Classical Fourier Transforms

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

2141/3140-543210 - Printed on acid-free paper

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).

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 -~ ,

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

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


-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

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.

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

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

§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

§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

(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)


(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


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


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

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


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


(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.

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

'" 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+£

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


~

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.

~

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

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.

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

(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

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.

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

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)


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)


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) ,


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.


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,

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 - 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 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 - 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 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

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

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.

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

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


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)

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) •

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,

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),

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


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 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),

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 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

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

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

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.

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