberian theorem 73
A
But 2n f KR g is the Fourier transform of f*HR*g (Theorem 2, §2). Hence by the uniqueness theorem (Theorem 7, §6), we have
where HR*g E L1 (-co,co), and f E Sf. By Lemma 12.1, it follows that
HR E Sf"
Proof of Theorem 23. If f(x) '" 0, -co < X < co, then by the immediately preceding Lemma (12.7), HRESf , for R = 1,2, ••• ,n, •• , hence, by
Lemma (12.6), Sf = L1 (-co,co).
§13. A general tauberian theorem
A tauberian theorem is the corrected converse of an abelian theorem.
The word "abelian", in this context, originates from Abel's theorem
on power series, which states that if (an) is an infinite sequence of 00
real numbers, and L a converges, and has the sum s, then the power 00 n=O n
series L anxn converges uniformly for 0 ~ x ~ 1, and n=O
co lim L anxn s. xt1 n=O
The direct converse of this theorem is false. If we
n = 1,2, ••• , then co
but L an is not n=O
co L a xn = 11 ' 0 < x < 1 and lim
n=O n x - xt1 convergent. Tauber proved (1897)
take an
; (_1)nxn n=O that the converse
is correct under the additional condition, known ever since as a
"tauberian condition", that n an->-o, n->-co. Tauber's theorem was later
sharpened by J.E. Littlewood, who showed (1910) that the condition
n an = 0 (1), as n ->- co, was sufficient to ·prove the converse of Abel's
theorem, and thereby provided the impetus for the remarkable work of
Hardy and Littlewood on a variety of special problems. Adopting a
totally different point of view, Wiener showed (1930-32) that "most"
tauberian theorems, like the converse of Abel's theorem, follow as
special cases of a "general tauberian theorem", which properly belongs
to the theory of Fourier transforms on L1 (-co,co), and of which the
following is the simplest version.
74 I. FOURIER TRANSFORMS ON L/-oo,oo}
Theorem 24 (Wiener). Let hex) be a bounded (measurable) function
defined for -co<x<co, and let K1 (X) EL 1 (-co<x<co), with K1 (a) *0 for
-co < 0. < co. If
co (13.1) lim f K1(x-~)h(~)d~ = A f K1(~)d~, for some complex A,
X-rCX) -co
then we have, for every K E L1 (-co,co),
co (13.2) lim f K(x-~)h(~)d~ = A f K(~)d~.
X-+co -00
The following is a kind of converse, which is easier to prove.
Theorem 25 (Wiener). Let K1 E L1 (-co,co), and let its Fourier transform
K1 have a real zero. Then there exist a bounded (measurable) function
h on (-co,oo), and a function KEL 1 (-co,co), such that (13.1) is true,
but (13.2) is false.
A
~roof. If K1 (c) = ° for a certain real c, choose K E L 1 (-oo,00) , such
- (1/2)x2 that K(C) *0, for example K(x) = e (Example 6, §1), and let
h(~) = e-ic~. Then we have
00 -ic~ f K1 (x-~) e d~
for every real x. But
f K (x-O e -ic~d~ -00
f -00
-icx e
-icx A
e K(c) ++ a limit, as x+oo.
n
0,
Proof of Theorem 24. If K(x) is of the form K(x) = L Ak K1(x+Ak)' k=l
where Ak is complex, and Ak real, then the theorem is trivially true.
By Theorem 23 on the L 1-closure of translations, given any
K E L1 (-00,00), there exists a function K3 of the form
such that
(13.3)
n K3 (X) = L Ak K1 (X+Ak ),
k=l
f I K (x) - K 3 (x) I dx < <:,
where <: is any strictly positive number given in advance. !f
Ih(x) 1< B < 00, then we have
§13. A general tauberian theorem 75
( 1 3 .4) IJOO K(x-~)h(~)d~ - Joo K3(X-~)h(~)d~1 < B'E. -~ -~
Since (13.2) holds with K = K3 , we have
( 1 3 .5) I( K3(x-~)h(S>d~ - A (K3(~)d~1 < E, x'::xO> 0, -~ -~
while (13.3) gives
( 1 3 .6) IA Joo K3(~)d~ - A Joo K(~)d~1 < IAIE. -00 -00
Combining (13.4), (13.5), and (13.6), we obtain
and hence the theorem.
Theorem 26 (J.E. Littlewood). Let f(x)
an is complex, and let
L n=O
n a x n
for I x I < 1, where
lim f (x) xt1
s, (s finite).
If an = 0(1/n), then L n=O
a n
s.
Proof. Let [x] S(x) = Lan' x>O,
n=O
where [x] denotes the greatest integer not exceeding x. Then we have
[x] S(x) - f(e- 1/ x ) = L an
n=O L
m=O a e-m/ x
m
[xl L a (1_e-n / x ) + L a e-m/ x
n=1 n m=[x]+1 m
and since nlanl <M<oo, for all n.:: 1, and [x].::x, [x]+1 >x, we have
(13.7) [xl
Is(x) - f(e- 1/ x ) I < L ~,~ + 0(1) = 0(1). - n=1 n x
By hypothesis, f(e- 1/ x ) is bounded as x -+ co, hence (13.7) implies that
(13.8) S (x) 0(1), as x-roo.
76 I. FOURIER TRANSFORMS ON L /-00,00)
00 n Consider the function f(x) = L anx, with x
n=O
( -I; 00
f e-e ) = L n=O
-ne-I; ane j e-~e-te-~S(t)dt
o
00
f ( I:- ) -e-(~-n)
e - ., -n e S (e n ) dn . -00
We note that ~ + 00 as x + 1, and if we choose K1 (S)
K1 E L1 (-00,00) ,. and
A 00
-~ e-~ e-e , then
K1 (a.) = f K1 (~) eia.~d~ = r (1-ia.) * 0, (Example 3, §1) -00
00
so that J K1(~)d~ = r(1) = 1. Our hypotheses imply that -00
00 00
(13.9) lim J K1(~-n) S(en)dn = s = s f K1 (n)dn ~ +co -00 --00
Because of (13.8) this implies, by Theorem 25, that
co 00
(13.10) lim J K(~-n) S(en)dn = s f K(n)dn , ~+oo -co
for every K E L1 (-00,00). If we choose for K the Littlewood function
given by
K(S)
we see that
-00
0, { -~ e ,
for ~.:. 0,
for 0 < ~.:. L,
0, for ~ > L,
~
f ~-L
where x = e~, so that x + 00 as ~ + 00, and
L J K(n)dn o
Hence (13.10) becomes
x
L > 0,
(13.11) lim ~ [ -L S(y)dy -L s (1-e ).
x+oo xe
-L Now for xe .:. y.:. x, we have
§14. Two differential equations
since
-L IS(x) - S(y)l..:. M x(1-e )+1
- -L xe
77
I [x] I M x(1-e-L )+1 IS(x) - S(y) I = L an < y [(x-y)+1] < M -L
n=[y+1] xe
On writing the identity
S(x)-s -L x(1-e )
x J _L{S(x)-S(y)}dy + -L xe x(1-e )
and making use of (13.11), we obtain
lim sup IS(x)-sl < lim M x(1-e-L )+1 x(1-e-L ) x .... co x .... co xe-L .x(1-e-L )
x J _LS(y)dy-s, xe
1-e-L M --=L
e
for every L>O. On letting L+O, we obtain the result: S(x) +s, as
§14. Two differential equations
To illustrate the application of Fourier transform methods in the
study of differential equations, we shall consider two simple cases:
the equation of heat conduction in an infinite rod, namely
(14.1) dU(X,t)
at
2 d u(x,t)
dX 2
under suitable conditions, and Laplace's equation
(14.2) 0,
which governs the distribution of temperature in an infinitely large
plate, under suitable conditions. The first equation is connected with
the Gauss-Weierstrass integral of f E L1 (-co,co), and the second with
the Cauchy-Poisson integral of f E L1 (-co,co), studied in §9 (see
Corollaries (7.16), (7.18), (9.5) and (9.6». These are given respec
tively by
78 I. FOURIER TRANSFORMS ON L/-oo,oo)
co
(14.3) U(f;x,t)::: U(x,t) J f(i;lW(x-i;,t)di;, fEL1 (-00,00), t>O, -00
W(x,t) U(1Tt)
-x2/4t e , -0:) < X < CD,
and
00 (14.4) V(f;x,t)::: V(x,t) = J f(i;)P(x-i;,t)di;, f E L1 (-00,00), t > 0,
1 P(x,t) = TI
-00
t
t 2+x2
We note that W(x,t) satisfies (14.1), and is the so-called fundamental
solution. Similarly P(x,t) satisfies (14.2).
We shall identify those properties of U(x,t) which will show that it
is the unique solution (almost everywhere) of equation (14.1). We
assume throughout that f E L1 (-00,00) •
U1 • For each t > 0, U(x,t) E L1 (-00,00) as a function of x, and
U2 • We have lim t+o+
as was proved in (9.5).
0,
au a2 u U3 • For t > 0, and -00 < x < 00, the partial derivatives - -- exist ax' ax2
and belong to L1 (-00,00) as functions of x.
This can be seen by differentiating under the integral sign and
noting that the resulting integral is the convolution of two
functions in L 1 (-00,00).
au U4 • For t> 0, at E L1 (-co,oo) as a function of x, and
lim II U(-,t+h)h- U(-,t) - aU(-,t) II 0_ h+o at 1
Since
U(x,t) J f(i;)W(x-i;,t)di;, f E L1 (-00,00), -00
§14. Two differential- equations
W(x,t) 2
, e-x /4t ( t) 21(nt) = W -x, , t > 0,
we have for h * ° (14.5) /"IU(X,t+h)h- U(x,t) _ aU~xt't) IdX
-00
< /" dx /''If(X-~) IlwU;,t+h)h- W(~,t) - aW(aSt't) Id~ -00 -00
< Ilfll, /"IWU;,t+h)h- W(S,t) - aW(aSt't) Id~. -00
But
I , W(~,t+h) - W(~,t) _ aW(~,t) ~m h - at
h+o
for every ~. This limit relation holds also in the L1-·norm, for
aW(a~t't) = 2jn {-~ t-3/2e-~2/4t + ~42 t-5/2e-~2/4t} ,
t>o, so that
for fixed t> 0, in the interval [t-h,t+h], with h sufficiently
small, and secondly
where
hence
W(~,t+h) - W(~,t)
h
W1 (~,y) aw(~,y)
ay
1 h = h f W,(~,t+y)dy,
°
By Lebesgue's theorem on dominated convergence, we deduce that
79
80 I. FOURIER TRANSFORMS ON L1 (-OO,oo)
lim Joo iW(i;,t+h)h- W(i;,t) _ aW(i;,t) Idi; 0, h+O -00 at
and (14.5) now yields U4 •
Us' The equation
ClU(x,t) at
holds for t > 0, -00 < x < 00, as can be verified directly by calcu
lation.
Theorem 27. Given f E L1 (-00,00), let U(x,t) be any function with proper
ties U1 to US' Then
U(x,t) J f(i;jvl(x-i;,t)di;, t> 0,
1 _i;2/ 4t where W(i;,t) = 21 (-rrt) e , for almost all x E (-00,00).
Proof. Let U(a,t) be the Fourier transform of U(x,t) considered as
a function of x. It exists because of property U1 • By property U2 ,
together with (1.12), we have
( 1 4 .6) D(a,t) + f(a), as t+O+, for all a, -oo<a<oo.
Because of property U3 , together with Theorem 3, we have
( 14 .7) 2 A
[~] (a) ax
On the other hand,
(-ia)2 U(a,t), t>O, -=<a<=.
A
U(a,t+h) - U(a,t) h
hence, by property U4 '
( 1 4 .8)
But
aat [D (a , t) ]
2 a U(x, t)
dX 2 aU(x,t)
at
§14. Two differentiaL equations
because of property US' hence (14.7) and (14.8) yield
Thus
'}t [U(ex,t) ] _ex 2 U(ex,t), for every real ex,
hence
U(ex,t) A(ex) for every real ex.
Since U(ex,t) +f(ex), as t+O+, by (14.6), we obtain
(14.9) U(ex,t) f(ex) t l> 0, -co < ex < co.
The Fourier transform of the convolution
co
(hW) (x) = f f(I;)W(x-l;,t)dl;, t> 0, -00
2 however, equals f(ex)e-ex t (Theorem 2). Because of (14.9), and the
uniqueness theorem (§6, Th. 7), U(ex,t) = (f*W) (ex), for t> 0, and for
almost all ex E (-co,oo) •
81
The Cauchy-Poisson integral V(f;x,t) of fEL 1 (-00,00) , given by (14.4)
is associated in a similar fashion to the Laplace equation in (14.2).
We list the characteristic properties of that integral:
V 1. For each t> 0, V(x,t) E L1 (-co,oo) as a function of x, and
V2 • We have
lim t+o+
IIV(·,t) - f(o) 111
as was proved in (9.6).
0,
av a2v V3 0 For t> 0, and -co < x < "", the partial derivatives - -- exist ax' ax2
and belong to L1 (-co,co) as functions of x.
82 I. FOURIER TRANSFORMS ON L/-oo,oo)
2 V For t> 0, the partial derivatives aV(x,t) a V(x, t) exist and
4· at ' at2
belong to L 1 (-00,00) as functions of x, and
lim II V(·,t+h) - V(·,t) - aV(·,t) II 0, h-+O h at 1
and
aV(·,t+h) aV(·,t) lim II ____ ~a~t __________ a~t~ ___
h-+O h O.
VS. The equation
2 a V(x,t)
ax2 o
holds for all x E: (-00,00) and t> o.
Theorem 28. Given f E: L1 (-00,00), let V(x,t) be any function with
properties V1 to VS. Then
00 V(x,t) f f(OP(x-s,t)d!;, t > 0,
t where P(s,t) = --- for almost all x E: (-00,00).
11 t2+s2 '
Proof. Because of property V1 ' the Fourier transform V(a,t) is de
fined for each t > 0, and -00 < a < 00. By property V 3' and Theorem 3 of
§1, we have
(14.10) (-ia) 2 V(a,t);
on the other hand,
[V(. ,t+h) h - V(· ,t) ] (a) V(a,t+h) - V(a,t) h
and property V4 implies that
and
aV(a,t) at
h '" 0,
§14. Truo differential- equations 83
a 2 V(a,t), by (14.10).
Hence
(14.11)
By property V2 , however, together with (1.12), we have
V(a,t) .... f(a), as t .... 0+, for all a E (-0:>,0:»,
hence
A(a) + B(a) = f(a).
By property V l' we have, for each t > 0,
which, in turn, implies that
because of (14.11). Hence
IIfll1 + IA(a)1 IB(a) I < at
- Ie I and on letting t .... 0:>, we get B (a) = 0, for a> 0; similarly also
A A la I t A(a) = 0 for a<O; so that V(a,t) = f(a)e- , for aE (_0:>,0:». But the
Fourier transform of the convolution
0:>
(hP)(x) = f f(~)P(x-~,t)d~, t>O,
equals f(a)e- Ialt (by Theorem 2, §1). Because of the uniqueness
Theorem (Th.7, §6) it follows that V(x,t) = (f*P) (x) for almost all
x E (-0:>,0:», and t > O.
§15. Several variables
The definition of Fourier transform is easily extended to functions
of several variables. If Ek denotes the real Euclidean space of
84 I. FOURIER TRANSFORMS ON L 1 ( -co, 00)
dimension k, and xEEk , we write x = (x 1 , •.. ,xk ), where -=<xr <=,
for r = 1,2, •.. ,k; we write Ixl = (x~ + •.. + X~)1/2, and if CLEEk'
we write <X,CL> = x 1a 1 + ..• + XkCLk •
For any p such that 1.:::.p < =, we denote by Lp(Ek ) the Banach space
of complex-valued, Lebesgue measurable functions f on Ek , relative to
the norm
( )1/P
II f II p = ~ I f ( x) I p dx < =,
k
modulo the subspace of functions which are zero almost everywhere.
Here dx (= dX 1 ..• dxk ) stands for the k-dimensional Lebesgue measure.
If f E L1 (Ek ), we define f, the Fourier transform of f, by the relation
(15.1) f(a) f f(x) ei<x,a>dx, with Hf] (a) = (2rr)-k/2 f (CL),
Ek
for all CL E Ek . We note that f is bounded, and continuous, and
(15.2) If(CL) I < IIfl11 <=,
as in (1.5) and (1.6).
If Th is the operator which takes f(x) into f(x+h), where x,h E Ek ,
then
( 1 5 .3) e -i<h,x>i (x) .
The Fourier transform of f(Ax 1 ,Ax2 , .•. ,Axk ), where A is a real
1 A(X1 Xk) number, and A * 0, is --k times the Fourier transform f T' .. ·'T ' I A I as in (1.10).
The Riemann-Lebesgue theorem holds: if f E L1 (Ek ), then f(x) ->- 0, as
I x I ->- =. The proof is similar to that of Theorem 1. We approximate
to f in the L 1-norm by box-functions in Ek • A box-function g is such
that g(x) = 1 in the box -=<ar.:::.xr.:::.br <=, for r = 1,2, ..• ,k, and
g(x) = 0 outside the box.
As in (1.13) the composition rule holds: if f,gEL 1 (Ek ), then
§15. SeveraL variabLes 85
(15.4) f f(x)g(x)dx = f f(y)g(y)dy. Ek Ek
As in §2, one defines the convoLution of f E L1 (Ek ) and g E L1 (Ek ) , written h f*g, by the relation
(15.5) hex) = f f(x-y)g(y)dy, Ek
and notes that hE L1 (Ek ). The operation of convolution is cOIllIllutative and associative, and
(15.6) (hg) f.g
as in Theorem 2.
As for pOintwise sUIllIl\abili ty, let K E L1 (Ek ) , and K = H. For R> 0,
let KR(X) = K(~), and HR(X) = RkH(Rx), as in (7.1). Then for every
f E L1 (Ek ) , we have, by the composition rule (15.4),
(15.7)
If we take, in particuLar,
(15.8)
then K is a radiaL function; that is to say, K (x) = K (y) if
Ixl = Iyl. If k = 1, a radial function is just an even function. By direct calculation (as a product of one-dimensional Fourier trans
forms, using Ex.6, §1), we see that
(15.9)
is also radiaL, so that HR(y-X) = HR(X-y), and (15.7) leads, for the
particular choice of K in (15.8), to the relation
(15.10)
which is analogous to
(15.11 )
f fca)e-i<a,x> 'K(~)da, R> 0, K(a) Ek
(7.1). We note further that
86 I. FOURIER TRANSFORMS ON L/-oo,oo)
where HR is defined as in (15.9), if we use Example 6, §1. Hence
(15.10) implies that
(15.12) (2rr)-k(f*HR) (x) - f(x) = (2rr)-k J {f(x+u) - f(x)}HR(u)du,
Ek
where HR (u) is a radial function, which depends only on I u I. Inte
grating first over the "surface" lui = t> 0, and then relative to the
"radius" t, for 0.:: t < 00, we obtain
( 15.3) -k (2rr) (hHR) (x) - f(x)
-k ooJ t k - 1 (J ) = (2rr) {f(x+tv) - f(x)}do v HR(t)dt, o 0
2 2 where ° denotes the "sphere": v 1 + .•. + vk = 1, and dov its (k-1)-
dimensional volume element.
We now define for x E Ek , k.:::. 2, and t > 0,
(15.14) J f(x+tv)do v ' wk - 1 °
where ° denotes the "sphere" vf + ... v~ = 1, wk - 1 its (k-1)dimen
sional volume, and dov its (k-1)-dimensional volume element.
For k 1, we define fx(t) = ~[f(X+t) + f(x-t) J. If
(15.15)
then relation (15.13) can be written as
(15.16) -k (2rr) (hHR) (x) - f(x) -k co k
wk _ 1 (2rr) £ R H(Rt)gx(t)dt,
which is the analogue of (7.2) in Theorem 8.
Following the sarne lines of proof as in Theorem 9, we can deduce
that
(15.17) lim ~ (f*H R) (x) R+oo (2rr)
at every point x at which
f (x) ,
§15. Several variables 87
(15.18)
By (15.10) it follows that if fEL 1 (Ek ), then
(15.19 ) f(a) . -IN 12/R2 e- 1 <a,x> e U da f(x)
for almost every x E Ek , since, by a Theorem of Lebesgue-Vi tali,
condition (15.18) is satisfied for almost every x, if fEL 1 (Ek ). Thus
the Fourier integral of any function in L l (Ek ) is Gauss-summable
almost everywhere (cf. (7.13)).
This ~ can again be used, as in Theorem 11, to prove that if fELl (Ek )
and fELl (Ek ), then the inversion formula holds almost everywhere, that
is to say:
(15.20) f(x)
for almos t every x E Ek .
1
(2"IT)k f f(a)e-i<a,x>da
Ek
Corresponding to Corollary (8.7), we obtain the result that if
~ ELl (Ek ), is non-negative, and continuous at the origin, then
fELl (Ek ) .
As for summability in the L l -norm, one can prove, using (15.12), the
analogue of Theorem 15, in an analogous manner: we have
(15.21)
for the particular kernel (15.9).
Instead of choosing K to be the Gauss kernel e- 1x12 as in (15.8), one
can choose K to be any radial function, such that K ELl (Ek ), K(O)= 1, ""
K is continuous at the origin, and (2"IT)-k f K(a)da = 1, and prove the -""
analogues of (15.7) and (15.10). We shall not elaborate here on
general summability.
On the other h~nd, it is worth noting that if fELl (Ek ) , and f is
radial , then f is also radial, and assumes a special form depending
on whether k is even or odd.
88 I. FOURIER TRANSFORMS ON L/-<»,oo}
For we have
f{o.) = f f{lxl)ei<o.,x>dx, o.EEk , Ek
(15.22)
and if T: x -+ y is any orthogonal transformation of
the matrix (bij ) with determinant +1, such that Yi
Ek , represented k I: biJ.xJ., j=1 a..
by
i 1, ••• ,k, we can choose the first row of the matrix b 1j = ----L- for
k 10.1 1, ••• ,k. Then we have <o.,x> = .I: o.ixi = lo.l·Y1' while
J.=1 j Ixl = IYI, since T is orthogonal. Hence
f{o.) = f ilo. IY 1
f{ IY I)e dy, o.EEk , Y = (Y1'···'Yk)' Ek
which shows that f{a) depends on 10.1 alone, hence is roadiaZ.
2 2 1/2 I Let p = (Y2 + ••• + Yk ) , so that P2:,0, and yl
integrate first
2 2 Y 2 + ••• + Yk
over the "surface" of the (k-1) dimensional sphere
p2, and then relative to the "radius" p from ° to 00.
Thus
where wk- 2 stands for the (k-2)-dimensional volume of the "sphere"
2 2 Y2 + ••• + Yk = 1. If we now use polar coordinates: Y1 = r cos a,
p = r sin a, and note that ° ~ a ~ 1T, since P':: 0, while r.:: 0, we obtain
j f{r)rk- 1 ei1o.lr cos a{sin a)k-2dr da.
° From Example 8, of §1, we have
where J v is the Bessel function of order v, hence
A c'Wk_ 2 00 k/2 f{o.) = 10.1 (k-2)!2 of f{r)r J 1 (Io.lr)dr,
2{k-2)
c =
1 v> - 2'
§15. Several variables 89
for k.:: 2, with wO:: 2.
Since the volume Vk (s) of the (solid) sphere defined by I y 12 < s
equals {nk/2sk/2/r(~k+1)}, which can be proved by induction o~ k, we
have wk- 1 = {2nk / 2/r(k/2)}, for k> 2. We thus have for any f E L1 (Ek ),
where f is radial,
(15.24) (2n)k/2 ~ k/2 lal(k-2)!2 of f(r)r J 1 (Ialr)dr.
2(k-2)
1 If k = 3, then (k-2)/2 = 2' and J 1 (x)
1 2 ( :X)1/2 sin b f " x, ecause 0
(15.23) with v = 2' hence
f(a) = ~ j rf(r)sin(lalr)dr. lal 0
If k 2, then obviously
" f(a) = 2n f rf(r)Jo(lalr)dr. o
Formula (15.24) holds good also for k = 1, since J 1(x) = (;x)1/2cos x, -2
as can be seen directly by taking v -~ in the defining series for
J v given in Example 8, §1.
" In general, a Bessel function appears in the integral for f if k is
even , and a trigonometric function if k is odd.
Chapter II. Fourier transforms on L2(-00, (0)
§1. Introduction
The Banach space L2(-m,~) is endowed with an inner produat. For any
two functions f,g belonging to L2 (-=,=), the inner product (f,g) is
defined by
= ( 1.1) (f,g) f f(x)g(x)dx,
-m
the integral existing because of Schwarz's inequality. (The bar "_"
denotes complex conjugation). It has the following properties:
(i) (f,g) = (g,f), (ii) (f,f) = Ilfll~, (iii) (f,ag) = a(f,g),
aEIC, (iv) (f,g1+g 2) = (f,g1) + (f,g2)' for f,g1,g2 EL2(-m,=),
(v) l(f,g)I.::llfI12 '1IgI1 2 , (vi) (f,g) is continuous in f, and in g.
A Banach space with such an inner product is referred to as a Hilbert
space, and the particular Hilbert space L2 (-m,m) is separabZe, in the
sense that it contains a aountabZe subset which is dense in L2 (-=,m).
If a function f belongs to L2 (-=,m), it does not necessarily follow
that fEL 1 (-=,=), as for example f(x) = 1+lxl • One has therefore to
define the Fourier transfo~m on L2 (-=,=) somewhat differently from
the way it was done in the case of L,(-=,=) in Chapter I, but make sure
that the two definitions coincide if f E L, (-=,=) .n·L2 (-ao,=).
We shall define a Fourier transform first on a dense subset of
L2 (-=,=), and then extend it uniquely to the whole of L2 (-=,m). The
extension will be defined aZmost everywhere, instead of everywhere as
in the case of L1 (-=,ao). One can start with the dense subset of bounded
functions belonging to L1 (-ao,=), or the subset L1 (-=,=)·n'L2 (-=,=), or
the subset of continuous functions which are of bounded variation over
a finite interval and vanish outside that interval, or the subset of
92 II. FOURIER TRANSFORMS ON Li-oo,ro)
step-functions, or the subset S of infinitely differentiable functions
which "decrease rapidly" (Schwartz's space, §3, Ch.I). For convenience
we shall choose the last alternative.
The Fourier transform will turn out to be a bounded, linear operator
on S, which is an isometry, and which can be uniquely extended to a
bounded linear operator on all of L2(-~'~)' whose range coincides with
L2(-~'~)' and which is "isometric", hence a unitary operator. Thus if
f E L2(-~'~)' its Fourier transform Hf] automatically belongs to
L2(-~'~)' Because of the symmetry typified by this fact, we define 1 A
Hf] = m:;r) f, for fELl (-oo,~) .n.L2(-~'~) (ef. (1.2) of Ch.I).
§2. Plancherel's theorem
Fourier transforms on S. If S denotes Schwartz's space (as in §3,
Ch. I), and f E S, we define the Fourier transform H f] of f by the
relation
(2.1) F[f] = ~ i, or
Hf] (a) 1(~1T) J co
iax f(x)e dx, -ex> < Ci. < co.
The map f + Hf] is a Zinear map of ~ onto itseZf, from what we have
already seen (in §8, Ch.I). If we define F[f] by the relation
(2.2) v Hf] (a) Hf] (-a)
~J -co
-iax f(x)e dx, -CD < a < 00 I
then (by 8.12), Ch.I, p. 50) we have
v (2.3) F[Ff] = f.
Further if f,gES, then (by (8.13), Ch.I, p. 50) we have (cf. (1.1»
(2.4) (f,g)
and
§2. PZanahere],'s theorem 93
(2.5) 1 7T2ifT F[f*g] = F[f]°F[g],
where the star "*" denotes convolution (see (8.14), Ch.I, p. 50).
Fourier transforms on L2(-~'~)
Since S is a dense subset of L2(-""'~)' if f E L2(-~'~)' there exists a
sequence (fn) of functions belonging to S, such that Ilf-fnI12+0, as
n+~. Hence Ilf -f 112+0, as m,n+~, but by (2.4), and the linearity m n of the map f +F[f ], IIF[f ] - F[f ]112+0, as m,n+~. Since the spaae m m m n L2 (-~,~) is aomp],e te, there exists a function FE L2 (-co,~), such that
IIF[fn] - FI12+0, as n+~. The function F is defined a],most every
where on (-~,co). We call F the Fourier transform of f E L2 (-~,oo), and
denote it by F[f].
We note that F does not depend on the approximating sequence (fn ).
Let (gn) be another sequence of functions belonging to S, such that
Ilgn-fI1 2 +0. Then fn-gn = fn-f + f-gn , and Ilfn-gn ll 2 ~ Ilfn-fl12 + 119n-fI1 2 +0, as n+~. But, by (2.4), IIF[fn ] - F[gnll12 = Ilfn-gn I1 2 +0,
hence IIF[gn] - FI12 ~ IIF - F[fn] 112 + IIF[fn] - F[gnlI12+0, as n+~.
We are thus led to the following
(2.6) Definition. If f E L2(-~'~)' the Fourier transform F[f] of f
is defined to be the limit, in the L2-norrn, of the sequence {F[fn ]} of
Fourier transforms, of any sequence (fn) of functions belonging to S,
such that fn converges in the L2-norm to the given function
f E L2 (-co,oo), as n +~. The function F [f] is defined almost everywhere
on (-co,co), and belongs to L2(-co,~).
The definition makes it clear that the map f + F[f] of L2(-~'~) into L2 (-00,co) is ],inear. That is to say, if f1 ,f2 E L2 (-co,co), and a,b E 11:,
then
(2.7)
We note further that for any f E L2 (-00,00), we have
since, by definition (2.6), and (2.4), we have
94 II. FOURIER TRANSFORMS ON Li-oo,oo)
(2.9) IIfl12 = lim [lfn [[2 = lim [1F[fn lI1 2 = [1F[fl[[2' n-+oo n+oo
If fl ,f2 E L 2 (-=,=), then we can apply (2.8) to the function f = fl + f 2 ,
and noting that l[f11[2 = [IF[fl11[2' Ilf21[2 = [1F[f211[2' deduce that
Replacing fl by if1 , we see that
Hence we see that (2.8) implies, in the notation of (1.1), that
(2.10)
v We can similarly define F[fl for any f E L2 (-=,=) by extending defi-
nition (2.2) from S to L2 (-=,=). Thus given any f E L2 (-=,=), there
exists a sequence (f), f ES for n = 1,2, ... , such that [[f-f -[[2-+0, n l1 n v n
as n -+ =. The sequence (t [f 1), where F [f 1 E S, converges in the L2-n n v
norm to a function in L 2 (-=,oo), which we define to be F[fl. It is
defined almost everywhere on (-=,=), and we have
v F[fl (a) = F[fl (-a)
for almost all a E (-=,=) .
The "inversion formula" holds almost everywhere, without any special
artifice, since we have
( 2 • 11 ) v F[F[fll
v f = F[ F[ f 11, for f E L2 (-=,00) .
For if fEL 2 (-=,=), and fnES, n = 1,2, ... , and [[f-fnI12-+0, as n-+=,
we have, by definition, lim IIF[fnl - F[fll[2 = 0, lim 11¥[fnl - F[fl[1 2 n-+oo n-+oo
= 0, where F[f], F[fl EL2 (-=,=), and since (2.11) holds for any f E S
(cf. (2.3», we see that
v v v l1 [I f- F [ F [ f 1111 2 < II f- f 11 2+ [[ f - F [ F [f II I [ 2+ [I F [ F[ f ll- t-[ F[ f II [1 2 -+ 0, - n n n n
v as n-+oo, hence f = F[F[fll almost everywhere, and similarly also
F[F[f]] = f, giving (2.11), which shows that the map f -+ F[fl is a
§2. Plancherel's theorem
linear map of L2 (-co,co) onto itself.
We shall now show that F[f] thus defined for f E L2 (-co,co) is related 1 A
to the integral defining 7T21iT f in the case of f E L1 (-00,00) •
Let f E L2 (-co,oo), and let f (x)
that
(2.12) F[f] (a) f(a) 7T21iT
0, for Ixl ~A > 0. Then we shall see
1 f f(x)eiaxdx, for almost all 7T21iT a E (-00,=) •
If we denote the integral on the right-hand side of (2.12) by f*(a),
we shall show that F[f] (a) f* (a) for almost all a E (-co,co) .
There exists a sequence (fn ), n = 1,2, .•. , with fn E S, such that
95
every member of the sequence vanishes outside the interval (-A,A), and
Ilf-fnI12-+0, as n-+co. That implies, by (2.7) and (2.8), that
IIF[f] - F[fn ]11 2 -+0, and, in particular,
R (2.13 ) f
-R
2 IF[f ](x) - F[f](x)I dx-+O, as n-+ co , n
for each R> 0. But
A (A 2 )1/2 I F[fn ](ai-f* (a) I < f I f (x) -f (x) I dx:: 2A f I f (x) -f (x) I dx
-A n -A n
-+ 0, as n -+ co,
hence F[f ] -+ f* uniformly over every finite interval, and, in parn
ticular,
(2.14 ) R 2 f IF[f ](x) - f*(x) I dx-+O, as n-+oo.
-R n
A comparison of (2.13) and (2.14) shows that f* = F[f] almost every
where on (-R,R), for each R> 0, and hence almost everywhere on (-co,co),
thus proving (2.12).
Finally let f E L2 (-co,co) without any further condition. We define the
function f R , for each R> 0, by the requirement
(X)' for Ixl < R,
0, for Ixl ~ R.
96 II. FOURIER TRANSFORMS ON Li-oo,oo)
Then fREL1(-co,co).n.L2(-co,00), and by what has just been proved in (2.12), we have
(2.15) 1 R .
7f2iTT f f (x) e1axdx, -00 < a < 00. -R
On the other hand, since f+ F[f] is a linear map of L2 (-00,00) onto
itself, we have, by (2.8),
(2.16)
Hence for each f E L2 (-00,00) , the integral F[fR] given by (2.15) con
verges in the L2-norm to F[f] EL2 (-00,00) , as R+oo. This implies, by
weyl's formulation of the Riesz-Fischer theorem, that there exists a
sequence (1\), with Rk > a for k = 1,2, ••• , such that
as 1\ + co, for almost every 0: E (-00,00). In particular, if the integral
(2.17 ) 00
1 f f(x)eixadx 7(27TT -00
exists as a Cauchy principal value for almost all a E (-00,00), it
equals F[f]. And if fEL 1 (-00,00).n.L2 (-00,00), then
(2.18) 00
F[f] (a) = l(i1f) f f(x)eio:xdx, -00 < 0: < 00. -00
We subsume the results of (2.6) - (2.11), (2.17) and (2.18) under the
following
TheoX'em 1 (P~anaheX'el). If f E L2 (-oo,co), then theX'e exists a funation
F[f] E L2 (-00,co), designated the FouX'ieX' tX'ansfoX'm of f, suah that, foX'
any X'ea~ a,
(2.19)
and
(2.20)
with
1 R . ~2) f f(x)e1axdx+F[f](a), in the L2-noX'm, as R+oo, "1.G1f1 -R
1 R -iax 7(2iiT £R F[f] (a)- e da + f(x), in the L2-noX'm,. as R+ co,
§2. Planeherel's theorem 97
(2.21)
Every funetion f E L2 (-00,00) is the Fourier transform of a unique ele
ment of.L2(-00,00).
As in the L1-case (see Ch.I, (1.8), (1.9», we have for f E L2 (-=,=),
and any real number a, the relations
(2.22)
while
(2.23)
-iya Hf('+a)](y) =e Hf](y)
v F[f('+a)](y)
Hf(')](Y) F [f] (-y) ;
v _ F[f(')](y) Hf] (y) •
To indicate the reasoning involved, let us consider the first relation
in (2.22). It follows from the fact that
where'
R . . R+a . J f(x+a)e1.Yxdx -R
R+a . ( ) J f(x)e1.Y x-a dx e-1.yaJ f(x)e1.Yxdx , -R+a
-R . J f (x) eJ.yxdx + 0, -R+a
R+a . and J f(x)e1. Yxdx + O,
R
-R+a
in the L2-norm, as R+oo, because of (2.9).
For f,gEL 2 (-00,00), we have already shown in (2.10) that (2.8) implies
that
(2.24) J F[f](x) F[g](x)dx.
On using (2.22) and (2.23) in (2.24), we get the following relations:
00 (2.25) J f(x)g(-x)dx J Hf] (y) Hg] (y)dy
-00
(2.26) J f(x)g(a-x)dx f -iax Hf] (x) F[g] (x)e dx
and
(2.27 ) J f(t)g(t)eixtdt J F [ f ] (t) F [ g] (x- t) d t ,
98 II. FOURIER TRANSFORMS ON Li-«>.oo)
all the integrals being absolutely convergent.
We may look upon (2.26) as the L2-analogue of formula (2.2) of
Chapter I.
If we choose
g(x) ga,b(X)
1 b i F[g](y) = 7T2nT J e xYdx
a
and (2.27), with x 0, gives
(2.28) b 1 00 J f(t)dt = 7T2nT J a -00
On the other hand, we may take
F[g] (y)
Then
{1' for a < x < b,
0, for x ~ a, x.:: b ,
1 7T21TT
eibY_eiay iy E L2 (-00 < y < 00) ,
e-ibY_e-iay F[f](y) . dy,
-l.Y a < b.
for a < y < b,
0, for y ~ a, y.::b.
g(x) -ixy F[ga,b 1 (y) e dy
1 b -ixy J e dv_ 7T2nT a
1 7T2nT
-ixb -ixa e -e -ix
and (2.25) gives
(2.29) b J F[f](x)dx a
E L2 (-00 < x < 00) ,
00
1(~1f) J f(x) -00
ibx iax e -e ix
dx, a<b.
Remarks. The operator F has been defined on L1 (-oo,oo) in Chapter I,
(1.2), and on L2 (-oo,oo) in Plancherel's theorem. The definition can
be extended to the space L1 (-oo,oo) + L2 (-oo,00) consisting of all func
tions f of the form f = f1 + f2' where fl E L1 (-00,00), f2 E L2 (-00,00) •
§ 2. PZanchere Z ' s theorem
For such an f we define F[f] = F[f 1 ] + F[f2 ]. This definition does
not depend on the particular decomposition f = f1 + f 2 . For if
99
f = gl + g2' gl ELl (-00,00), g2 E L 2 (-co,oo) is a different decomposition,
then f 1-g1 = g2-f2EL1(-oo,oo)onoL2(-oo,oo) on which, by (2.18), the L1-
definition and the L2-definition of F coincide, so that
F[f 1-g1 ] = F[f 1 ] - F{gl] = F[g2] - F[f2 ], and we have
F[f 1 ] + F[f2 ] = F[gl] + F[g2]' Thus the operator F is defined on the
space L1 (-co,co) + L 2 (-oo,oo) , which, in fact, contains all the spaces
L (-00,00), for 1 < P < 2. For if gEL (-00,00), 1 < p < 2, then g = gl + g2' p - - p where g1 ELl (-co,co), and g2 E L2 (-00,co). We have only to define
= {g (x), if I g (x) 1 ~ 1, gl (x)
0, if 1 g (x) 1 < 1 , and
= {g(X), if Ig(x)1 <1, g2(x)
0, if Ig(x)l~l,
so that
fig 1 (x) 1 dx f Ig(x) Idx [xilg(x) 1~1]
while g2 is bounded·, and belongs to Lp(-co,oo), since
f Ig(x)IPdx<co, [xilg(x) 1<1]
and hence
< f Ig(x) IPdx < co. [xll g (x)l<l]
The operator F so defined on L (-oo,co), 1 < p < 2, may be called the p
Fourier transform as well.
100 II. FOURIER TRANSFORMS ON Li-oo,co)
§3. Convergence and surnmability
The problem of convergence of the "Fourier integral"
(3.1) _ 1 R -iC(x
SR(f;x) = 7T2iTT J F[f](a) e da, R> 0, f E L2 (-eo,eo), -R
-co < x < co,
as R+eo, can be handled in the same way as .in the L1-case, (Ch.I,
§§4, 7), once the basic formula (cf. (4.2), Ch.I)
(3.2) 3. coJ (tl ~dt 7f gx t ' o
gx(t) = ~[f(x+t) + f(x-t) - 2f(x)]
is established, which can be done by using Plancherel's theorem.
For let
F[4>] (y)
Then we have
-iyx r ' 0,
Iyl < R,
lyl2. R , R > 0, -eo < X < co.
( ) 1 J F[u,] (y). e-iyudy 4> u = 7T2iTT 't'
1 JR -iy(x+u)d 7(2iiT -R e y
-00
and formula (2.25) gives
R J Hf] (al. e-iaxda -R
(~) 1/2 sinR(x+u) (x+u)
J f (u) 4>( -u) du J (2)1/2 sinR(x-u) du feu) TI (x-u)
-co
( _rr2)1/2 Jeo sinRt f (x-t) --t- dt -eo
( 2)1/2 eo . Rt TI b [f(x+t) + f(x-t)] s~~ dt,
which leads to (3.2). The following analogue of Theorem 5 of Chapter I
§3. Convergence and sUTl1l'l'rlbiUty
is a consequence, the proof being similar.
Theorem 2. If f E L2 (-co,co). and f is of bounded variation in a
neighbourhood of the point x E (-co,co). then
101
1 JR -iax 1 lim SR(f;x) = lim ~ F[f](a).e da = 2[f(x+0) - f(x-O)]. R+co R+co-R
The problem of (C,1) summability, for example, can again be handled
without any new difficulty once the pasic formula (cf. (7.2), Ch.I)
= -1T
co sin 2 (Rt/ 2) Jg-(t) dt o x R(t/2) 2
is established. Let
F[q>] (y) = [(1 - l]l)e-iXY ,
0, Iy I > R.
lyl2. R ,
Then we have
q>(u) 1 JR (1 _ hl)e-i(x+U)YdY = 4 sin2 [R(x+u)/2] = ~ -R R ~ R(x+u) 2
(cf. Example 2, §1, Ch.I), so that formula (2.25) again gives
1 R ~ J F[f](a) ... (21T) -R
(1 lal) -iax 2 J"" sin2 [R(x-u)/2] - ~ e da = 7T feu) du -co R(x-u) 2
co = ~ J
1T
sin2 (Rt/2) f(x-t) dt -co Rt2
coJ [f(x+t) + f(x-t)] sin2 (Rt/2) dt, = 21T 0 R(t/2) 2
which leads to (3.3). Following the same lines of proof as in Theorem
10 of Chapter I, we obtain, for example,
Theorem 3. IffEL2 (-co,co). then
lim /(~1T) JR F[f] (a) (1 - I~I )e-iaxda = f(x), R+co -R
102 II. FOURIER TRANSFORMS ON £2(-00,00)
for almost all x E (-00,00).
We note that formula (2.24) plays the same role in the L2-theory as the
composition rule (cf. (1.13), Ch.I) did in the L1-case. The problem of
inversion in L2 (-00,00) does not arise in the same form as in Ll (-00,00),
and the theorems on point-wise surnrnability of the Fourier integral,
while true, do not carry the same import.
To consider the question of surnrnability in the L 2-norm, we need an ex
tension of Theorem 2 of Chapter I, on the convolution of two functions
in Ll (-00,00) •
(3.4) If fELl (-00,00), and gEL (-00,00), where 1 < P < 2, then for p - -
almost every x E (-00,00), the function f(x-y) g(y) belongs to
Ll (-co < y < (0), and if we define f*g by the relation
(f*g) (x) = J f(x-y)g(y)dy,
then
(3.5)
so that f*g E L (-co,oo). p
This is proved in the same way as Lemma 2.1 and Theorem 2, of Chapter
I, except that one has to use Holder's inequality instead of Schwarz's.
In particular, if p = 2, and h == f*g, then hE L2 (-00,00) , so that F[h]
is defined, as well as F[f] (cf. (1.2), Ch.I), and F[g]. We shall now
prove
Theorem 4. If fELl (-oo,ob), and g E L2 (-00,00), and h
hEL2 (-00,co), and
(3.6) F[h] (x) ,Ie 2n) F[f] (x) F[g] (x) ,
for almost all x E (-00,00).
Proof. Obviously F[f] is bounded; in fact, ,I(2n) IF[f](a)I .::. Ilflll =
M < 00, and if (gn) is a sequence of functions belonging to Schwartz's
space S, (cf. §3, Ch.I), such that Ilg-gn I1 2 +o, as n+co, and if we set
§3. Convergence and summability 103
h n = f*gn' then h n E L1 (-00,00) on oL 2 (-oo,oo), and F[hnl = /(2'Tf) F[fl F[gnl
by Theorem 2 of Chapter I. Further, because of (3.5), Ilh-hnl12 =
Ilf*(g-gn) 11 2 + 0 , as n+oo. Since IIF[hl - F[hn J112 = Ilh-hnI12+o, as
n + 00, and
as n+oo, we obtain (3.6).
The question of summability in the L2-norm is answered by the following
Theorem 5. Let f E L2 (-00,00) , K E L2 (-00,00) , and F[Kl :: HEL 1 (-=,oo)on>L2 (-=,oo), H even. For R>O, let HR(y)
Then we have
( 3 .7)
Proof. By formula (2.25), we have
J F[f] (y) K(~)e-iYXdY = J f(x+y) HR(y)dy -00
RH(Ry) .
By (3.5) we see that f*H R E L 2 (-oo,oo), for each R> O. By using the
properties of the L2-modulus of continuity (see (1.17), Ch. I), we
deduce as before (cf. Th.16, Ch.I) that
Remark. Clearly we can take for K(a) the Abel kernel e- 1al , and the -a 2
Gauss kernel e , as we did in Chapter I.
§4. The closure of translations
Let f E L2 (-oo,oo), and let S denote the closure, in the L 2-norm, of f m
the set Sf of all "translations" of the form L ck f (.+tk ), where tk k=l
104 II. FOURIER TRANSFORMS ON L2 (-ro.oo)
is real, and c k complex (cf. §12, Ch.I). Then we have the following
Theorem 6 (Wiener). If f E L2 (-00,00). with F [f] (a) '*' 0 for almost all
aE (-00,00). then Sf = L2 (-oo,oo).
This can be obtained as a special case of another theorem which
provides a sufficient condition for a function in L2 (-oo,oo) to belong
to Sf.
LetlR1 denote the set of all real numbers, and let ~ = F[f] where
f E L2 (-00,00). Let E~ denote a set in JR1 with the property that, except
for null sets (i.e. measurable sets of Lebesgue measure zero),
~(a) '*' 0 for a E E~, and ~(a) = 0 for a EJR1 - E~. The set E~ is, of
course, measurable.
If F(x)
then
m L c k f(x+tk ), F is not identically zero, and ~
k=l
{ -iatl -iatml
~(a) = c 1e + .•. + cme J~(CL),
and clearly E~ E~, except for null sets, and if g E Sf' with
F[g] = \jJ, then
F[F],
which is defined to mean that E\jJ = E 1 ·U·E2 , where El CE~ and E2 is a
null set. But the converse is also true, as shown by the following
Theorem 7. If f,gEL 2 (-oo,oo). and F[f] ~, F[g] \jJ. and if
(4.2)
then g E Sf"
Proof. Clearly Sf is a closed, linear subspace of L2 (-oo,oo), which is a
separable Hilbert space. Given any gEL 2 (-oo,oo), there exists an ele-o -ment g ES f , such that
(4.3) g
with
(4.4) 0,
§4. The closure of translations
for every hE Sf' If !pO F[fO], we have again
(4.5)
because of (4.1) and (4.2). If we now choose hex) = f(x+t), for a
fixed, real t, in (4.4), then we have, by (2.24) and (2.22),
00
f -00
where !p0.!p E L1 (-00,00). Hence, by Theorem 7 of Chapter I,
(4.6)
105
for almost all aE (-00,00). But we have (except for sets of measure zero)
!p°(a) = ° for a ElR1 - E , by (4.5), whi'le !pea) '* ° for a E E • Hence O!P ° !p !p (a) = ° for almost every a. By Plancherel'~ theorem, f (x) = 0, for
almost every x, which gives, because of (4.3), g(x) = gO(x) for
almost every x, and hence g E Sf'
Proof of Theorem 6. If E!p = lR1 , then automatically we have Elj! < E!p I
for every g E L2 (-00,00). Theorem 7 then implies that g E Sf' for every g E L2 (-CD,00).
§5. Heisenberg's inequality
We shall now prove an L 2-analogue of what is referred to as Heisen
berg's inequality, originally proved by Weyl under somewhat stronger
assumptions.
Theorem 8. Le t f E L2 (-00 ,(0). Then We have
(5.1 )
and the equality takes place only in ease f(x) 2
c e -kx , k > 0, cEil:.
Proof. We may assume that
106 II. FOURIER TRANSFORMS ON Li-OO,oo)
f 2 2
x I f (x) I dx < co, f 00 2 2
a I F [ f] (a) I da < co,
for otherwise (5.1) is trivially true since neither term can be zero.
Let f*(a) denote the inverse transform of (-ia F[f] (a», that is to
say
(5.2) * v f (a) = F[-ia F[f] (a)].
* By Plancherel's theorem, f E L 2 (-co,co). The left-hand side of (5.1)
equals
2 2 a I F[ f] (a) I da
-00
f I (- i a F[ f] (CJ.) ) I 2 da
by (5.2) and Plancherel's theorem. By schwarz's inequality, we have
(5.3) foo x 2 If(x) 1 2dx foo If*(x) 1 2dx ~ [fOO X(f*1; ~f) dX]2, -00 _00
since Re [x f*(x)1(x)] = ~ x(f*1 + 1*f).
To prove the theorem it suffices therefore to show that
(5.4) [( x (f*1 + 1* f) dx t = II f II i . -co
Let fn E S - Schwartz's space - be so chosen that
(5.5) lim f o. n-+oo _00
This is indeed possible, since S is a dense subset of L2 (_00,00). Since
alF[f](a) I EL 2 (-00,00) by assumption, and F[f] EL2 (-oo,oo), we have
(1+a 2 ) 1/2 r [f] (a) E L2 (-00,00). There exists a sequence (gn)' gn E S, such
that
f 2 1/2 2 Ig (a) - (1+a) F[fl(a) I da-+o, as n-+ oo , n
§5. Heisenberg's inequaZity
or co 2 \gn(a) \2 f ( 1 +a ) 2 1/2 - F [ f] (a) da -+ 0, as n -+ oo.
(1+a )
Choose fn such that
noting that F[fn] E S implies that fn E S (c£. Ch.I, (3.7)).
Now by Plancherel's theorem, and (5.5),
where f~ denotes the derivative of f n , so that F[f~](a) and f* is defined as in (5.2). It follows that
(5.6)
Now
00
00 (1+a 2) 1 /2 f IF[ f ] (a) - F[ f] (a) I da
(1+a2) 1/2 n
as n-+- oo ,
[CO 1/2 [ = ]1/2
< f d a ] f ( 1 +a 2 ) I F[ f n ] (a) - F[ f] (a) I 2 da -co (1 +a 2 ) -co
= Bn' say,
where Bn -+- 0, as n -+- co. Hence
(5.7) 1 ,2 , •••
But fn - f E L2 (-co,oo), so that
107
108 II. FOURIER TRANSFORMS ON L2 (-oo,oo)
lim R->-co (L 2-norm)
1 f -iax 7T2ifT [ FE f n] (a) - FE f] (a) ] e da , -co
almost everywhere, by (5.7). Hence we have, almost everywhere,
(5.7) ,
Therefore there exists c> 0, independent of x, such that
(5.8) Ifn(x)-f(x) 1< c < co, almost everywhere.
Since f*EL 2 (-co,=) by definition, we have f*EL 1 (-R,R) for O<R<co.
And
(5.9)
R < f
-R
R < f
-R
(f'I - f*I) (X)dXi n n
R If' 1.1 1 -Ildx + f
n n -R
Since fEL 2 (-co,=), and Ilf~-f*112->-0, as n->-co, by (5.6), the second
term on the right-hand side of (5.9) tends to zero as n->-co. The first
term on the right-hand side is
(5.10) R
B f If'-f*+f*ldx n -R n
R < B f
n -R
R If'-f*ldx + B f
n n -R If*1 dx
->- 0, as n ->- =, (for fixed R> 0) ,
since Ilf~-f*112->-0. From (5.9) and (5.10) we have
(5.11) R
lim f n->-co -R
and similarly also
f'I dx n n
R f f*I dx, -R
§5. Heisenberg's inequaZity
R R f x f' (x)i (x)dx + f x f*(x)i(x)dx, as n + co, -R n n -R
and
R R f x f~(x)fn(x)dx + f x f*(x)f(x)dx, as n + co. -R -R
Hence
R lim lim f
co (5.12)
R+co n+co -R x(f'i + flf )dx
n n n n f x(f*i + f*f)dx,
where
(5.12) , I~: x(f*i + f*f)dXI < co,
by (5.3) together with the hypotheses: x I f (x) I EL2 (-co < X < co) ,
a.1 F[ f] (a.) I E L 2 ( -co < a. < co) •
Now
R
f -R
x(f' (x)i (x) + n n
fI(x)f (x»dx n n
the dash denoting the derivative, and
R lim lim f x(lfn(X) 12) 'dx R+co n+co -R
(5.13 )
R 2 ' f x( Ifn(x) I ) dx -R
109
since fn + f almost everywhere, as n + co, by (5.7)', and II fn-f 112 + 0,
as n+co [assuming, as we may, that R+co through values which are not
in the exceptional null set in (5.7) ']. The left-hand side of (5.13)
is finite by (5.12) and (5.12)'; so is IIf1l2; hence
is finite, and :> O. If the limit is <5 > 0, then there exists RO such
that we have
110 II. FOURIER TRANSFORMS ON Li-=,oo)
which contradicts the assumption: f E L2 (-00,00). Hence
lim R{lf(R) 12 + If(-R) 12} = 0, R-~oo
and this, taken together with (5.13) and (5.12), yields
[fOO x(f*(x)f(x) + f*(X)f(X))dxt = Ilfll~, -00
so that (5.4) is proved, hence the inequality in the theorem.
In order to determine when the inequality becomes an equality, we note
(by the first application of Schwarz's inequality just before (5.3))
that
if (and only if) f*(x) = K x f(x) almost everywhere, K being a complex
constant. Here f* is defined as in (5.2). In fact, f* is the deri
vative almost everywhere of f. For by (5.6), we have
which implies that
x lim f f' (y)dy n+oo ° n
x f f*(y)dy
° over any finite interval [O,x]. But the left-hand side equals
lim [fn(x) - fn(O)] = f(x) - f(O), almost everywhere, by (5.7)'. n+oo Hence f equals, almost everywhere, an absolutely continuous function,
and
(5.14) f* (x) d~ [f(x)], for almost all x E (-00,00).
With this identification, (5.1) becomes an equality if
d dx (f (x) ) K x f (x) , (K a complex constant)
§5. Hei8enberg'8 inequality 111
1 d 1 d (1 d ) 1 d - 2 or x dx(f(x» = K f(x), or x dx x dxf(x) = x dx(K f) = IKI f(x),
( 1 d)2 2 2 2 that is to say, x dx f(x) - IKI f(x) = 0, or (D - IKI )f = 0,
where D = (~~x), which implies that (D - IKI) (D + IKI)f = o.
1 df f' If (D + IKI)f = 0, then x dx = - IKlf, or jf = - IKlx, or x 2
log f = - IKI :r + c, or
(5.15)
+IKlx2/2 If (D - IKI)f = 0, then f(x) = c 2 e ¢ L 2 (-co,co).
2 The function f(x) = e- IKlx /2 actually satisfies the equality in
(5.1), for if f*(x) = K xf(x), then we have
co 2 2 co 2 2 ( co )2 J x If(x) I dx J a. I F[f] (a.) I da. = J Ixf*(x)f(x) Idx -~ -~ -~
( CO 2 2)2 2 (CO 2 2)2 = £co IKlx If(x) I dx = IKI £co x If(x) I dx ,
-IKlx2/2 and if f(x) = e , the last expression is
(CO -IKlx2 2
I K 12 J e 21 K I dx ) ~by part~al -co ~ntegrat~on)
2( 1 co t 2 )2 IKI 3/2 J e- dt 21K I -co
while
112
§6. Hardy's theorem
II. FOURIER TRANSFORMS ON L2 (-<»,oo)
1 ( 1 "4\1(IKI) f
_t2 )2 e dt
1 I 2 7f 4TKT ( 7f) = 4TKT
2 We have noted that the function f(x)
that
e-x has the special property
(§1, Ch.I), which implies that
-co < Ct < OJ,
2 -(1 /2 e .
2 Hardy has shown that this property of the function e-x /2 , together
with its order of magnitude, characterize the function in the sense
of the following
Theorem 9. Let f(x) be a measurable function defined on -=< x < =. Let 2
f(x) = O(e-x /2)
F[ fJ (a)
as I (1 I -+ =. Then
, as Ixl
f(x) = c
-+ =,
where c is a complex number.
and
For the proof we need to apply the principle of Phragmen-Linde16f,
which may be viewed as an extension of the maximum principle.
(6.1) Theorem of Phragmen-Lindelof. Let f(z) be a non-constant
holomorphic function of the complex variable z (= re i8 ), in the domain
D defined by the relations
§6. Hardy's theorem
D: r_>O, _..2!:..< a <..2!:.. a > -21 20. - - 20.'
and z.et
and I f ( z) I .::. M < co, for r 2. 0, a
r S If(z) 1< K e , S < a, zED,
113
'IT ± 20. '
where 0 denotes the cZ.osure of D3 and the constant K is independent of z. Then we have
If(z)! <M, zED.
Proof. Consider the function
We have
On the lines a = ± 2'ITa ' we have cosya > 0, since y < a. Hence on these
lines
IF(Z) I .::. If(z) I < M.
Further on the arc defined by I z I = R > 0, ! a I 'IT
< 20.' we have
y 1 'IT -ER cos (2 y-) < e a If(z)! ! F(z) I
RS-ER Y cos (1. Y'IT) < K' e 2 a .... 0, as R .... co,
since S < y < a. Hence for R2. RO > 0, we have IF(Z) I.::.M, on that arc. By
the maximum principle, we have
IF(z) I
It follows that
< M, for 0.::. r.::. R, 'IT < 20.
ErY If(z) I .::. M'e , zED, E > O.
On letting E .j. 0, we get the required result.
114 II. FOURIER TRANSFORMS ON Li-«>,oo)
(6.2) Corollary. Take a 2, S 1.
Proof of Theorem 9. Let z = x + iy, where x and yare real, i !=T. Consider the function defined by
(6.3) fez) , fco izt 7T2nT f(t)e dt. -co
The integral converges absolutely and uniformly in every strip
-co < -A< lm z:,:A< co, since
co
1(~7f) J , 2
co --t -yt < K, f e 2 dt
, 2 2 co -"2(t+y) +y /2 K, f e dt
-co
(6.4)
Hence fez) is an entire function of z.
Since
f(x) ~(f(X) + fe-x»~ +~(f(X) - fe-x»~ = flex) + f 2 (x),
say, where f1 is an even function, and f2 is odd, and f1,f2 satisfy
the same conditions as f, we shall consider the case f even, and f
odd, separately.
Let f be even. Then f is also even, so that
f (z)
and
2n L anz
n=O
co
(6.5) (j)(z) := f(lz)
is an entire function.
co
L n=O
i8 Set z = r e ,so that x r cos8, y
lm(lz) = r 1/ 2sin(8/2).
From (6.4) we have
(6.6 )
r sin8, Iz "8/2 Ir e~ ,and
§6. Hardy' $ theorem
for all z.
For z real, and positive, say z = r> 0, we have
1
(6.7) IqJ(r) I = If(/r) I < K3 e - 2"r
by hypothesis. Choose M>max (K2 ,K 3 ), and a such that O<a<Tf, and
define the function
Now
w(z,a) _ w(r,e,a)
1 2"r
Iw(r,O,a) I = e
[ iZ e-ia/2 ] exp ""2 ° sin(a/2)
[ir ei(e-a/2)]
exp ""2 ° ';;'s""i-n"7(-a-rI":'<2":'")-
[ r sin(e-a/2) + i r cos(e-a/2)] exp - 2" ° sin(a/2) 2 sin(a/2) .
1
Iw(r,a,a) I = e - 2"r
Hence (on writing qJ(r,e) = qJ(z), qJ(r) = qJ(r,O», by (6.7), we have
Iw(r,O,a) oqJ(r,O) I < M,
and
Iw(r,a,a) °qJ(r,a) I < M, by (6.6).
By the Phragmen-Lindelof principle, we obtain
that is to say
I I [ r Sin(e-a/ 2)] qJ ( z) .2. M exp 2" ° sin (al 2 )
If we keep e fixed, and let at Tf, then
r 2" cose , for 0 < 8 < Tf.
115
116 II. FOURIER TRANSFORMS ON L2 (-=.00)
By continuity this holds also for e = n.
Similarly we consider the half-plane -n < e < 0, and obtain
1
le2z\p(z) I .s. M, for all (finite) z •
1 "2z
Since \p is entire, e \p(z) reduces to a constant, therefore
-\p (z) or f (z) and the theorem follows.
- f (z) If f is odd, then f is odd, hence f(O) = 0, so that is an even, z 1 2
entire function. By what we have just proved, f (z) - 2 z
c 2 z e
1 2
for z = x real, f(x)
only if c 2 = ° or f(z)
o(e- "2x ), by hypothesis. This is possible
= f(z) :: 0.
§7. The theorem of Paley and Wiener
But
A fundamental theorem due to Paley and Wiener enables us to give a
characterization of the Fourier transforms of functions belonging to
L2(-~'~) -which vanish outside a finite interval, in terms of entire
functions of exponential type in the complex plane.
An entire function f(z) of the complex variable z is said to be of
exponential type, if
(7.1)
for some (finite) A> 0. The lower bound of such numbers A is called
the type of f; it is non-negative.
We denote by EO, 02.°, the class of entire functions of type at most
o. Thus if f E EO, then
(7.2) f(z)
for every £ > 0.
§? The theorem of Paley and Wiener
Theorem 10 (Paley and Wiener). Let ° < A < 00. Then we have
(7.3) 1 A .
F(x) = 1(211) J f(u)e1Xudu, -oo<x<oo, -A
117
for some fEL 2 (-A,A). if and only if F(x) EL2 (-00<X<00) and F aan be
extended to the aomplex plane as a funation of the alass EA.
Proof (First part). If (7.3) holds with f(u) EL2 (-A<U<A), then
f(u) E L1 (-A < u < A), and for complex z,
1 A izu F(z) = 1(211) iA f(u)e du ( z x+iy)
is an entire function of z, with
IF(z)l.:: 1 e Alzl t If(u)ldu = o(eAlzl ), 7T2iTT -A
so that FE EA. Further the Fourier transform of f, where f (u) is
defined to be zero for I u I > A, is F, so that by Plancherel' s theorem,
IIFI12 = Il f I1 2 <00.
(Second part). Let F (x) E L2 (-00 < x < (0), and for complex z, let
F ( z) E EA. Then by P lancherel' s theorem,
(7.4) f(x) lim R->-oo (L2-norm)
1 R . J F(u)e-1Xudu 1(211) -R
belongs to L2 (-00 < x < (0). We shall prove that f (x) vanishes almost
everywhere for Ixi >A>O, so that f(x) EL 2 (-A<X<A).
For complex z, let
(7.5) g(z)
Then
1 2
J F(u-z)du. 1
-2
(7.6) g(z) is an entire function of z,
such that, for E > 0,
118
1 2
II. FOURIER TRANSFORMS ON L2 (-OO,oo)
1
Ig(z) I < f IF(u-z) Idu 1
O(f~ e(A+€) (lzl+lul)dU)
-2 -2
(7.7)
1 2
o(e(A+€) Izl f1 e(A+€) lu 1dU )
-2
o(e(A+€) Izl),
since F E EA. Further, for real x, we have
(7.8)
and
co f -co
(7.9)
2 Ig(x) I
Ig(x) 12dx < co
J -co
1 2
f 1
-2
1 1
{J~IF(U-X) 12dU}dX 2 co
IF(U-X) 12dx f du f 1 -co
-2 -2
IIFII~dU = IIFII ~ < co,
so that g(x) is bounded, with g(x) EL2 (-co<X<co).
Next let
(7.10) iBz G(z) = e g(z), z complex, B>A>O.
Then G(z) is an entire function of exponentiaZ type, by (7.6) and
(7.7). Further if z is real, and z = x, then
(7.11 ) IG(x)1 = Ig(x)1 =0(1), by (7.8),
and if z = ib, b > 0, then by (7. 7), we have
( -b(B-A-e)) o e ... 0,
as b ... +co ,
if € is chosen sufficiently small, since B > A. In particular,
§? The theorem of Paley and Wiener 119
(7.12) IG(ib)1 = 0(1), as b++"', for B>A.
Because of (7.11) and (7.12) and the Phragmen-Lindelof principle (6.1),
'B l'e TI it follows that le l Zg(z)1 = 0(1), for Z = Re , 0::e:: 2 ' or
1 (R ie) 1 < BR sine TI, , 1 ie 1 BR sin8 g e c 1e ,O::8::2.Slmllarly g(Re ) <c2e ,for
TI 2 < 8 :: TI. Hence we have
(7.13)
Let L > 0, and x > B > A> 0, and consider the integral
R eixu f 1-Liu g(u)du, ""R
where R is large enough to ensure that LR> 1.
ib
-R o R
By Cauchy's theorem applied to the semicircular contour defined by
IRe zi :: R, 1m z = 0; and 1m z > 0, Izl = R; (see Fig.), we have
R ixu f ~-Liu g(u)du -R
so that
(7.14) IfR eixug(u) dul _< c'R TIf -xR sine + BR sin8 d8 +0, 1-Liu LR-1 e
-R 0
since LR> 1, x>B, by choice, sin8>0 for 0::8::TI, and
(7.15 ) TI f e-aR sin8d8 o
1 OCR)' for any a>O,
which can be seen as follows. We have
as R .... "',
120 II. FOURIER TRANSFORMS ON Li-=,oo)
1T J e- aR sin8d8 o
1T/2 1T J + J o 1T/2
11 + 1 2 , say,
where
11T/2 I -CR1T/2
1111.::. J e-CR8d81 = -e cR + c~1 = O(i), c = 21Ta> 0, o
since; < si8n8 .::. 1, for 0'::' 8'::'~' so that -aR sin8 < -;a R8 = -cR8.
And
Let
(7.16)
1T J e-aR sin8d8 1 1 , (with 8 = 1T-\p,
sin8 = sin\p). 1T /2
g (u) = ~ , L> 0, -00< u< 00. -L 1-Ll.u
Then g-L E L1 (_00 ,(0), since it is the product of two functions each of
which belongs to L2 (-oo,oo). From (7.14) we deduce that the Fourier
transform of g-L vanishes for L>O. x>B;
(7.17 ) O,L>O,x>B.
We have, however,
(7.18) J I 2 g(u) 1-Liu - g(u) I du -+ 0, as L -I- 0,
-00
by Lebesgue's theorem on dominated convergence (cf. (7.8)). Hence,
by Plancherel's theorem,
(7.19) J IF[g_L](u) - F[g](u)1 2du-+O, as L-I-O.
From (7.17) it follows that
(7.20) F[g](x) = 0, almost everywhere, for x>B.
We have, however, for real v,
1/2 (7.21 ) g(v) = J F(u-v)du
-1/2 v
by (2.26). (Note that F[f](x) F[ f] (-x) ). Since
§7. The theorem of Paley and Wiener
sinx/2 f(x) x/2 ELl (-00,00) .n'L2 (-00,00), we deduce from (7.21) that
F[g] (x) = f(x) sinx/2 x/2
121
for almost all x E (-00,00). Since (7.20) holds for every B > A, we con
clude that f (x) = ° for almost every x> A.
Similarly we show that f (x) = ° for almost every x < -A. For we have
only to consider G(-ib), b> 0, instead of G(ib), and note that,
after (7.7),
(7.22)
for every E: > 0, and if B < -A < 0, then for E: sufficiently small, we
have
(7.23) I G (- ib) I = 0 ( 1 ), as b ->- +00, B < -A < 0,
corresponding to (7.12). Again by the Phragmen-Lindelof principle
applied to a semi-circular domain in the lower half-plane, we obtain
-R o R
(7.24) Ig(Re i8 ) I -- o(eBR sin8) , -1r~82.0' R>O,
corresponding to (7.13). And we have
I R ixu () I I ° ixRei8 (R i8) R i8 I J e g, u du < i J e gee d8 -R 1+L1.u - -1T 1+LiRei8
° < ~ J e-(X-B)R sin8 d8 LR-1
-1T
cR 1TJ e(X-B)R sin8 d8 , LR-1
° which tends to zero as R ->- 00 if x < B, c being a constant. This corre-
122 II. FOURIER TRANSFORMS ON Li-oo,ooj
sponds to (7.14). And we deduce (as in (7.17)) that
(7.25) F[gL](X) = 0, for L>O, x<B<-A<O,
where gL(x) = il~~xEL1(-=<x<=). It then follows, as before, that
f (x) = ° for almost every x < -A, and the second part of the theorem
is proved.
Combining the Paley-Wiener theorem with the Riemann-Lebesgue theorem
(Ch.I, Th.l), we obtain the following
(7.26) Corollary. If F(z) is an entire function of exponential
type, with z = x+iy" and F(x) EL2 (-=<X<=), then F(x) -+0 as Ixl-+=.
§8. Fourier series in L2 (a,b)
Let ~O(x), ~1 (x), ~2(x) , ... , be complex-valued, non-null functions
(that is, functions which are not almost everywhere equal to zero)
defined on the interval (a,b) of the real line. We say that {~n} is
an orthogonal set if
b J ~ (x)\ii (x)dx m n a
{a,
A > ° m '
for m * n; m,n 0,1,2, ... ,
for m = n.
We note that ~m (x) E L2 (a < x < b), and that no ~m (x) can vanish
identically, since that would imply that Am= 0.
If, in addition, Am
set.
1 for m 0,1,2, ... , we call {~n} an orthonormal
If {~n} is orthogonal, then clearly {A~~72} is orthonormal.
n
Given f E L2 (a,b), let
1 X-
n
b J f(x)\iin(x)dx, a
th for any integer n.::.O, We call c n the n Fourier coefficient of f,
and write
§8. Fourier series in L2(a,b) 123
( 8. 1 )
We call the series
the Fourier (orthogonal) series of f, relative to (~n)' and indicate
that relation by writing
(8.2) f(x) ~ l: n=O
c n
The set {einx}, n = 0, ± 1, ± 2, ... , is orthogonal on (0,2~), while the
set {einx (2~)-1/2} is orthonormal, since
a+2~ . . f elmx·e-lnx dx a
for any real a.
In order to have the equality
f(x)
it is necessary that the set {~n} should be "complete", in the sense
that if we add a new, non-null function ~ to the set, then the re
sulting set is no longer orthogonal. Otherwise there would exist a
non-null function, namely ~ itself, with all its Fourier coefficients
equal to zero.
We thus define a set (~n)' ~n E L2 (a,b), to be complete, if there
exists no non-null function in L2 (a,b) which is orthogonal to ~n for
every n > O. In other words,
b f f(x) (j) (x)dx
n 0, n=0,1,2, ... ,
a
implies that f(x) 0, for almost every x E (a,b) .
It follows that if {~n} is complete, and orthonormal, and the functions
f,gEL 2 (a,b) have the same Fourier coefficients, then f is equivalent
to g (cf.§l, Ch.I), that is to say that f equals g almost everywhere in
124 II. FOURIER TRANSFORfrlS ON Li-oo,oo)
(a,b). In this sense, the Fourier series of f is unique.
( b 2 )1/2 Let fEL 2 (a,b), with IIfl12 = ~ If(x) I dx • Let ((jJn) be an ortho-
normal set in L2 (a,b), and let cn denote the nth Fourier coefficient
of f relative to ((jJn). By a polynomial in the (jJn' of rank k, where
k is a non-negative integer, is meant an expression of the form
(8.3) k
wk(x) = r Ym (jJm(x), m=O
where all y's are complex numbers. We call
(8.4) k
fk(x) = r C (jJ (x) m=O m m
the Fourier polynomial, of rank k, of the function f.
With this notation we have the following identity:
(8.5)
and, in particular,
(8.6)
n Ic 12 + r I 12 m cm -Ym '
2 Icml .
m=O
To prove (8.5) we note that
and
(8.7)
b f f(x) ¢n(x)dx = a
b
b f I W (x) 12dx n a
n r
m=O
f {f(x) - Wn(x)} {f(x) - ¢n(x)}dx a
n n n
m:o c m Ym - m:o cm Ym + m:o
n r
m=O
2 Ic I + m
n r
m=O (c -Y ) (c -y ).
m m m m
n By (8.6), r Icml2 < Ilfll~, and on letting n-->-oo, we obtain Bessel's
m=O inequality, namely
§8. Fourier series in L2(a,bJ 125
(8.8) b 2 f If(x)1 dx. a
From (8.5) we conclude that
(8.9) the best approximation, in the L 2 -norm, to fEL 2 (a,b) by
polynomials ¢n' of a given rank n, is provided by the Fourier poly
nomial fn of f.
An important result on the Fourier series of functions in L2 (a,b) is
the following
(8.10) The Riesz-Fischer theorem. Given any sequence (cn ) of complex 00
numbers, such that L Icnl2<oo, there exists a function fEL 2 (a,b), n=O
such that the cn's are the Fourier coefficients of f, relative to the
given orthonormal set (~n):=o. We have further:
( 8 • 11 )
and
(8.12)
b f If (x) - f(x)1 2dx-+O as n-+ co ,
n a
m
L n=O
Ic n l 2 .
m Proof. Let f =: f (x) ----- m m L c k ~k (x)
k=O - L c k ~k. If n>m, then
k=O
n L
k=rn+1
by the hypothesis on the c n . Since the function space L2 (a,b) is
complete (cf. §2) (which is not to be confused with the set (~n) being
complete, which is not assumed here), there exists a function
f E L2 (a,b), such that
(8.13) b f If (x) - f(x) 12dx-+o, as n-+oo.
n a
It follows that
b b f f(x) ~ (x)dx = lim f
m a n-+oo a lim cm n-+oo
so that c m is the mth Fourier coefficient of f, relative to (~n).
126 II. FOURIER TRANSFORMS ON L2 (-«>,00)
(Note that if Ilf-fnI12+0, and Ilg-gn I1 2 +0, then Ilfg - fngnl11 +0.)
Hence, by (8.5), we have
and (8.13) now implies (8.12).
Remark. We note that by (8.6), (8.11) is equivalent to (8.12). We
next make use of the completeness of (~n)'
(8.14) ParsevaZ's theorem. (A) If fEL 2 (a,b), and (~n) is a aompZete, orthonormaZ set in L2 (a,b), and c n denotes the nth Fourier
aoeffiaient of f, reZative to the (~n)' then
(8.11) , b f Ifn(x) - f(x) 12dx+0, as n +00, a
and
(8.12) , b 2 f If(x) I dx = a
00 L
n=O
(B). If f,gEL2 (a,b), f ~ (cn ) and g ~ (dn ), (as defined in (8.l)), then
00
L n=O
c d n n
the series on the right aonverging absoZuteZy.
00 2 Proof. (A) By Bessel's inequality we have Lie I < 00, and the Riesz-
n=O n Fischer theorem gives an fEL 2 (a,b) satisfying (8.11) and (8.12). The
completeness of (~n) implies that that f differs from the given f only
on a set of measure zero in (a,b), and (8.11)' and (8.12)' follow.
(B) We have
b f fn(x)g(x)dx a
n b L c f ~ (x)g(x)dx
m=O m a m
Since Ilf-fn"2+0, as n+oo, by (8.11)' it follows that
b b f fn(x)g(x)dx + f f(x)g(x)dx, as n+ oo ,
a a
§8. Fourier series in L2(a.bJ
and since
00
co 2 L I c I , and
n=O n
L m=O
I c d I < co. m m
2 L Idnl , converge by (8.12)', we have n=O
co
The closure and completeness of (~n) in L2 (a,b)
127
(8.15) A set of functions (~n)' ~nEL2(a,b), is said to be alosed in L2 (a,b), if the polynomials in ~n (see (8.3» form a dense subset
of L2 (a,b) in the L2-norm.
(8.16 ) The orthonormal set (~n) is closed in L2 (a,b), if and only
if it is complete in L2 (a,b).
Proof. If (~n) is complete, then given any f E L2 (a,b), there exists,
by (8.11)', a sequence (fn) of Fourier polynomials of f, such that
Ilf-fnI12+0, as n+oo. Hence the set (~n) is closed.
Conversely let the set (~n) be closed in L2 (a,b), and f E L2 (a,b) ,
f ~ (cn ), and c n = 0 for all n::.. O. Then there exis ts a sequence of
polynomials in the ~n' say lPn' such that IIIPn-fI12+0, as n+co. By
(8.9), we have IIfn-fI12+0, as n+ co , where fn is the Fourier poly
nomial of f of rank n. But fn = 0, since all the cn's are zero, hence
IIfl12 = 0, or f is zero almost everywhere in (a,b). It follows that
(~n) is complete in L2 (a,b).
Remarks
1. Let ( ) 1 e inx I I 0 1 ~n x = 7T2iTT ' x 2. n, n = ,:1:, :1:2, ••• ,.
I t is a classical result that (~n) -is complete in L1 (-n , n), hence
also in L2 (-n,n).
2. Let {
+ 1, for 0 < x < ~ ~O(x) = 1
-1, for '2<x< 1,
and ~O(O) = ~O(~) = 0, with ~(x) = ~(x+1). Let ~n(x) = ~0(2nX) for
n = 0, 1, 2, Then (~n) is orthonormal over (0,1).
The function ~n(x) takes alternately the values +1, and -1 in the -n-1 -n-1 -n-1 -n-1 -n-1 intervals (0,2 ), (2 ,2,2 ), (2·2 ,3·2 ), ••.. If
128 II. FOURIER TRANSFORMS ON Li-oo,w)
m> n, the integral of (j)m (j)n over any of these intervals is zero,
and the set «(j)n) is obviously orthonormal. It is not complete,
since the function ~(x) = 1 may be added to it. (Note that the set
{~}.U·{(j)n}n>o is not complete.) If we define
sign CI.
sign CI.
for CI. > 0, sign CI. = -1 for CI. < 0, and
° for CI. = 0,
then we have (j)n(x) = sign sin 2n+ 1TIx. The functions «(j)n) are known
as Rademacher's functions.
3. The condition of orthonormality in (8.16) is not necessary.
§9. Hardy's interpolation formula
As an application of the Paley-Wiener theorem one can obtain some
special formulae of interpolation at integer points for entire func
tions of exponential type, which help to determine the value of the
function at an arbitrary point in the complex plane in terms of its
values at the integer points on the real axis. The notions concerning
Fourier series outlined in the previous section enter into the proofs.
If F(z) is an entire
as defined in (7.2),
in ECY , CY > 0, may be
function of z belonging to the class ECY , CY > 0,
then F (TI z) E E TI, so that the study of functions CY
reduced to that of functions in ETI.
Theorem 11 (Hardy). LetF(z)EETI , z
(9.1) f 2
IF(X) I dx < 00.
Then we have
(9.2) F(z)
Proof. If we define
(j)n (x)
sinTI z 11 n=-oo
1 inx
~e 0, Ixl > TI,
x+iy, and
F(n) z-n
§9. Hardy's interpolation formula 129
for n = 0, ±1, ±2, ... , then the set (~n) is orthonormal over (-~,oo),
but not complete on (-00,00) (since any function which vanishes on
(-n,n) has all its Fourier coefficients equal to zero). It follows
that the set {F[~nJ} is also orthonormal (by (2.24)). Now
y 1 -iax n ei(n-a)xdx F[~n J (a) 7T21iT J ~n(x)e dx 2n f -n
(9.3) sin(a-n)n (-1) n sin(an) n(a-n) n(a-n)
Since F(x) E L2 (-00,00) , let
y a (_1)n sin (nx) (9.4) F(x) ~ L a F[~nJ(x) L n n n (x-n)
where
a n
n=-oo
1 7T21iT J
n=-oo
-y--
F(x) n~ 1 (x)dx n
is the nth Fourier coefficient of F relative to (F[~nJ). [Since
(F[~ J) is not complete, the series in (9.4) does not, in general, n
represen t Fl.
By the Paley-Wiener theorem, F[FJ (x) = 0 for I x I > n, hence y
F[FJ E L1 (-00,00) .n'L2 (-00,00). (Note that F[FJ (a) = F[FJ (-a)). Since (~n)
is complete in L2 (-n,n), given E > 0, there exists a polynomial in the
~n's, with complex coefficients an' such that
N II F[FJ - L an~n 112 < E,
-N
since ~n vanishes outside (-n, n) for every n > O. By Plancherel's
theorem we have
This is equivalent to saying that
(9.5) IIFII ~ = L n=-oo
2 I a I < 00. n
(See (8.16) and the Remark after the Riesz-Fischer theorem).
The series
130
n=-oo
{sin(lIx)}2 2 (x-n)
II. FOURIER TRANSFORMS ON Li-=,oo)
is uniformly convergent on (-=,=). By Schwarz's inequality, and (9.5),
it follows that the series
n=-oo a (_1)n sinllx
n 11 (x-n)
converges uniformly on (-=,=). Since FE ElI by hypothesis, F is
continuous, hence
---11
(9.6) F(x) L n=-=
a (_1)n sinllx n 11 (x-n)
sinllx
for -= < x < =.
If we put x n, we get
F (n), n 0, ±1, ±2, ...•
n=-co
n an (-1 )
x-n
The series on the right-hand side of (9.6) converges absolutely and
uniformly also with a complex z in place of the real x, provided that
-= < -A~ 1m z ~ A < +=, hence represents an entire function of z, which
coincides with F(Z) because of (9.6), and (9.2) follows.
Theorem 11 can be used to prove an interpolation formula where the
assumption that F (x) E L2 (-00 < x < (0) is replaced by the boundedness
of F(x).
Theorem 12. If F(z) E E'rr, z
then we have
x+iy, and F(x) is bounded for -00 <; x < 00,
(9.7) F (z) sinllz
11 {F' (0) + F(zO) + n:-= (_1)nF (n) (z2n + ~)},
n*O
where F' denotes the derivative of F.
Proof. If we set G(z) = (F(Z) ~ F(O)), then G(z) E Elf, and
G (x) E L2 (-00 < x < (0). Theorem 11 then gives
G( z)
so that
sinll z 11 n=-co
§1O. TbJo inequalities due to s. Bernstein
(9.8) F(z) - F(O) z simrz 7T
00 (-1)n(F(n)-F(O) )
n(z-n) + F' (0) sin7Tz
7T
since G(O)
7T sin7Tz
so that
F(O)
F' (0). We have, however,
- + z
F(O)
(_1)n{_1_ + .l} l: z-n n n=-oo n*O
sin7Tz 7T --7T- sin7Tz = F(O)
-z
sin7Tz 7T
+
and if we use this in (9.8) we obtain (9.7).
§10. Two inequalities due to S. Bernstein
z 00 (-1) n l: ,
n=-oo n( z-n)
n*O
00 (_1)n) + z 1:
n=-oo n (z-n) ,
n*O
131
The interpolation formula obtained in Theorem 12 can be used to prove
two important inequalities originally proved by S. Bernstein.
Theorem 13 (5. Bernstein). If F(z) E Ea , a> 0, z = x+iy, and F(x) is
boundedfor-oo<·x<oo~ withM=sup iF(x)l, then we have
(10.1) IF'(x) 1:5. aM, -oo<x<oo,
the dash denoting the derivative. EquaLity occurs in (10.1) if and onLy if
(10.2) F(z) = a e iaz + b e-iaz ,
where a and b are compLex numbers.
Proof. It is sufficient to prove the theorem for a otherwise study F(Z7T) •
a
7T, for we can
By taking z
( 10.3)
x in (9.7), and differentiating once, we get
F' (x) sin7Tx COS7TX F 1 (X) + 7T 00
n=-oo
(_1)n-1 F (n) 2 (x-n)
132 II. FOURIER TRANSFORMS ON L2 (-oo,oo)
where
F' (0) + F(O) + L x n=-oo
(-1)nF (n)fl --1-- + ~} x-n n'
the differentiation being justified by the fact that the differen-1 tiated series converges uniformly. On setting x = 2' we get
(10,4) F I (.1) 4 L
(_1)n-1 F (n) 2 1T (2n-l)2 n=-oo
so that
I F' (~) I ~ M 1 4M 2 (10.5) < L
1T M1T. - 'IT 2 1T 4"" n=-oo ( 2n-1 )
For any real xo ' consider
( 10.6) G (z) F(XO + z -
1T We have: GEE , I G(x) I < M, 1
F(XO + n - 2)' and
(10.5) implies that
which proves the first part of the theorem.
To prove the second part, we apply formula (10.4) to the function G in
(10.6), qnd obtain
On replacing
(10.7)
F' (x ) = .1 o 1T
Xo by x, and
F' (x) 4 1T
n=-oo
n by
L n=-co
(_1)n-1 G (n) 2 (2n-1)
n+l, we get
n 1 (-1) F(x+n+2)
(2n+1) 2
4 1T
00
n=-oo
Let us suppose that for x = xl we have the equality
1TM e ia a real.
Formula (10.7) then gives
(2n-1)2
§10. TWo inequaLities due to S. Bernstein 133
n 1 e ia } 4 co {(-1) F(x1+n+2) - M
(10.8) L (2n+1)2
= 0, 'IT n=-co
the series on the left-hand side being absolutely convergent, since
F is bounded on the real axis.
Since IF(x) I ~ M, we have Re[M-(-1)ne-iaF(x1+n+~)]~ 0, and if n 1 +ia [ n -ia 1 ] (-1) F(x 1+n+2) *M e , then Re M-(-1) e F(x1+n+2) >0, hence (10.8)
implies that
(-1 ) n F (x1 +n+~) M e ia , for n 0, :1:1, :1:2, ...
If we set H(z) = F(X1+z+~), then H(O)
1 H(n) = F(x1+n+2), so that
( 10.9) (-1)~(n) = H(O).
M e ia , and
If we apply Theorem 12 to the function H(z), we get
H(z) = sin'ITz {HI (0) + H(O) + 'IT z
because of (10.9). We have, however,
'IT cot 'ITZ ! + ~ (z~n + ~). n=-co n*O
Hence
H(Z) Si~'ITZ [HI (0) + H(O) "'IT cot 'ITz]
A cos 'ITZ + B sin 'ITZ, say,
Since F(z) = H(Z-x1-~)' it is proved that if equality occurs in (10.1) ~ then (10.2) holds.
We have to show that if (10.2) holds, then equality occurs in (10.1)
for some x, -co < X < co.
If F(x) a e iox + b e-iox and a IbleiS , 0>0, then
134 II. FOURIER TRANSFORMS ON L2 (-OO,oo)
F(x) = ei(a+ox) (Ial + Ibl e i (B-a)-2iox), so that for
x = x, (B7;-1T, we have IF'(x,)1 = o(lal + Ibl) = 0 Max IF(x)l. x
Remarks. By letting 0'" ° in ('0.'), we deduce that the only functions
FE EO, which are bounded on the real axis, are constants.
On the other hand, it follows from a theorem of Siegel that if FE EO,
and F(x) ELl (-0:> < x < 0:», then F :: 0.
Theorem 14 (S. Bernstein). Let T(x)
and ck complex. Then the inequality
IT(x)I~M impZies that
IT' (x) I ~ nM,
n r c k e ikx where x is real
k=-n
and this inequality becomes an equality if and only if
T(x) = a e inx + b e- inx
where a and b are complex numbers.
Proof. We have only to note that for complex z, T(z) is an entire
function, wi th TEEn, while T (x) is bounded, and apply Theorem 13.
§11. Several variables
The proof of Plancherel's theorem in several variables is not
essentially different from that in one variable. If Ek denotes the
real Euclidean space of k dimensions, let S denote the Schwartz space
of infinitely differentiable functions on Ek , such that for any
a = (a" ••• ,ak ), B = (B 1 , ••• ,Bk ), where a"a2 , •.• ,ak and B"B 2 , ••• ,Bk are non-negative integers,
sup I xa (OBf) (x) I < 0>, xEEk
§11. SeveraZ variabZes 135
where Cl.k Q
~ , and O»f =
13 1 13k
(a!1) ••• (a~) f. Then S is a
dense subspace of L (Ek ), 1 < p < co. P - For f E S we define the Fourier
transform F[f] by the relation
F[f](x) = (27f)-k/2 J f(t) ei<x,t>dt, xEEk •
Ek
Then F[f] E S, and f + F[f] is a one-to-one mapping of S onto S (as in
the case of one variable, cf. §2). If f,g E-S, then hg E S. We further
have
and
J f(x) F[g](x)dx Ek
J F[f](y) g(y)dy, Ek
(f,g) = (F[f], F[g]).
If f E L2 (Ek ), there exists a sequence (fn) of functions belonging to S,
such that Ilf-fnI12+o, as n+ co , and IIF[fn ] - F[fm]11 2 = Ilfm-fnIl2,
which implies that IIF[fm] - F[fn]1I2+o, as m,n+co. Since L2 (Ek ) is
complete, there exists gEL2 (Ek ), such that Ilg - F[fm]11 2 +o, as
m+ co • And IIgl12 = lim IIF[fm]11 2 = lim IIfml12 = Ilf112. We define g to m+co m+co
be the Fourier transform of f E L2 (Ek ). It is independent of the
approximating sequence, and is defined almost everywhere. We denote it
by F[f].
We similarly define
F[f] (x) = (27f) -k/2 J Ek
f(t)e-i<x,t>dt, for fES,
and extend the definition to all of L2 (Ek ). It follows, as in (2.11),
that
so that f .... F[f] is a linear mapping of L2 (Ek ) onto itself; it is also
isometric. The proof that the definitions of the Fourier transform on
L2 (Ek ), and on L1 (Ek ) coincide on L1 (Ek )·n.L2 (Ek ) follows as in the
case of E 1 •
Chapter III. Fourier-Stieltjes transforms (one variable)
§1. Basic properties
We assume as known the fundamentals of the theory of Riemann-Stieltjes
integrals.
Let F(y) be a function of bounded variation for -00< y < co. For x real,
let
( 1.1) ~(x) = foo eixYdF(y) R
_ lim f eiXYdF(Y), (R>O). -00 R+co -R
We call ~ the Fourier-StieZtjes transform of F, or the Fourier trans
form of dF, and denote it sometimes by the symbol dF.
If, in particular,
Y (1 .2) F(y) f f(t)dt, fEL 1 (-00,00) ,
-co
then (1.1) reduces to the Fourier transform on L1 (-00,00) studied in
Chapter I.
The integral in (1.1) converges absolutely and uniformly and ~(x) is
a bounded, continuous function of x defined for every x in (-oo,co).
We have only to note that
I~(X) I ~ f I dF (y) I < co, -00
and for any real h * 0,
00 le ihY-11·ldF(y) I I~(x+h) - ~(x) I ~ f
-00
< Ihl f lyl·ldF(y)1 + 2 f IdF(y) I 11 + 1 2 , - IYI<R IYI~R
138 III. FOURIER-STIELTJES TRANSFORMS
say, where R> O. Given any E > 0, one can choose R so large that
II21 < E/2, and h so small that II11 < E/2.
But unlike the Fourier transform on L1(-~'~)' ~(x) does not necessarily
tend to zero as Ixl .... ~. For example, if F(x) = 1, for x>O; F(x) = -1,
for x < 0; and F(O) = 0, then ~(x) = 2.
Theorem 1. Let F(x) be of bounded variation in (-~,~), with
( 1 .3) F(x) ~ {F(x+O) + F(x-O)}, for aZZ x,
and
( 1 .4) ~(x)
Then we have
(1.5) F(x) - F(O) R e-itx_1
2n lim f ~(t) -it dt R .... ~ -R
OD
- 2n J
so that ~ determines F up to an arbitrary, additive constant.
Proof. If h is real, and fixed, and h '*' 0, then (1.4) gives
(1 .6) -ihx e
OD f eix(y-h)dF(y) -~
From this and (1.4) we get
(1.7) ~(x) [e- ihX_1] = J~ -~
J -~
G(y) = F(y+h) - F(y).
Here G is of bounded variation in (-~,~), and G E L1 (-~,~). For if
we suppose h 2. 0, h fixed; and suppose that F is non-decreasing (since
F is expressible as the difference of two non-decreasing, bounded
functions, if it is real-valued; and if it is complex-valued, one can
consider the real and imaginary parts separately), then G(y) 2. 0, and
we have for R> 0,
R J G(x)dx -R
R J {F(x+h) - F(x)}dx -R
R+h -R+h (J - J )F(X)dX
R -R
§1. Basic properties 139
h 0 f F(x+R)dx - f F(-x-R)dx o -h
+ h{F(+~) - F(-~)},
as R++~, and the limit is finite by hypothesis.
We note that G(x) +0, as Ixl +~, (by definition and by the hypothesis
on F), and by partial integration,
[ R. ] [ . ]y=R lim f e~xYdG(y) = lim e~xYG(y) R~ -R R+~ y=-R
which gives
co
R - ix lim f
R~ -R
eiXYdG(y) ~
eiXYG(y)dy, f -ix f -~ -~
so that, by (1.7), we have
tp(x) [ e-i~x_1 ] ~
eixYG(y)dy. f -~x -~
ixy e G(y)dy,
Since F(x)
that
~[F(X+O) + F(x-O)], this implies (by Theorem 5, Ch.I),
G(x) R e- ihY_1 -ixy
2n lim f tp(y) -iy e dy, R+~ -R
for each x. For x = 0, this gives (1.5).
Theorem 2. Let F be of bounded variation in (-~,~), and let ~
F(x) = ~ [F(x+O) + F(x-O)], for all x, and Zet tp(x) = f eiXYdF(y).
Then we have
(1.8) j {F(y) - F(-y)}dy = 1 f~ tp(y) 1-CO~ xy dy, o n _~ y
the integral on the right-hand side converging absolutely {since tp is
bounded},
Proof. We have, for R > 0,
~
Rn f tp(y) (1-C~S Ry) dy -~ y
00
1 f n
~
dF(y) f e ixy 1-cos Ry dy
Ri
140 III. FOURIER-STIELTJES TRANSFORMS
R = f (1 - hl) dF (x)
-R R
1 R R I [F(y)-F(-y)]dy.
a
(see Ex.2, §1, Ch.I; also Ex.l,§a, Ch.I).
Remarks. Let F 1 (X), F 2 (X) , ... , be non-decreasing, and bounded functions
in (-00 < x < (0), and let
( 1 .9) n=1,2, ....
The following examples show the difficulty in preserving the equality
sign in (1.9) after letting n + +00.
(1.10) Let Fn(X) =
for each x, as n + +00,
n + +00, except for x
0, for x<n; Fn(X) = 1, for x.:::.n. Then Fn(X) +0,
but ~ (x) = e inx does not tend to a limit as n
2~k, where k is an integer.
(1.11) Let Fn(X) be continuous, Fn(X) = ° for x.::. -n, Fn(X) = 1 for
x.:::.n, Fn(X) 2xn + ~ for -n<x<n (linear). Then Fn(X) .... ~' for each
x, as n -->- co, and
n eixy ~n(x) = f ~ dy
-n
so that ~n (x) -->- 0, as n-+ oo , for each
n-+ co , so that
sin nx nx
x * 0, and ~n(x) -->- 1,
lim ~n(x) * f e ixy d (lim F n (y» . n-->-oo n-->-oo
§2. Distribution functions, and characteristic functions
at x 0, as
In order to avoid some of the difficulties pointed up by the above
examples, we shall now consider non-decreasing functions F(x), which
have finite limits at x = ±oo. We suppose further that
(2.1) F (-00) 0, F(+=) 1 •
We call such a function F a distribution function. We call the Fourier-
§2. Distribution functions, and characteristic functions
Stieltjes transform of such an F, namely
00
~(x) = f eixYdF(Y), -00
a characteristic function, corresponding to the given distribution
function F.
It is clear that ~ is continuous and bounded. (Note that
I~(x) I ~ F(+oo) - F(-oo) = 1, and ~(O) = 1). It is also hermitian, in
the sense that ~(-x) = ~(x) •
Theorem implies that given ~, the distribution function F is
141
defined uniquely at the points of continuity of F, because F(-=) = 0,
F(+=) = 1. The next theorem shows how the convergence of a sequence
of distribution functions implies the convergence of the corresponding
sequence of characteristic functions.
Theorem 3. Let (Fn) be a sequence of distribution functions, and (~n)
the corresponding sequence of characteristic functions, so that
(2.2) ~n(X) = f eiXYdFn(Y). -=
If Fn converges to a distribution function F as n->-oo, at the points
of continuity of F, and if ~ is the characteristic function of F, then
(2.3) ~n(X) ->- ~(x) as n->-=,
and the convergence is uniform in every finite intervaZ.
Proof. We note that the set of discontinuities of F is countable.
Given £ > 0, we choose R> 0, such that F(x) is continuous both at
x = R and x = -R, with F(-R) < £, F(R) > 1-£, and such that
Fn(±R) ->- F(±R), as n->-=. Then we have, for h~nO' also
Now
- f eiXYdFn(Y) IYI>R
142 III. FOURIER-STIELTJES TRANSFORMS
Then we have
II21 < (1 - F(R) + F(-R)) < 2E, for all x,
while
On the other hand,
. y=+R I1 = [(F(Y) - Fn (y))e1XY ] - ix J {F(y) - F (y)}eiXYdy.
y=-R I y 12. R n
Since F n (±R) ->- F (±R), as n ->- =, the first term on the right-hand side
tends to zero, uniformly in x. And
l-iX J {F(y) - F (y)}eiXYdyl2. Ixi J IF(y) - F (y) Idy->-O, lyl'::'R n lyl'::'R n
uniformly in each fini te x-interval since I F I 2. 1, IF n I 2. 1; hence so
does I 1 . Thus we have altogether
and for x in any finite interval.
The next result is a kind of converse to Theorem 3.
Theorem 4. If tOn (x) ->- to(x), for each x, as n ->- =, where tOn is the
characteristic function corresponding to the distribution function
Fn , and if to is continuous at the origin, then Fn converges to a
distribution function F, at the points of continuity of F, and to is
the characteristic function of F.
Proof. Since Fn(X) is a non-decreasing function of x for n = 1,2, ... ,
and since 02. F n (x) .::. 1, for all x E (-=,=), there exists (by Helly IS
theorem) a subsequence F of Fn of non-decreasing functions (of x) n k
which converges, as n k ->- =, to a non-decreasing function F (x). Clearly
02. F (x) 2. 1. We shall see that F is a distribution function, and that
Fn ->- F, at every point of continuity of F.
By definition, we have
J e ixy dF (y). n
§2. DistI'ibution functions, and characteristic functions 143
If we use formula (1.8), we get, for R> 0,
R 1 0:> 1 R J {F (y) - F (-y)}dy = - J ~n (y) -co~ Ydy. ° n k n k 1T_0:> k Y (2.4)
Since I~n(x) 1:5. 1, for all xE (-0:>,0:», and ~n(x) +~(x), for each x, as
n+o:>, we get on letting nk+o:> in (2.4),
R 1 J [F(y) - F(-y) ]dy 1T
° or
R 1 (2.5) R J [F(y) - F(-y)]dy = 21T
°
0:> 1-cos Ry f ~(y) 2 -0:> Y
0:>
f ~(Y)HR(y)dy, -co
dy,
HR(y) sin2 (Ry/2)
R(y/2)2
Since ~n(O) = 1 for n
0:>(. )2
1,2, •.• , and ~(x) = lim ~n(x), we have ~(O) = 1,
and since J s~na da 1 co n+co
1T, we have 21T J HR(y)dy = 1. (cf. Ex.1, Th.13, -0:> -co
Ch. I) .Hence, for a suitably chosen 0 > 0, we have
QD QD
-QD -co
= 11 + 1 2 , say.
Because of the continuity of ~ at the origin, given E > 0, there exists
a 0 > 0, such that
while
1111 :5. sUI? I~(Y)-~(O) 1·1 < E, Iy I <0
I I 2 1 J R[sin(RY/2)]2 dy =]. J (sin t/2)2 dt + 0, 1 2 :5. 21T IYI~o (Ry)/2 1T Itl~RO t/2
as R+o:> for a suitably chosen, but fixed, 0>0. Hence
co lim ~ J ~(y)HR(y)dy = 1, R+co 21T -co
which implies, because of (2.5), that
R lim R f {F(y)-F(-y)}dy R+o:> ° (2.6) 1.
144 III. FOURIER-STIELTJES TRANSFORMS
The left-hand side equals F (+00) - F (-00). Since 0 ~ F (x) ~ 1, and F is
non-decreasing, we deduce that F(+oo) = 1, F(-oo) = O. Hence F is a
distribution function. By Theorem 3, we have
00 ~ (x) = ~(K) = f eiXYdF(y).
nk -00
If there exists a pOint of continuity Xo of F, such that F (xO) ~F(x ) ·n 0
then we can find a subsequence F , nk
, which converges evepywhepe to a
distribution function F*, and such that F*(XO) *F(XO)' If, for example,
F*(XO) >F(XO)' then since F is continuous at xo ' and F* is non
decreasing, we have F*(x) >F(x), for xE (XO,XO+11), for some 11>0,
sufficiently small. (Similarly if F* (xO) < F(XO) , then F* (x) < F(x) for
x E (XO-11 , ,xC) for some 11' > 0). This is impossible since, by Theorem
3, ~ is also the characteristic function of F*, and the characteristic
function determines the d~stribution function up to an additive constant
(Th.1) and F*(-oo) = F(-oo) = o.
§3. Positive definite functions: the theorems of Bochner and of F. Ries2
We shall consider classes of functions f which can be represented as
Fourier-Stieltjes transforms.
Lemma 1. If f(t) is a compZex-valued function which is measupable
and finite fop -00 < t < oo~ and satisfies the condition
(3.1) m L
~=1
m L
v=1 f(t -t )p P > 0
~ v ~ v
fop any integep m.:. 1~ and apbitpapy peal numbeps t 1 ,t2 , ... ,tm and
apbitpapy complex numbeps P1,P2, ..• ,Pm' then we have
(3.2) f(O)':'O; f(-t) = f(t); If(t) I ~f(O).
Proof. On taking m = 1 in (3.1), we obtain
2 Ip11 f(O)':'O,
so that f(O) > o.
§3. Positive definite funcnions: the theorems of Bochner and of F. Riess 145
On taking m = 2, tl = t, t2 = 0, we obtain
(3.3)
If we choose P2 = 1 in (3.3), we get
f(O)[lp112+1J + f(t)P1 + f(-t)P 1 > 0,
and if in this we set P1 = l,i respectively, we see that f(t) + f(-t)
is real (since f(O) is real), and that f(t) - f(-t) is purely ima
ginary, hence f(-t) = f(t).
If f(O) = 0, P1 = 1, P2 =< -f(t), then (3,3) gives -2If(t) 12 ':'0, since
f(-t) = f(t), hence f(t) 0, for all t.
If f(O) > 0, P1 = f(O), P2 = -f(t), then (3.3) gives
which implies that If(t) I ~f(O), and (3.2) is proved.
Lemma 2 (F. Riess). For any complex-valued measurable function f,
condition (3.1) implies that
(3.4) f f f(t-s)P(t)P(s)ds dt .:. 0,
for any P E L1 (_co,co), provided that f (0) is finite.
Proof. If A>O, and pEL2 (-A,A), we take Pj.l = P(t]..l) in (3.1) and
integrate with respect to t]..l' ]..I = 1,2, ... ,m, where m> 1. We get from
each of the diagonal terms (in which j.l = v)
A (2A)m-1 f (0) f Ip(t) 12dt,
-A
while each of the remaining terms gives
Hence we have
A A (2A)m-2 f f f(t-s)P(t)p(s)ds dt.
-A -A
A A A m(2A)m-1 f (0) f Ip(t) 1 2dt + m(m-1) (2A)m-2 f f f(t-s)P(t)P(s)ds dt > 0
-A -A -A
146 III. FOURIER-STIELTJES TRANSFORMS
m-2 On dividing throughout by m(m-1) (2A) , and then letting m -+ co, we
get
A A (3.5) J J f(t-s)p(t)p(s)ds dt2. 0 ,
-A -A
for any P E L 2 (-A,A). If p E L1 (-A,A), (3.5) still holds good, for if
we define, for any integer n2. 1,
Pn (t) fP(t), if Ip(t) l2. n ,
In, if I p (t) I > n,
then we have I Pn (t) I 2. I P (t) I E L1 (-A,A), and 1 Pn (t) I 2. n, for all
tE (-A,A). Hence PnEL2(~A,A).n.L1(-A,A), so that (3.5) holds with
Pn(t) in place of pit). Since Pn(t)-+p(t), as n-+co, and If(t)l~f(O),
and f(O) is finite, we obtain (3.5) for any pEL 1 (-A,A). If we then
let A-+co, we obtain (3.4).
Lemma 3. If a measurable function f (as in Lemma 1) satisfies (3.1),
and f (0) is finite, then for any E > 0, the function
(3.6) 2
-EX e f(x), -00 < x < co, E > 0 I
also satisfies (3.1); and fE(x) EL 1 (-co<x<co).
Proof. We have
rn rn L L fE (tlJ -tv) PlJ Pv
lJ =1 v =1
rn rn -E(t -t )2 L L PlJP v f ( tlJ - tv ) e lJ v
lJ =1 v =1
2 rn rn 1 -t /4E
(3.7) L L PlJP V f (tjJ -tv) 2(1TE) 1/2
f e lJ =1 v=1
if we note that
itt -t )t + lJ v dt,
(see § 1, Ch. I )
and replace x by x/(2/E), and a by 2a/E.
§3. Positive definite functions: the theorems of Bochner and of F. Riesz 147
Now the expression in (3.7) equals the integral
00 2 ( m f e -t /4e: L
-00 \1=1
since f satisfies (3.1) by hypothesis. It follows from (3.7) that fe:
satisfies (3.1). Since f(O) is finite, and (by Lemma 1),
if(t)i.::.f(O), we note that f is bounded, hence fe:EL1(-oo,oo) for every
e: > O.
Lemma 4. Suppose that f is a complex-valued measurable function on
(-=,00) which satisfies (3.1), with f(O) finite. Let f (x) = 2 e:
e-e:x f(x), e: > 0, so that (by Lemma 3) fe: E L1 (-00,00). Then we have
" " f (a) e: .:: 0, fe: E L1 ('-00,00) , -co < a. < 00,
and
(3.8) f (x) J " -iax
e: 21T fe: (a) e da,
for almost all x E (-"",,,,,). In particular, (3.8) holds at every point
of continuity x of fe:' and therefore of f.
Proof. By Lemma 2, f satisfies condition (3.4). In it we choose p,
such that
p (t) -2e:t2 iat e e -ex> < t < 00,
where a is real and fixed. Then (3.4) becomes
2 2 f f f( ) -2e:(x +y) ia(x-y)d d x-y e e x y > o.
If we make the substitutions
x - y u, x + y v,
:then we have
148 III. FOURIER-STIELTJES TRANSFORMS
1 2 2 eiuadu
'2 J J f(u}e-E(u +v } dv
1 (£: 2 2
'2 J e-€V dV) e-€U f(u)eiuadu
1 CO(CO 2) . '2 J J e-€V dv f€(u}e 1uadu > O.
-00 -00
Now f€ E L1 (-co,co), and f€ is bounded on (-co,ao), since f(O} is finite,
and the last inequality shows that f has a positive Fourier transform. ~ €
By Theorem 12 of Chapter I, f€ E L1 (-co,co) and (3.8) follows.
Lemma 5. Let K(a}
so that K E L1 (-co,co), and let K HR(a} = RH(Ra}~ then
for
for
H. If, for R> 0, KR(a} := K(~}, and
J f(x}eiaxdx, as in Ch.I, (8.1)). If f€ is
defined for every € > 0 as in Lemma 3, then we have
(3.9 )
Proof. Since K is even, H is even, so that the left-hand side of (3.9)
equals
By the composition rule (Ch.I, (1.13}), this equals
sinc: f€ E L1 (-co,co) by Lemma 4, KR E L1 (-ao,co), HR E L1 (-co,co), KR is even,
and KR = HR'
Lemma 6. Let f be a complex-valued measurable function defined on
(-oo,ao), which satisfies condition (3.1), and is continuous for x = O.
Let
§3. Positive definite funa"twns: the theorems of Boahner and of F. Riesz 149
(3.10) 2
fn(x) = e-x /nf(x), n~ 1, n integral, -co < X < co.
Then there exists a non-deareasing, bounded funation Vn (t), -co < t < co,
suah that
co co (3.11) 2TI J HR(X-y)fn(y)dy = J -co < x < co.
-00 -co
Proof. If f is continuous at x = 0, then f(O) is finite, and by
Lemma 4, fn(-a)~o, for -co<a<co, f n EL1 (-00,co). If we define
(3.12) 1 t
Vn (t) = 2TI J fn (-a)da, -00 < t < 00, -00
then Vn(t) ~O, and Vn(t) is a non-decreasing function of t for
n = 1,2, •••• If f is continuous at the origin, then, by Lemma 4, we
have
00
fn(O) = 2TI J fn(-a)da, so that
(3.13 ) o ~ V n ( t) ~ f n (0) = f ( 0), f or -co < t < co, n 1 ,2, •••
Now (3.9 ) gives (3.11).
Theorem 5. Let f be a aomp~ex-va~ued measurab~e funation defined on
(-co,co), whiah is aontinuous at the origin, and whiah satisfies aon
dition (3.1). Then there exists a non-deareasing, bounded funation
V(t), suah that
(3.14 ) co
f(x) = f eixtdV(t), -co < x < co
-co
for a~most aZ ~ x E (-00,00). If f is aontinuous everywhere, then (3.14)
ho~ds for every x E (-00,00) •
Proof. We have to let n-+oo in relation (3.11) of Lemma 6 just proved,
and then let R-+oo. Since {Vn(t)} is uniformly bounded, by (3.13), by
a theorem of Helly there exists a subsequence (nk ), and a non
decreasing function V, such that IV(t)1 <f(O), and V -+V pointwise on - nk (-ClO,OO) •
Since KR(a) in Lemma 5 vanishes for lal ~R, and is continuous, we
have, by a theorem of Helly-Bray (see the Notes)
150 III. FOURIER-STIELTJES TRANSFORMS
co
(3.15 ) f -co
for each R> 0, on the right-hand side of (3.11). If we take the left
hand side of (3.11), and let n + co, through the same subsequence (nk )
as in (3.15), we obtain, since Ifn(y}I 2. f n(O} = f(O}, and f(O} is
finite, and HR ELl (-co,co),
(3.16) 1 co
lim --2 f HR(X-y}f (y}dy k+co 1T -co nk
co
21T f HR(x-y}f(y}dy, -co
by Lebesgue's theorem on dominated convergence. From (3.16) and (3.15),
and (3.11), we thus obtain
(3.17) 1 co co iax
21T f HR(x-y}f(y}dy = f KR(a}e dV(a} , (R> O) • -~ -m
We now let R + co. The function KR (a) e iax is continuous in a, and
vanishes outside (-R,R), and KR(a} + 1, as R+ co, for each a, while
co
J IdV(a} I 2. V(+co} - V(-co} < f(O} < co. -co
Hence
co (3.18) lim f
R+co -co
on the right-hand side of (3.17). We shall show that the left-hand
side of (3.17) gives
(3.19) 1 co 1
lim 21T f f(y}HR(X-y}dy = lim 21T (f*HR) (x) = f(x}, R+co -co R+co
for almost all x E (-co,co), so that (3.14) follows, and the theorem is
proved.
Let w > 0, arbitrarily chosen and kept fixed, and let fw (y) = f(y},
for Iyl < Wi while f (y) = 0, for IYI > w. Since f is bounded and w -measurable in (-w,w), fELl (-w,w) i hence fw E Ll (-co,co). By Theorem 10
of Chapter I, we have
lim ~ (fw*HR) (x) R+co 21T
That is to say that
fw (x), for almost all x E (-co,oo) •
§3. Positive definite functions: the theorems of Bochner and of F. Riesz 151
w (3.20) lim 2~ f f(y)HR(x-y)dy = fix),
R+oo -w
for almost all xE [-w,w]. By a further conclusion in Theorem 10 of
Chapter I, this relation holds for every xE (-w,w), if f is everywhere
continuous.
On the other hand, if Ixl < w, we have
(3.21 ) 1--2
1 f f(Y)HR(X-Y)dyl < ~(OR) f Sin2[R(X-Y)~2] dy ~ Iyl.::w - ~. Iyl.::w [(x-y)/2]
-7- 0, as R -7- 00.
Thus (3.21) and (3.20) lead to the relation
( 3.19)
for almost all x E [-w,w], where w > 0 is arbitrary; and for every
xE (-w,w) if f is continuous everywhere. This combined with (3.18) and
(3.17) proves the theorem.
The next result is a kind of converse and easier to prove.
Theorem 6 (Bochner). Let Vet) be a non-decreasing function of t,
-00 < t < 00,. which is bounded everywhere. Then the function
(3.22) fix) = f eixtdV(t), -00 < x < 00,
is continuous, and bounded, and satisfies condition (3.1).
Proof. As already remarked in § 1, f is bounded and continuous. To
verify ( 3 . 1 ) we have only to note that
m m m m 00 it(t -t ) L L fit -t )p P L L P)1P V f e )1 v dV(t)
)1=1 v=1 )1 v )1 v )1=1 v=1 -00
()1~1 it t)( m -it t f P)1 e)1 L Pv e v )dV( t)
'v= 1
m it t 2 f
1)1:1 P)1 e )1 I dV(t) > 0,
152 III. FOURIER-STIELTJES TRANSFORMS
since V is non-decreasing, and the integrand is positive.
(3.23) Definition. A complex-valued function f(x) defined for
-eo<x<eo is said to be positive definite, if it is continuous, and
bounded, and satisfies condition (3.1).
Theorems 5 and 6 yield the following
Theorem 7 (Bochner). In order that a function f(x) defined on (-eo,eo)
may be written as
eo f(x) f
_00
where V is a non-decreasing, bounded function on (_eo,eo), it is
necessary and sufficient that f be positive definite.
Since a positive definite function is, by definition, continuous in
(_eo,=), the proof of Theorem 7 does not require the full force of
Theorem 5. The following theorem is sufficient.
Theorem 8. If f is a complex-valued function defined on (-oo,eo), which
is continuous, and satisfies condition (3.1), then
eo (3.24) f(x) = f
-eo
where V is a non-decreasing, bounded function of tE (-eo,oo) .
Proof. Define for each integer n, n':: 1,
Then we have fn E L1 (-eo,eo), for ~very n, since f is bounded, and by
Lemma 4, with £ = 1/n, we have fn E L1 (_eo,eo), and
(3.25)
A
where fn(-o.),::o, for _eo < 0. <00, for every xE (_eo,eo), since f is con-
tinuous (see Th.11', Ch.I, p. 45).
If we define, as in (3.12),
§3. Positive definite functions: the theorems of Bochner and of F. Riesz 153
t v (t)
n 21T f fn(-a)da, -co < t<co,
then (3.25) becomes
(3.26) f
f(O) .
We consider two cases
Case (i). Let f(O) = 1. Then Vn is a distribution function, for each
n, with fn as its characteristic function. Since fn(x) ->- f(x), as
n->-co, for each x, and f is continuous (at the origin), Theorem 4
implies that Vn converges to a distribution function V (at the points
of continuity of V), whose characteristic function is f. That is to
say, (3.26) implies that
as claimed.
Case (ii). Let f(O) '*' 1. Condition (3.1) implies that f(O) 2.0. If
f(O) = 0, then f is identically zero, and V is a constant. If
f (0) '*' 0, then f (0) > a, and the function g (x) = ~ ~~~ is continuous,
and satisfies condition (3.1), with g(O) = 1. By what has been proved
in Case (i), it follows that
f(x) co
f eiaxd(f(O) .V(a)) f iax e dV 1 (a) ,
where V1 (a) = f(O) .V(a), and V1 is non-decreasing, and bounded.
The above proof yields the following
(3.27) Corollary. Positive definite functions f, with the property
f(O) = 1, are characteristic functions.
An easy extension of Theorem 7 is the following
Theorem 9 (Bochner). Let g E L2 (-co,co), and let
154 III. FOURIER-STIELTJES TRANSFORMS
co
(3.28) fix) I g(x+t)g(t)dt. _co
Then we have
co
fix) = I eixudV(u), -co
where V is non-decreasing, and bounded, in (-co,co).
Proof. Since
which is finite, since IIgl12 is finite, f is bounded.
We also have, for any fixed x,
If(x)-f(x 1 ) 12 ~ Jco Ig(x+t)-g(x 1+t) 1 2dt I co Ig(t) 12dt,
-00
and if we let xl -+- x, then the right-hand side tends to zero, so that
f is continuous.
Since
f(x-y)
we have
m L
)1=1
m L
\!=1
J g(x-y+t) g(t)dt I g(x+t)g(y+t)dt,
fix -x )p p )1 \! )1 \)
co \ m \2 I L g(x +t)p dt~O. -co )1=1 )1 )1
It follows that f is positive definite, and hence, by Theorem 7, that
§4. A uniqueness theorem
For a special kind of Fourier transform, namely
I ixy e (p(y) dy, -co
§4. A uniqueness theorem 155
where tp E L1 (-w, w), for each finite w > 0, we shall prove a theorem of
uniqueness, which generalizes Theorem 7 of Chapter I.
Theorem 10 (Offord). If tp(u) E L1 (-w < u < w), for every finite w > 0, and
if
(4.1) w
lim J tp(u)eiuxdu 0, w-+oo -w
for every real x, then tp(u) = 0, for almost every u E (-00,00).
We first prove three preliminary lemmas.
Lemma 1 (Schwarz). If f(x) is real-vaZued and continuous for a.:::.x::'b,
b> a, and
(4.2) lim f(x+h) + f(x-h) - 2f(x)
h-+O h 2 0,
for every x E [a,b], then f is linear in (a,b); that is to say,
f(x) = Ax+B, where A and B are some constants.
Proof. Let
F(x)
where e ±1, and let
G(x)
[ x-a ] e f(x) - f(a) - - {feb) - f(a)} , b-a
1 F(x) - ~E(x-a) (b-x),
where E > O. Then G is a continuous function of x for x E [a,b], with
G(a) = G(b) = 0; so is F, with F(a) = F(b) = O.
Case (i). If F(x) = 0, for every x E [a,b], the lemma is immediate
since then we have
f(x) feb) - f(a).x + bf(a) - af(b) b-a b-a
Case (ii). If there exists a point C in (a,b), such that F(c) * 0,
choose e in such a way that F(c) >0, and choose E so small that
G(c) > O. Since G is continuous in [a,b], it attains its maximum M,
say, at a point Xo E (a,b), and M> 0, since G(c) > O. We have, because
of (4.2),
156 III. FOURIER-STIELTJES TRANSFORMS
E > O.
But G(xO+h) .::. G(xo )' and G(xO-h) .::. G(xo ) , so that the above limit is
negative or zero, which implies a contradiction. Hence F(x) = 0 for
all xE [a,b], and the lemma follows.
Lemma 2. Let lP(u) E L1 (-w < u < w), for each finite w> 0, and ~et
w (4.3) lim l(w):= lim f lP(u)du o.
w~~ w~~ -w
If lP(u) -+0, as lul-+ oo, then
(4.4) rOO () (sin uh)2 du ) lP.U uh -oo
exis ts for every h > 0, and tends to zero as h -I- o.
w w Proof. Since f lP(u)du
-w assumption (4.3) implies
f w(u)du, where w(u) = lP(u) + lP(-u) , o w that I (w)
in (4.4) equals
f w(u)du-+o, as w-+ oo • The integral o
and this, in turn, equals, by partial integration,
- f l(S) H'(u)du, where H' EL 1 (0,00), H(u) = (Si~ U)2 . o
Since l(S) is bounded, we have by Lebesgue's theorem on dominated
convergence,
= lim f l(S) H' (u)du h-l-O 0
f lim l(S) H' (u)du = o. o h-l-O
Lemma 3. Let lP(u) EL1 (-w<U<w), for each finite w>o; ~et
w lim f lP(u)eiuxdu = 0, W+<Xl -w
for every Y'ea~ x, and lP(u) -+ 0, as I u I -+ oo. Then lP(u)
all u E (-oo,oo).
o faY' almost
§4. A uniqueness theorem 157
Proof. We take, as we may, ~(u)eiux instead of ~(x) in Lemma 2. Then
we have
(4.5) . (sin Uh)2 f ~(u)elUX uh du+O, as hi-O.
Define
F(x) J
where
L (x, u)
~(u) e iUx _ L(x,u)
2 -u
(+iUX, for lui < 1,
0, for I u I > 1 .
du,
For a fixed h '" 0, consider the function
(4.6)
F(x+h)+F(x-h)-2F(x)
h 2 f
f
f
~(u) {ei(X+h)U + ei(x-h)u _ 2 eiUx}du (iuh)2
~(u)
(iuh) 2
iux e { i ~h - i2Uh} 2
e - e du
(Sin(U2h) )2 iux
~(u) e (uh/2) duo
Because of (4.5), and Lemma 1, it follows that F is linear, hence
F(x+h) + F(x-h) - 2F(x) = 0, which implies, in turn, that
. uh 2 (Sln "'2
J ~(u)eiux Uh/2) du =0.
But the integral on the left-hand side is the Fourier transform of a
function belonging to L1 (-00,=). Hence by Theorem 7 of Chapter I, ~(u)
is zero for almost all u E (-=,=) .
We remark that this method of proof goes back to Riemann in his
treatment of the uniqueness of trigonometric series.
Proof of Theorem 10
Assumption (4.1) holds with y+x, and y-x, in place of x (where y is
real), so that it implies that
(4.7) w
f iuy ~(u)e cos ux·dx+O, as w+=,
-w
158 III. FOURIER-STIELTJES TRANSFORMS
for every real y and x. If we set, for fixed y,
iuy -iuy ~(u) = ~(u)e + ~(-u)e ,
then (4.7) becomes
(4.8) w J ~ (u) cos ux·du -+ 0, as w -+ 00, a
for every real x. Now let
v ~l(v) = J ~(u)du
a
v J ~(u)eiuYdu, v> o. -v
Then (4.8) implies, in particular, that
(4.9) ~ 1 (v) -+ 0, as v -+ 00.
Since
w w J ~(u) cos ux·du = ~1 (w) cos wx - ~1(0) + x J ~1 (u) sin ux·du, a a
where ~ 1 (0) 0, (4.9) implies that
w x J ~ 1 (u) sin ux·du -+ 0, as w-+ oo ,
a
which implies that for every x * 0, we have
w J ~ 1 (u) sin ux·du -+ 0, as w-+ oo ,
a
where ~1 (u) -+0, as U-+ oo , by (4.9). By Lemma 3, ~1 (u) = 0, for almost
all uE (-00,00). Since ~1 is absolutely, continuous, ~t follows that ~(u)
is zero for almost all u, hence ~(u)elUY + ~(_u)e-1UY = 0, for almost
all u. On setting y = 0, we have: ~(u) + ~(-u) = 0, or ~(u) = -~(-u) ,
for almost all u. Hence
~(u) 2i ~(u) sin uy,
and therefore ~(u) = a for almost all u E (-00,00) •
, 2 Remark. If ~(u) lU .t e then ~(u) 'I'Ll (-00 < u < 00), although
w lim J ~(u)eiuxdu W-+OO -w
§4. A uniqueness theorem 159
is finite for all x E (-00,00). (See Ex.10, Step (iii), §1, Ch.I).
Another such example is provided by
(jJ(U) =exp(a.u+ieu ), O<a.<1.
Notes
Chapter I
§1. Theorem 1 is a generalization of the basic result on Fourier
series, which states that the Fourier coefficients of an integrable
function tend to zero, which was proved by Riemann for Riemann
integrable functions, and extended by Lebesgue to Lebesgue-integrable
functions. The idea of the second proof sketched in the Remarks follow
ing Theorem 1 is due to Lebesgue, Bull. Societe Math. de France,
38(1910), 184-210.
A peculiar generalization of Theorem 1 has been given by Bochner and
Chandrasekharan in [1], Th.46, Ch.III. See the later definition of
pseudo-characters by Bochner [5] Ch.3, p.53.
A characteristic function in the sense used here is not the same as
in Chapter III, hence the alternative term "indicator function".
For the role of Hermite functions in Fourier analysis, see, for
instance, N. Wiener [3].
The standard work on Bessel functions is Watson's [1]. A short intro
duction is given in Ch.XVII of Whittaker and watson [1].
Example 10 is due to S. Rarnanujan, J. Indian Math. Soc. 11 (1919),
81-87; Coll. Papers, No. 23 [1]. The integral in Step (i) is evaluated
by Cauchy's theorem in Linde16f's book [1]; see p. 49 for the moti
vation of the proof.
§2. For the notion of an algebra, and basic facts about algebras,
see, for instance, G. Birkhoff and S. MacLane: "A survey of modern
algebra", p. 225.
Notes on Chapter I 161
For an introduction to Fourier analysis on groups, see the classic
by Weil [1]; also Loomis [1], Naimark [1], the Appendix in Goldberg
[1], Rudin [1], and Reiter [1], where the contributions of A. Beurling,
I.E. Segal, and others, are described.
§3. For the theory of distributions, in general, with applications,
see the classics by L. Schwartz [1], and I.H. Gelfand and G.E. Shilov
[1]. For distributions in connexion with Fourier transforms, in par
ticular, see Ch.I of Hormander's book [1], also Yosida [1], Ch.VI,
and Donoghue [1].
The function w in (3.12) was introduced by Wiener [1], p.562.
§4. The localization theorem here is motivated by the one on
Fourier series due to Riemann, see Hardy and Rogosinski [1], pp. 39-42,
and Zygmund [1], Ch.II, §6, p. 53, §8.
The examples of Mellin transforms given here are of frequent occurrence
in analytic number theory, see, for instance, the author's book [3].
Several more are given by Titchmarsh [3].
§5. For Poisson's summation formula, see Bochner [1]; his proof
is also given in the author's book [3]. For some special applications
see, for instance, Zygmund [1], Vol.I, Ch.II, §13.
For the theta-relation (5.8), in the general setting of theta-functions,
see, for instance, the author's book [4], where the connexion with
elliptic functions, and the theory of numbers, is elucidated on an
elementary level.
There is also an L2-version in one variable, see Boas [2].
§6. The proof of the uniqueness in Theorem 7 (without the use of
summability and general inversion) can be effected by the use of a
piece-wise linear (trapezoidal) function instead of the function w
see Bochner and Chandrasekharan [1], Ch.I, §6, Th.5.
For a sharper version of the uniqueness theorem due to A.C. Offord
[1], see Th.10, Ch.III.
c,S
§7. The motivation for the summability theorems here is again supp-
162 Notes on Chapter I
lied by the theory of Fourier series, see, for instance, Hardy and
Rogosinski [1], Ch.V, p. 70; Bochner and Chandrasekharan [1], §7.
For properties (7.6) and (7.20) of integrable functions, and for the
definition of the "Lebesgue set", see, for instance, Titchmarsh [2],
§11.6.
Convolution integrals of the type (7.1) are of importance in the
theory of approximation. See Butzer and Nessel [1], where generalized
singular integrals of the type of Cauchy-Poisson, Gauss-Weierstrass,
Fejer, and Bochner-Riesz, are dealt with in detail. For more general
methods, see Stein [3], Stein and Weiss [1].
§8. Example 3, following Theorem 11', is used by C.L. Siegel in his
proof of Hamburger's theorem on the Riemann zeta-function. See, for
instance, the author's book [3], Ch.II, §5.
Theorem 12 is due to Bochner and Chandrasekharan [1], Ch.I, Th.9,
p.20; also p.211, where it is commented upon. A generalization was
later given by Bochner in his book [5], p.25, Th.2.2.1.
Theorems 13 and 14 make it possible to define the Fourier transform
on L2 (-oo,oo) and prove Plancherel's theorem [cf. Ch.II] by starting
from the subspace L1 -n-L 2 .
§9. The systematic use of summability in norm seems to have
originated with Wiener. For Theorem 17 see Bochner and Chandrasekharan
[1], Ch.I, §10, who also proved further results in that direction.
On Weyl's form of the Riesz-Fischer theorem, see !'leyl [1], and Wiener's
[3] remarks; also Stone's [1], p.26; and J. von Neumann's [1], p.
109-111 .
§10. For the central limit theorem, see Cramer [2], Ch.17, §4, and
Feller [1], Ch.VIII, §4. According to Cramer, the theorem was first
stated by Laplace in 1812; a rigorous proof under "fairly general"
conditions was given by Liapounoff; and the problem of finding the most
general conditions of validity was solved by Feller, Khintchine, and
Levy. The proof given here differs only in detail from that given, for
instance, by Dym and McKean [1], Ch.2, §7.
Notes on Chapter I 163
§ 11. Theorem 21 is the analogue, for Fourier transforms, of a
classical theorem on the absolute convergence of Fourier series due to
N. Wiener [2] and P. Levy [1]. See Zygrnund [1], Vol.I, Ch.VI, §5. The
proof given here differs only in detail from that of R.R. Goldberg [1],
Ch.2, §9, which is itself closely modelled on Bochner's proof [2] of
the Wiener-Levy theorem.
§12. Wiener was the first to study "closure" properties of functions
in L1(-~'=) and in L2(-~'~)' and relate them to Fourier transform
theory. See Wiener [2]. The proof given here of Theorem 23 is the same
as Bochner's [2]. An algebraic reformulation of the theorem would be
that every proper closed ideal of L1 (-~,~) is contained in a maximal
ideal. The problem of characterizing the sub-class of functions f in
Ll(-~'ro) which have the property that Sf is the intersection of the
maximal ideals containing it has received attention. For the work of
Beurling and others, see Pollard [1], and Reiter [1]. For generalizat
ions of Theorem 23, see Ch.4 of Goldberg [1], Reiter [1], where further
references can be found, e.g. to Agrnon and Martdelbrojt [1], Malliavin
[ 1 ], and others.
§13. Theorems 24 and 25 are due to Wiener [2]. Theorem 26 is due
to Littlewood [1], and forms the prototype for many of the tauberian
theorems of the period before Wiener. Littlewood's theorem can be
proved directly, and simply, as Ka~amata [1] has shown, by the use of
Weierstrass's theorem on the approximation of continuous functions,
instead of Wiener's theorem on the L1-closure. See Wiener's own re
marks [3], and Wielandt's [1] arrangement of Karamata's proof.
Wiener's work on tauberian theorems has been carried forward notably
by H.R. Pitt [1].
Albert Stadler [1] has recently proved a tauberian theorem, with re
mainder, of the Wiener-Ikehara type (see, for instance, the author's
book [2]), which yields the more refined forms of the prime number
theorem as corollaries. See Wiener's remarks [2], p.93, on this
possibility.
Bochner and Chandrasekharan [1], Th.29, p.54, subsume Karamata's
theorem as part of another theorem which characterizes what they call
the Karamata extension of the kernel e- a . The nature of this extension
in the case of general kernels seems not to be known.
164 Notes on Chapter II
§14. Theorems 27 and 28 are due to Bochner and Chandrasekharan [1],
Ch.I, §15. See Butzer and Nessel [1] Ch.7, for later developments.
Equations (14.1) and (14.2) arise in connexion with the problem of
conduction of heat, see Carslaw [1], §§16,45.
§ 15. The proof of (15.24) given here is the same as the one given
by Bochner in his book [1]; he comments that according to Burkhardt
[1] pp.1165-1173, the cases k = 2,3 are due to Poisson and Cauchy, and
that the "theorem is also not new for k arbitrary". A second proof is
given by Bochner and Chandrasekharan [1], pp.71-74.
The introduction of the spherical mean fx(t) is due to Bochner. He
carried the idea further into the study of multiple Fourier series.
See Bochner [4], followed by Chandrasekharan [1], Chandrasekharan and
Minakshisundaram [1], and [2], Ch.IV, and H. Joris [1]. Important work
with quite different techniques has been accomplished on topics in
multiple Fourier series by E.M. Stein, and others. See, for instance,
Ch.VII of Stein and Weiss [1].
The evaluation of Vk(s) by induction is done, for instance, by Walfisz
[1], p.41.
Chapter II
§2, §3. Different proofs of Plancherel's [1] theorem have been
given by Titchmarsh [2], Bochner [1], F. Riesz [1], Wiener [3],
M.H. Stone [1], p.l04, and Bochner and Chandrasekharan [1]. For the
Remarks following (2.29) see Stein and Weiss [1], p.18. For (3.5)
see Bochner and Chandrasekharan [1], p. 99 .
§4. Theorem 6 is due to Wiener; for Theorem 7 see Bochner and
Chandrasekharan [1], Ch.IV, §10.
§5. Weyl's proof of the inequality under somewhat stronger hypotheses
is given in Appendix I to his book [2]. His proof in the second edition
differs in detail from the one given in the first. The proof given
here differs only in detail from the one given in their book by Dym
and McKean [1].
§6. The Phragmen-Lindelof [1] principle takes many forms. See, for
instance, Calder6n-Zygmund [1], Littlewood [2], p.l07, Titchmarsh [2],
Notes on Chapter II 165
§5.71. For Hardy's theorem, see Hardy [1], and Titchmarsh [3], p.174,
where further references are given.
§7. Paley and Wiener [1] were the first to make a systematic study
of Fourier transforms in the complex domain (one variable). The proof
given here differs only in detail from the one presented by Dym and
McKean [1], Ch.3. For functions of exponential type see, for instance,
Boas [1].
§8. For generalities on Fourier orthogonal series see Kaczmarz and
Steinhaus [1], Ch.II, where several examples of orthogonal systems are
given, including Rademacher's [1], and Walsh's [1] which can be defined,
after Paley [1], in terms of Rademacher's functions. Let XO(t) = 1,
and if N is a positive integer, expressed in the binary scale as n 1 n 2 .
N = 2 + 2 +. .. , Wl. th n 1 > n 2 > ••• , then XN (t) = (jl (t)·(jl (t) ... , n 1 n 2
where the ((jln) are Rademacher's functions. The system (Xn ) is ortho-
normal over (0,1), and complete, and is known as Walsh's.
Bessel's inequality (in several variables) has been shown by Siegel
[1] to yield ~1inkowski's first theorem on lattice points in convex
sets [cf. the author's book [2], p.99]. Atle Selberg [1] has shown
that Bombieri's large sieve inequality can be viewed as a form of
Bessel's inequality in a Hilbert space, cf. H.L. Montgomery [1].
For a concise introduction to Fourier trigonometric series in L2 (O,2n)
see Hardy and Rogosinski [1]. Series and integrals can be treated
together on a group space. See Butzer and Nessel [1]. For Lebesgue's
proof of the completeness of the trigonometric system, see Hardy and
Rogosinski [1], or Zygmund [1], Vol.I, Ch.I, §6.
§9. Hardy's [2] interpolation formula is also treated in Zygmund
[1], Vol.II, p.276. As Dr. Albert Stadler has remarked, the condition
of boundedness in Theorem 12 can be replaced by one of polynomial
growth, in which case formula (9.7) will assume a more general form.
§10. S. Bernstein's work [1] is also presented in Zygmund [1], Vol.
II, p.11, Ch.X, and p.276, Ch.XVI. Zygmund's inequality for the inte
grated derivative of a trigonometric polynomial, as a generalization
of Theorem 14, is given by him immediately after Bernstein's result.
N.G. de Bruijn [1] has given generalizations of Bernstein's theorem for
166 Notes on Chapter III
polynomials in the complex domain. For the Remark preceding Theorem
14, see Siegel [1]. See also Stein [2].
§11. For the extension of the Paley-Wiener theorem to k dimensions,
k > 1, see Plancherel and Polya [1], Stein [2], Stein and Weiss [1],
Ch.III, Th.4.9. The last-mentioned reference connects the theorem with
the analysis of HP-spaces. See Narasimhan [1], Ch.3, for the role of
the Fourier transform in analytical problems on manifolds; also
Ehrenpreis [1] in connexion with several complex variables.
Chapter III
§1. For the basic theory of Stieltjes integrals see, for instance,
Burkill and Burkill [1], Ch.6, and Widder [1], Ch.I.
§2. As Zygrnund has remarked, the essence of Theorems 3 and 4 is a
classical result of the calculus of probability, in a form
strengthened by Cramer. See Zygmund [1], Vol.II, Ch.XVI, Th.{4.24),
p.262. See also Cramer [2], Ch.10. Bochner has a generalization to Ek ,
see Th.3.2.1 of his book [5], p.56.
For Helly',s theorem used in the proof of Theorem 4, see, for instance,
Widder [1], Ch.I, §16, Th.16.2.
§3. Theorems 6 and 7 are due to Bochner, see [1], Th.23. He refers
to previous work by F. Bernstein and M. Mathias. The generalization,
without the assumption of continuity, is due to F. Riesz [1], who
uses for the proof, however, his theorem on the representation of
positive linear functionals, which is not used here in the proof of
Theorem 5. For the Helly-Bray theorem used, see, for instance, Widder
[1], p.31, Th.16.4. It is not necessarily true when the interval of
integration is infinite, as Widder makes clear, hence the introduction
of the kernel KR{X). Carleman [1], p.98, gives a proof of Bochner's
theorem using the Poisson integral representation of functions which
are positive and harmonic in a half-plane. A proof of the latter (see,
for instance, Verblunsky [1]) can be obtained by using Herglotz's
theorem [1] on the representation of positive, harmonic functions in
a circle (which is stated, for instance, in Stone [1], p.571), or more
directly, as has been done by Loomis and Widder [1] using the theorems
of Helly, and of Helly-Bray. It should be remarked, however, that all
Notes on Chapter III 167
these representation theorems are, more or less, of the same order
of difficulty as Bochner's theorem, or Stone's spectral theorem [1],
p.331, as was early recognized by F. Riesz.
Apropos Corollary (3.27), see Cramer [1]. For Theorem 9 see Bochner
[3], p.329. Bochner has also a generalization to Ek , [5], Theorem
3.2.3, p.58.
For a generalization to distributions, see Schwartz [1], Vol.II, p.132,
Th.XVIII; Schwartz makes a reference to Weil [1], p.122.
§4. Lemma 1 is due to H.A. Schwarz. It is quoted by G. Cantor,
J. fur Math. 72 (1870), 141; and is given by Schwarz himself in his
Ges. AbhandZungen, II (1890),341-343, with a reference to Cantor's
quotation. Prof. Raghavan Narasimhan has remarked that a rearrangement
of Schwarz's argument is better adapted to generalization. "If f is
real-valued, and continuous on (a,b), and lim sup (~~f) (x) = 0, for all 2 -2 h+O
xE (a,b), where ~hf(x) = h {f(x+h)+f(x-h)-2f(x)}, then f is linear. To
prove this, it is sufficient to prove that if lim sup ~~f ~ 0, then f h+O
is convex (i.e. if R.(x) = cx+d, and f(a).s. R.(a), f(B).s. R.(B), where
a < a < B < b, then f (x) < R. (x) for a < x < B) ,since one can argue similarly
with lim inf ~~(-f). ;ince ~~R. = ~ f~r any linear function R., it is h+O
enough to prove that if lim sup ~~f~O, and f(a) .s.0, fiB) .s.0, then h+O
f(x) < 0 on [a,B], i.e. that f has no maximum on (a,B). - 2
f (x) + £x , £ > 0, one has only to show that if lim sup h+O
Replacing f(x) by 2
~hf > 0, on (a,b),
then f has no local maximum on (a,b). But this is obvious, for if Xo is
a local maximum, then ~~f (xO) .s. 0 for h small enough, since f (xO) ~ f (xO +h), f (xO) ~ f (xO-h) ." Cf. Narasimhan [2], p. 21-25.
Theorem 10 is due to A.C. Offord [1], and is the integral analogue of
Cantor's fundamental theorem on the uniqueness of trigonometric series,
which asserts that if a trigonometric series converges everywhere to
zero, it vanishes identically; all its coefficients are zero. See, for
instance, Zygmund [1], Vol.I, Ch.IX, p.326. Offord also proved [2] a
stronger theorem in which the hypothesis of convergence of the inte
gral in (4.1) is replaced by (C,1) surnrnability. Offord shows that the
stronger theorem is a "best possible", in the sense that even one
exceptional point cannot be permitted, and (C,1) surnrnability cannot be
relaxed to (C, 1+£) surnrnability for any £ > O. Zygrnund' s proof [1], Vol.
168 Notes on Chapter III
II, Ch.XVI, §10, of Offord's first theorem is based on an equicon
vergence theorem for trigonometric integrals and series which he
treats in Vol.II, Ch.XVI, §9, and on results from Riemann's theory of
trigonometric series which he treats in Vol.I, Ch.IX. For the use of
equiconvergence theorems in analytic number theory, see, for instance,
the author's book [3], Ch.VIII. A generalization of Offord's stronger
theorem to several variables would be of interest, though perhaps not
easy. An equiconvergence theorem for trigonometric integrals in two
variables has been given by H. Keller [2]. Functions of bounded
variation in two variables come into play, and it is a moot question
whether the notion of Vitali variation could be replaced by that of
Frechet, as Morse [1] and Transue did in another context.
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Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong
K. Chandrasekharan
Elliptic Functions 1985. 14 figures. XI, 189 pages. (Grundlehren der mathematischen Wissenschaften, Band 281). ISBN 3-540-15295-4
The first part of the book provides a selfcontained account of the fundamentals of the theory of elliptic functions of Weierstrass and of Jacobi. The close connection with the theory of theta functions and Dedekind's 17-functions is also explained. The proofs ofthe arithmetical results in the second part are so modelled as to exhibit clearly the analytical relations on which they are based: examples are Euler's theorem on pentagonal numbers, and Gauss' law of quadratic reciprocity. The proofs are arranged so as to enable the reader to recognize some of the motivation behind Siegel's analytic theory of quadratic forms, which in addition requires his theory of arithmetical reduction. No special knowledge of the theory of numbers is assumed. Only an acquaintance with the elementary theory of analytic functions and the theory of groups and matrices is presupposed. Both as a text that may be used by students and as a reference for researchers, this volume provides a wealth of relevant and useful material.
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong
K. T.Smith
Power Series from a Computational Point of View Universitext 1987.2 figures. VllI, 132 pages. ISBN 3-540-96516-5
Contents: Taylor Polynomials. -Sequences and Series. - Power Series and Complex Differentiability. - Local Analytic Functions. - Analytical Continuation. - Index.
The purpose of this book is to explain the use of power series in performing concrete calculations, such as approximating definite integrals or solutions to differential equations. This focus may seem narrow but, in fact, such computations require the understandirlg and use of many of the important theorems of elementary analytic function,theory, for example Cauchy's Integral Theorem, Cauchy'S Inequalities, and Analytic Continuation and the Monodromy Theorem. These computations provide an effective motivation for learning the theorems, and a sound basis for understanding them.