Introduction to Fourier Analysis and Generalised Functions

INTRODUCTION TO

'\

FOURIER ANALYSIS

J

AND

GENERALISED FUNCTIONS

BY

M. Di~ecto~,

J. LIGHTHILL, F.R.S.

Royal

Ai~c~aft Establishment,

Farnborough

CAMBRIDGE AT THE UNIVERSITY PRESS

Ig64

PUBLISHED BY THE SYNDICS OF THE CAMBRIDGE UNIVERSITY PRESS

Bentley House, :l00 Euston Road, London, N.W. 1 American Branch: 3:l East 57th Street, New York 2:l, N.Y.

© CAMBRIDGE UNIVERSITY PRESS

1958

TO

PAUL DIRAC who saw that it must be true, Fi~st P~inted Rep~inted

1958 1959

LAURENT SCHWARTZ

1960 1962 196 4

who prooed it, AND

GEORGE TEMPLE who showed how simple it could be made

First

p~inted

R~~inled

in Great

B~itain

at the

by offset-litho by Lowe

Unive~sity P~ess, Camb~idge


Ltd, London

CONTENTS Introduction page

1.1

Scope and purpose of the book

1.2

Knowledge assumed of the reader

1.3

Fourier series: introductory remarks

3

1.4

Fourier transforms: introductory remarks

8

1.5

Generalised functions: introductory remarks 2

J

2

10

The theory of generalised functions and their Fourier transforms

2.1

Good functions and fairly good functions

J5

2.2

Generalised functions. The delta function and its derivatives

16

2.3

Ordinary functions as generalised functions

21

2.4

Equality of a generalised function and an ordinary function in an interval

25

2.5

Even and odd generalised functions

26

2.6

Limits of generalised functions

27

3

Definitions, properties and Fourier transforms of particular generalised functions

3.1

Non-integral powers

30

3.2

Non-integral powers multiplied by logarithms

34

3.3

Integral powers

35

3.4

Integral powers multiplied by logarithms

40

3.5

Summary of Fourier transform results

42

4.1

The Riemann-Lebesgue lemma

46

4.2

Generalisations of the Riemann-Lebesgue lemma

47

4.3

The asymptotic expression for the Fourier transform of a function with a finite number of singularities

5I

4-

The asymptotic estimation of Fourier transforms

Vlll

CONTENTS

5 5·1 5:Z

.

5·3

Fourier series

C:~~~g:~;en:~~Ii~~~:~~~n;f trigonometrical

CHAPTER I

series as

page 58 . e ennmatlon of the coefficients in at' ngonometncal series . 60 EXistence of Fourier-series re . generalised function presentation for any periodic D t

.

.

5·4

Exam.ples. Poisson's sununation fonnula

5·S

Asymptotic behaviour of the coeffic' t ' F' . len sma OUner senes

71

INDEX

77

INTRODUCTION x.x. Scope and purpose of the book

This book grew out of an undergraduate course in the University of Manchester, in which the author attempted to expound the most useful facts about Fourier series and integrals. It seemed to him on planning the course that a satisfactory account must make use of functions like the delta function of Dirac which are outside the usu~l scope of function theory. Now, Laurent Schwartz in his The01'ie des distributions'" has evolved a rigorous theory of these, while Professor Temple has given a version of the theory (Generalised functions)t which appears to be more readily intelligible to students. With some slight further simplifications the author found that the theory of generalised functions was accessible to undergraduates in their final year, and that it greatly curtails the labour of understanding Fourier transforms, as well as making available a technique for their asymptotic estimation which seems superior to previous techniques. This is an approach in which the theory of Fourier series appears as a special case, the Fourier transform of a periodic function being a 'row of delta functions'. The book which grew out of the course therefore covers not only the principal results concerning Fourier transforms and Fourier series, but also serves as an introduction to the theory of generalised functions, whose general properties as well as those useful in Fourier analysis are derived, simply but without any departure from rigorous standards of mathematical proof. On the other hand, the application of Fourier transforms, or of generalised functions, to the solution of differential, integral or other functional equati:>ns is not explicitly treated, nor is the extension to Fourier transforms or generalised functions of more than one variable. .. Volumes

t

I

and

2

(1950-1). Paris: Hermann et Cit:.

Proc. Roy. Soc. A, 228, 175-90 (1955). LFA

2

FOURIER ANALYSIS AND GENERALISED FUNCTIONS

1.2.

Knowledge assumed of the reader

The book is written for mathematical readers with some interest in, and general knowledge of, methods of mathematical proof (particularly of results concerned with limiting processes). However, little detailed knowledge of particular topics is assumed. The Lebesgue theory of integration is not required; the expression 'absolutely integrable' is frequently used below (where in the Lebesgue theory one would write' integrable ') to remind the reader familiar only with a simpler approach to integration that the integral of the modulus is assumed finite. The theorems that the modul~s of an integral is less than the integral of the modulus, and that orders of integration and/or summation can be interchanged in an expression in which convergence is retained when each term is replaced by its modulus, are constantly used, however; and one or two other basic integration theorems are quoted occasionally. The 0 and 0 notations are assumed known and understood. * The functions dealt with are functions of a real variable only (though they have complex values). On the other hand, some knowledge of functions of a complex variable is desirable, as use is made on a few occasions of the evaluation of simple definite integrals by contour integration. Familiarity with a few other standard integrals, like the error integral, is also assumed. All this material is in the ordinary undergraduate course of analysis in British universities. Accordingly, the book should be intelligible both to undergraduates and to working mathematicians whose study of analysis may not have gone much beyond the undergraduate level. It is,'perhaps, unlikely that anyone will come to the book without any previous knowledge of Fourier series or integrals. However, the account of these topics in chapters 2 to 5 is completely selfcontained, and in addition some indication is given, in the following sections of this chapter, of some of the principal uses to which Fourier analysis may be put. .. Those remembering these notations imperfectly are recommended to read (; as 'a term of order at most ... '. and 0 as . a term of order less than ... '. This will remind them that 'f = G(g)'. meaning that Ifl < A Ie I for some positive A as' the limit is approached. includes the possibility that 'f=o(g)'. meaning that If I < € Ie I (sufficiently near to the limit) for atlY positive € (however small).

INTRODUCTION

3

1.3. Fourier series: introductory remarks A Fourier series is a representation of a periodic function f(x) (say, of period 21)* as a linear combination of all those cosine and sine functions which have the same period, say as 00

f(x) =tao+ L

n=1

n1TX

n1TX

00

a.. cos -1- + L

(I)

bnsin -1-'

n=1

Fourier series in this sense are used for analysing oscillations periodic in time, or waveforms periodic in space, and also for representing functions of plane or cylindrical polar co-ordinates, when x in (I) b'ecomes the polar angle 8, and the period 21 becomes 21T. The series can be written, more compactly, in the complex form 00

f(x) =

L n=-c:o

Cn

e i1U1x/l,

(2)

where cn = Han - ibn) for all n, if a-n signifies an and b_n signifies -bn (so that bo=o). One great advantage of expressing a function in terms of cosines and sines, or (even more) in terms of exponentials, is the simple behaviour of these functions under the various operations of analysis, notably differentiation. For example, if a linear partial differential equation has coefficients independent of x, and solutions periodic in x with period 21 are required, the series (2) may be used as a solution, and the Cn determined by solving differential equations in which derivatives with respect to x no longer occur. For example, solutions of Laplace's equation in plane polar co-ordinates r, 8, which are periodic in 8 with period 21T (thus, representing solutions which are one-valued in an annulus with 00

centre the origin), may be written as

L cnCr) einQ • If we substitute n=-oo

this in

c2j

1

of

1

o2f

or2 + r or + Y2 082 =0, and assume that we can differentiate term by term, we obtain

;

~

n=--(X)

(d Cn ~ dCn_~ ) inQ_ dr2 + r dr r2 Cn e - 0 . 2

• This means thatf(x}=f(x+2l) for all x. 1-2

4

5

FOURIER ANALYSIS AND GENERALISED FUNCTIONS

INTRODUCTION

If we assume that expressions of functions by such trigonometrical series are unique, then a series which vanishes identically must have vanishing coefficients, which gives a differential equation for cn' whose gener~l solution is n Cn =A"r + Bnr"'· (5) If boundary conditions are available on circles r = constant, for examplef=g(O) on r=a, and say of/Or = h(O) on r=b, we may use these conditions to determine An and B n, provided that g(O) a;d h(O) can be represented as Fourier series, say

x = 0 and x = l, then the argument is usually presented as follows. The only cosines or sines which satisfy the boundary condition f=o both at x=o and at x=l are sin1l1Tx/l (for n= I, 2, 3, ...), so that the half-range series (or' Fourier sine series ') 00 n7TX (8) f(x)= 1: b"sin -l- (o<x
Then

00

00

g(O)= 1: gnelnO,

h(O)= 1: h"e inO•

n=-oo

n=-oo

A"an+B"a-n=gm

nA"b'lt.-l_ nB n b-n-l=hn ,

(6) (7)

whence A" and B" may be determined. This example is so simple that it could be treated in many different ways (and, in fact, the conclusions just obtained agree with those given by Laurent's theorem), but, clearly, the procedure is directly applicable in more complicated problems, provided always that the boundary conditions from which the c" are to be determined are given on boundaries where the argument of the Fourier series (here 0) varies independently of the other variables. The example makes it clear that a satisfactory Fourier-series theory will be one in which term-by-term differentiation and unique determination of the coefficients for a given function are both possible. These two requirements had never been simultaneously satisfied by any of the Fourier-series theories until the 'generalised function' approach given in chapter 5 was developed. There is a different, and perhaps even commoner, way in which Fourier series are used, namely to represent a function which is not periodic, but instead is defined in the first place only in a restricted interval. The period of the Fourier series is usually taken as twice the length of the interval, and then the series are called' half-range series'; also, 'quarter-range series' are sometimes used. It will be seen that these series form a restricted class of Fourier series, in which only a proportion of the terms in the' full-range series' (I) is used. For example, if a partial differential equation is to be solved in a region part of whose boundary consists of the lines (or planes)

'11.=1

is appropriate when these are the boundary conditions. Alternatively, if of/ox=oatx=o andx= l, the half-range series (or 'Fourier cosine series ') 00 n7TX f(x)=tao+ 1: ancos-(o<x
z

(

l) o<x<,

(10)

containing an even smaller selection of the terms in a Fourier series of larger period fl, is appropriate. These series can be approached in a slightly different but more useful manner as follows. To satisfy boundary conditions f( 0) = 0 and f(l)=o, an odd periodic function f(x) of period 2l (that is, a function satisfyingf( -x)= -f(x) andf(x+ zl) =f(x), which by the substitutions x = 0 and x = -l imply the stated boundary conditions) is introduced. Its Fourier series (I) then contains only odd terms and reduces to (8). Similarly, the boundary conditions of/ox = 0 at x = 0 and x = l can be satisfied by using an even periodic function. Finally, the boundary conditions f=o at x=o and of/ox = 0 at x = l can be satisfied by using an odd periodic function, of period 4l, which is also an even function of x -l; note that only those sine terms of period 4l which satisfy the latter condition appear in (10). In each case, naturally, it is the value of the periodic function in the range 0 < x < l that represents the solution to the problem. A simple example is now given to show the advantage of replacing boundary conditions wherever possible by conditions of periodicity and parity, even in a case where Fourier series are not used. If a string is stretched between two fixed points x = 0 and x = l and

6

FOURIER ANALYSIS AND GENERALISED FUNCTIONS

INTRODUCTION

plucked (that is, released from rest in a given distorted shape), we can imagine an infinite stretched string whose traAsverse displacement y=f(x) is periodic with period 21 and odd and agrees with the given shape for 0 < x < I (see fig. I). If this infinite string were released from rest in this position, the displacement y would remain odd and periodic, and so continue to satisfy the boundary condition y = 0 at x = 0 and x = l. Hence the simple solution of the initial-value problem for the infinite string, namely

sufficient condition for convergence in the sense of generalisedfunction theory is that cn = 0(111 IN) as 111 I~0Ci for some N. Next, if an equation such as (2) is gi·ven, one may ask how the cn may be expressed in terms of the function f(x). For example, when the series is absolutely and uniformly convergent, one may multiply both sides by e-imrrx/l and integrate term by term from -I to I, giving c = 2. ~(x) e-imrrx/l dx, (12) m 21 -z"

(II)

y=t{f(x+ct)+f(x-ct)},

can be used to give the solution, avoiding the need to consider multiple reflexions from the ends. Similar advantages accrue if other problems of this kind are treated in this way.

I!

since all the terms of the series vanish except that for which 11 = m. The corresponding problem in the theory of generalised functions is solved in § 5.2. Note that equation (12) implies am =

(x)

7

II!

II!_lex) cos -1- dx,

I

m1TX

.

m1TX

bm=l _/(x)sm -1- dx.

Note also that iff(x) is even then 2J!

x -21

-I

o

m1TX

am=Z /(x)cos-I-dx, 21

which are the equations for a Fourier cosine series, while if f(x) is odd then 2J! . m1TX am=o, bm =7 /(x)sm -1- dx, (15) Fig.

I.

Illustrating the construction of an odd periodic function lex) taking given values in 0 < x < I.

Accordingly, it is best to consider every Fourier series as the Fourier series of a periodic function, even in cases where one is interested in the first place only in values over a half- or quarterperiod. This is especially advisable in constructing any general theory of Fourier series, since the sum of such a series, if it exists, is certainly periodic! We may conclude this section by listing the principal aims of a theory of Fourier series. First, one must obtain conditions under which a trigonometrical series like (I) or (2) converges (in the sense which one is using). For example, a sufficient condition for absolute uniform convergence is that Cn = 0(lnl-1-C) as /nl ~co for some e> o. In §5. I, however, it will be shown that a necessary and

which are the equations for a Fourier sine series. Next, one may choose an arbitrary periodic function f(x), and ask under what conditions equation (2) will hold with the Cn given by (12), or by the equivalent expression in the theory one is using. Elaborate tests must be satisfied in the ordinary convergence theory for this to be so, but it will be proved in § 5.3 that the equation holds for all periodic generalised functions. Finally, one may ask the questions already mentioned concerning term-by-term differentiability and uniqueness. These are the properties which are most notably lacking in the ' convergence' and 'summability' theories, respectively;* but both results are almost trivial in the generalised-function theory. " For these classical theories see, for example, G. H. Hardy and W. ,V. Rogosinski, FOl/rier Series (2nd edition, 1950), Cambridge University Press.

8

INTRODUCTION

FOURIER ANALYSIS AND GENERALISED FUNCTIONS

x+ Fourier transforms: introductory remarks The Fourier integral may be regarded as the formal limit of the Fourier series as the period tends to infinity. Thus, if f(x) is any function of x in the whole range ( - co, (0), one can form a periodic function fi(x) of period 2i which agrees with f(x) in the range ( -I, l). The Fourier series (2) of fi( x), with the expressions (12) for the en, can be written in the form

fi(x)=

i

n=-oo

eintTXIZgZ(!!.-) 2.-, 2i 2i

where gb)=fl e-ZrriXYf(x).dx. -I

(16) A formal limit as /-+co, where in the series n/2i is written y, and the difference between successive values of y is written dy, is

f(x) = f~co e2rriXVg(y)dy,

where

g(~)=f~co e-2rriXYf(x)dx, (17)

since in the limit the periodic function fi(x) becomes f(x) everywhere. Under these circumstances the function g(y) is often called the Fourier transform (F.T.) of f(x). Then the first of equations (17) may be regarded as stating thatf(y) is the F.T. ofg( -x). The reader should be warned, however, that no general agreementhas been reached on where the 27T'S in the definition of Fourier transforms should be put. They can be taken out of the exponentials in equations (17), anJ a I/27T factor inserted before the first integral but not the second (or vice versa), or else a factor I/.J(21T) can in this case be inserted before each (to maintain as much symmetry as possible). All the different notations which have been used have some advantages. Here we follow Temple, and many modern authors, in including the 21T in the exponent, so that the exponential multiplying g(y) represents an oscillation with y as frequency lor wave number, according as x represents time or space), rather than as 'radian frequency' or 'radian wave number'. However, the results in this book are easily changed into one of the other notations by a slight change of variable. A vast literature* has been devoted to the determination of con• See, for example, E. C. Titchmarsh, Introduction to the theory of Fourier integrals (J937), Oxford University Press.

9

ditions on f(x) sufficient for equations (17) to be true with a gi.ven interpretation of the integrals. Even for a fixed interpret~tlOn, many alternative sets of sufficient conditions are necessary ~f one wishes to apply the equations at all widely, because relaxatIOn of one condition to admit some desired function requires usually the strengthening of some other condition, which in turn excludes s~me other function. All these difficulties disappear when generalIsed functions are used, since every generalised function f(x) has a Fourier transform g(y) which is also a generalised function, ~nd the F.T. of g( -x) isf(y). In the latter theory it is fo~nd ~onvement to proceed in an order different from that adopted m. thIs chapter, and to treat the properties of Fourier series as a special case of the properties of Fourier transforms. ..' The Fourier integral is used to analyse non-penodlc functlO~s of x in the range ( - co, co), as linear combinations of exponential functions. Such an analysis is useful for much the same reasons as with Fourier series. For example, it is effective in treating linear partial differential equations with coefficients indep~ndent of x, subject to boundary conditions given on boun~anes where x varies from - co to co independently of the other vanables. To apply these boundary conditions, it is necessary to be able to express any functions occurring in them as Fourie~ integrals. Here, a difficulty used to be that quite simple fun~tlOns (for ~xample, a constant!) have no Fourier transform in ordmary. functIOn ~heor~. This difficulty disappears in the theory of generalised functIOns, In which, for example, the F.T. of I is the delta function of Dirac. 'Half-range' Fourier integrals are also used, along much. the same lines as with Fourier series. Thus, iff(x) is to be determmed in the range (0, co) subject to a condition f( 0) = 0: one ~a! seek .an odd functionf(x) in the full range (-oo,co), whIch cOlI1cldes With f(x) in (0,00). Its F.T. g(y), by (17), may be written as

g(y) =

-

2i

tXt f(x) sin 21Txydx,

(18)

which is also an odd function, so that the expression for f(x) in terms of g(y) becomes f(x) = 2i f: g(y) sin 21Txydy. The integrals in (18) and (19) are called Fourier sine integrals.

(19)

\

! 10

Similarly, ifj(x) is to be determined in (0,00) subject to a condition oj/ox=o when x=o, one may seek an even function j(x) in the full range (-00,00) which coincides withj(x) in (0, eo). Its F.T. may be written as

g(y) =2 J~ j(x) cos 21TXydx,

II

INTRODUCTION

t

FOURIER ANALYSIS AND GENERALISED FUNCTIONS

I

6

5

(20)

which is also an even function, and the expression for j(x) in terms of g(y) becomes j(x) =2 J~ g(y) cos 21Txydy. (21) The integrals in (20) and (21) are called Fourier cosine integrals. As with Fourier series, no special theory is needed for Fourier sine integrals and Fourier cosine integrals. They should be regarded simply as what is obtained by taking Fourier transforms of odd and even functions respectively. In many cases, especially when it is not possible to evaluate a Fourier transform explicitly in terms of tabulated functions, it is useful to have a technique for evaluating the asymptotic behaviour of the F.T. g(y) as Iy 1-+00, in terms of the behaviour of j(x) near its singularities. It is difficult to find a comprehensive account of this technique in the literature, and since the theory becomes particularly simple when generalised functions are used, a substantial fraction of this book has been devoted to expounding it. This theory can also be applied without change (§ 5.5) to the problem of deterr.:ining the asymptotic behaviour as In 1-+00 of the coefficients en in the Fourier series for a given function.

1.5. Generalised functions: introductory remarks 1.()

The first' generalised function' to be introduced was Dirac's 'delta function' 8(x), which has the property

J:<X> 8(x) F(x) dx=F(o)

(22)

-1,0

-0,5

0

0·5

( h ample 6) used to define Sex); Fig. 2. Functions in the sequen)c,: \:P~:/;o e;he graph of the 11th function. the number" (for 11=4,20,100 IS a ac

every point, except x = 0 where it tends to infinity. In the limit, this sequence would have the property (22). . . Again, one might 'differentiate' 8(x) to obtam a functiOn 8 (x) I

for any suitably continuous function F(x). No function in the ordinary sense has the property (22), but one can imagine a sequence of functions (see, for example, fig. 2) which have progressively taller and thinner peaks at x = 0, with the area under the curve remaining equal to I, while the value of the function tends to 0 at

with the property

reo

8'(x)F(x)dx= -

reo

o(x)F'(x)dx= -F'(o)

12

FOURIER ANALYSIS AND GENERALISED

FUNCTIONS

INTRODUCTION

ongm. Similarly, 8'(x) corresponds to a dipole of unit electric moment, since as a special case of (23) we have

60

f:", 40

..---==~:!:2:':=Z::::~~-=::::::::;;1::::::::::;;:~~===---2i1'0 -0'5

-1{,

13

4

-60

Fig. 3· Functions"in b" th " the sequence used to define 8'(x) ,emge d envatlves of those graphed in fig. 2.

for any con~inuousl~ d~erentiable function F(x). Behaviour like (23) can again be.rea~Ised In the limit of a sequence offunetions (for ;(xa)mple, the denvatIves of those in the sequence used to represent x ; these are graphed in fig. 3). al Physically, ~(x) c~n be regarded as that distribution of charge ong the X-axIS whIch one speaks of as a unit point charge at the

x8'(x) dx= -

I.

Thus, these generalised functions correspond to familiar physical idealisations. The definition by means of a sequence is in fact that to be adopted in chapter 2, following Temple (who in this point follows Mikusinski). Alternative definitions are not considered here; the reader is referred for a historical account of the subject to Temple, j. Lond. Math. Soc. 28,134-48 (1953). In defining generalised functions by means of sequences, one must define under what circumstances two sequences constitute the same generalised function. For this purpose one multiplies each member of a sequence by a 'test function' F(x) , as in (22) or (23), integrates from - 00 to 00, and takes the limit. If the same result emerges for each sequence whatever' test function' is used, the sequences are said to define the same generalised function. Different classes of test functions, and of functions admitted for membership of the sequences, can -be used in different versions of the theory, but there is only one satisfactory choice when Fourier transforms are to be used. We will not tolerate any restrictions on the differentiability of generalised functions; therefore, as an equation like (23) indicates, both the functions admitted as members of sequences, and the test functions, must have derivatives of all orders. But the classes must be such that the Fourier transform of a member of either class is also a member of that class. The widest class satisfying both these conditions is that introduced in definition I (at the beginning of the next chapter). Therefore, we allow both the members of the sequences and the test functions to be arbitrary members of this class. We call members of this class' good functions', as a graphic term, which it seems desirable to introduce in preference to 'test functions' since it is not only the test functions which are to be restricted to membership of this cbss. The development of the theory of generalised functions in chapter 2 follows closely the lines suggested by Temple, but with the omission of a clause in the definition of a generalised function

14

IS

FOURIER ANALYSIS AND GENERALISED FUNCTIONS

which requires the integral of its product with a test function to be a •continuous linear functional' of the test function in a certain sense. To keep this restriction would introduce considerable complication~ into the proofs in the earlier stages of the theory, and it appears possIble to proceed satisfactorily without it. Only at one point (theorem 24 in chapter 5) has it been necessary to prove a result which is actually a special case of Schwartz's general theorem tha~ a linear functional must (in fact) be continuous in this sense, and the short hold-up here seems amply compensated for by the increased simplicity of the general theory. Another small difference from Te~ple's ~ap:rs is that the author has not found the concept of the mdefilllte mtegral of a generalised function valuable, and has accordingly omitted it. Although the delta function was the first generalised function to be introduced, the methods of attaching values to integrals and series which are introduced in the theory had much earlier forerunners, like Cauchy's •principal value' and Hadamard's •finite part' ?f an improper integral, and the theories of 'summability' of senes. The connexions with these topics are briefly mentioned in chapters 3 and 5.

CHAPTER

2

THE THEORY OF GENERALISED FUNCTIONS AND THEIR FOURIER TRANSFORMS

Good functions and fairly good functions DEFINITION 1. A good function is one which is everywhere differ-

2.1.

entiable any number of times and such that it and all its derivatives are 0(1 x

I-N ) as I x I-+CXJ for all N.

EXAMPLE

1.

e-x ' is a good function.

DEFINITION 2. A fairly good function is one which is everywhere differentiable any number of times and such that it and all its derivatives are 0(1 x IN) as I x 1-+00 for some N.

EXAMPLE 2. Any polynomial is a fairly good function. THEOREM 1. The derivative of a good function is a good function. The sum of two good functions is a good function. The product of a fairly good function and a good funct·ion is a good function. The proof is left to the reader. THEOREM

2.

If f(x) is a good function, then the Fourier transfomx

(F.T.) off(x), namely g(y) =

J:J(x)

e-27TjXll

dx,

(I)

is a good function.

PROOF. Differentiation p times and integration by parts N times shows that I g
~fOO (27TlY)

-<X)

N

dd X

N

{( -

27Tix)17 f(x)} e-21TlxlI dx

~ (I;~:N f:J:;'{xPf(x)} \ dx=O(ly 1-"'),

I (2)

which proves the theorem. THEOREM 3. If f(x) is agoodfunction with F.T.g(y), then the F.T. off'(x) is 27Tiyg(y), and the F.T. off(ax+b) is I a l-le27Tlblllag(y/a).

GENERALISED FUNCTIONS AND FOURIER TRANSFORMS 17

FOURIER ANALYSIS AND GENERALISED FUNCTIONS

16

The proof is left to the reader, who should note the special cases 1 and b=o.

a= 1, a= -

THEOREM: 4 (Fourier's inversion theorem for good functions).

Ifg(y) is the F.T. of agood function f(x) , thenf(y) is the F.1·. ofg( -x). PROOF. Any ofthe standard proofs of Fourier's inversion theorem applies without any difficulty to good functions. A simple version is to prove by elementary manipulation that the F.T. of e-ex'g( -x) differs fromf(y) by

If~", (~r e- ,-f(y)} dt 1 ~max If'(x) If:", (~) e- ,
l

1T

and then let

(3)

DEFINITION 4. Two regular sequences of good functions are called equivalent if, for any good function F(x) whatever, the limit (6) is the same for each sequence. EXAMPLE 4. The sequence e--
f:cof(x) F(x) dx

6-+0.

THEOREM 5 (Parseval's theorem for good functions). If flex)

andf2(x) aregoodfunctions, andgl(y) andg2(y) are their F.T.'S, then f:", gl(y)g2(y) dy= f:cofl( -x)f2(x) dx.

(4)

PROOF. Both sides can be written as the absolutely convergent double integral

(5) by theorem 4.

of the product of a generalised function f(x) and a good function F(x) is defined as (8) This is permissible because the limit is the same for all equivalent sequences f,,(x). EXAMPLE 5. The sequence e-x'/n' and all equivalent sequences define a generalised function I(x) such that (9)

f:co I(x)F(x)dx= f:co F(x)dx.

NOTE. This theorem will also be used (in theorem 11 b.e1ow) in a case when flex) is a good function and f2(x) is any function absolutely integrable from -co to co. The proof stands word for word in this case, since the double integral remains absolutely convergent.

Generalised functions. The delta function and its derivatives DEFINITION 3. A sequence fn(x) of good functions is called regular if,for any good function F(x) whatever, the limit

2.2.

:!.:,

f:/n(X) F(x) dx

(6)

exists. EXAMPLE 3. The sequence fn(x) = e-x'!n' is regular. (The limit in this case is f:co F(x) dX.)

This generalised function I(x) will be denoted more simply by

1.

EXAMPLE 6. The sequences equivalent to e--n:t' (n/1T)! define a generalised function o(x) such that

f:""O(x) F(x) dx =F(o).

(10)

PROOF. If F(x) is any good function,

If:co e-1lX' (n/1T)l F(x) dx -F(o) i = If:co e-nX'(n/1T)!{F(x)-F(o)}dx! ~max I F'(x) If:co e--n:t'(n/1T)! Ix I dx =(1T1z)-tmaxIF'(x)j-+o as n-+co. LFA

18

GENERALISED FUNCTIONS AND FOURIER TRANSFORMS 19

FOURIER ANALYSIS AND GENERALISED FUNCTIONS DEFINITION 6. If two generalised functions f(x) and hex) are

defined bysequencesfn(x) and hn(x), then their sumf(x) + hex) is defined by the sequence fn(x) + hn(x). Also, the derivative f'ex) is defined by the sequence f~(x). Also, f(ax + b) is defined by the sequence fn(ax+ b). Also, ¢(x)f(x), where ¢(x) is a fairly good function, is defined by the sequence ¢(x)fn(x). Also, the F.T. g(y) off(x) is defined by the sequence gn(Y), where gn(Y) is the F.T. offn(x). PROOF OF CONSISTENCY. In each item of this definition we must verify (i) that the sequence named is a sequence of good functions, but this follows at once from theorems I and 2; (ii) that the sequence named is a regular sequence; and (iii) that different choices of equivalent regular sequences to define the generalised functions f and h lead to equivalent sequences defining the new generalised function. Now, as regards the first item, for any good function F(x)

where in (IS) G(y) is the F.T. of F(x) and theorem 5 has been used. This completes the proof that addition, differentiation, linear suJ--stitution, multiplication by a fairly good function and Fourier transformation can each be applied to any generalised function, and that the result in each case is still a generalised function. * EXAMPLE 7. The F.T. of 8(x) is 1. tr PROOF. The F.T. of e-nx' (n/1T)! is easily found to be e- 'II'ln, which is obviously one of the sequences defining the generalised function I. THEOREM 6. Under the conditions of definition 6, we have for any

good function F(x) (with F.T. G(y)) I:J'(x)F(X) dx= - I:<x>f(x)F'(X) dx,

I:J(ax+b)F(X)dx=~I:J(X)F(x:b) dx,

!~rr;,I:<x> Un(x) + hn(x)} F(x) dx =!~I:<x>fn(X)F(X)dx+:~I:<x>hn(X)F(X)dx,

Again,

I<X>

I<X> n~~ _<x>f~(x)F(x)dx= -nl~~ _<X>fn(x)F'(x)dx,

I:

lim I<X> fn(ax+b)F(x)dx=-, I ,lim I'" fn{X)F(X-b) dx, n-+-CX) -co a n-+-co a -0:)

(13)

2~~I:<x> {¢(x)f.,(x)}F(x) dx= !~n;,I:<x>J,.(x){¢(x)F(x)} dx

<x>g(y) G(y) dy= I:J(x)F(-x)dx.

PROOF. These equations follow at once from equations (12) to (IS)' EXAMPLE 8. If..{'(x) is any good function,

(12)

and, since F'(x) is a good function (by theorem I), the limit on the right exists and is the same (by definitions 3 and 4) for all equivalent regular sequences fn(x). Hence all the sequences f~(x) are equivalent and regular, as was to be proved. Precisely the same argument applies to

I:'", Q(nl(x) F(x) dx =

(IS)

I)n .F
(-

(17)

PROOF. This follows by n-fold application of the first of equations (16) (' integration by parts '), followed by equation (10). At this stage one can, if thought necessary, prove a whole corpus of results like

i.. {I(x) +h(x)} =f'(x) + h'(x), dx

i.. {¢(x)f(x)} = ¢'(x)f(x) + ¢(x)f'(x), dx

¢(ax+ b)f(ax+b) =h(ax+b)

(14) and

I:.,., {¢(x)f(x)}F(x) dx= I: <X>f(x){¢(x) F(x)} dx,

(II)

and so the limit on the left exists, verifyiog (ii). Also, the limits on the right are independent of which of the different equivalent sequences of good functions fn and hn are used to define f and h. Hence all resulting sequences f" + hn are equivalent, verifying (iii).

(16)

if

dd f(ax+ b) = af'(ax + b), x

¢(x)f(x)=h(x),

• On the other hand. there is no satisfactory definition of the product of two generalised functions; for example, in the notation of definition 6.Jn(x) hn(x) is not in general a regular sequence.

20

FOURIER ANALYSIS AND GENERALISED FUNCTIONS

etc., which are hardly worth dignifying with the name of theorem since it is almost impossible to imagine reasonable definitions under which they would not be true; they will be used without reference, and the proof is short and easy in each case. The longest is as follows. If F(x) is any good function, then d F(x) dx{¢(x)f(x)}dx

co -co J

J:co F'(x) ¢(x) f(x) dx co d Jco-co F(x)¢/(x)f(x)dx = - _codx{F(x)¢(x)}f(x)dx+ J = J:co F(x) {¢(x) f'(x) + ¢/(X) f(x)} dx. =-

The more useful results on Fourier transforms are, however, collected into a theorem. THEOREM 7. If f(x) is a generalised function with F.T. g(y), then the F.T. of f(ax+b) is Ia 1-1e2"lb lllag(y/a). Also, the F.T. of f'(x) is 2myg(y). Finally (Fourier's inversion theorem for generalised functions),f(y) is the F.T. ofg( -x).

This powerful theorem follows at once from definition 6 and theorems 3 and 4. for example, to prove the inversion theorem, let the sequence f ..(x) define f(x); then, by definition 6, g..(y), the F.T. of f..(x), defines g(y), whence g..( -x) defines g( -x). But, by theorem 4, the F.T. of g..( -x) is f..(y). Hence the F.T. of g( -x) isf(y)· PROOF.

9. The F.T. of o(x-c) is e-2 ,,1Cl/, by example 7 and theorem 7. Hence, by the last part of theorem 7, the F.T. of e2 ,,1e:t is o(y-c). EXAMPLE

THEOREM

8. If f(x) is a generalised function and f'(x) = 0, then

f(x) is a constant (that is,f(x) is equal to a constant times the generalised function I). PROOF.

If F(x) is a good function, then clearly

~(x)= J:coF(x) dx

GENERALISED FUNCTIONS AND FOURIER TRANSFORMS 21

is a good function if and only if

J:co F(x) dx = o. It follows that

F2(x) = f:JF(x)-

~-: f:co F(t)dt}dx

(18)

is always a good function, since the function in c.urly brackets is a good function whose integral from - 00 to 00 vamshes. Hence, f:!(x)F(x)dx= (f:/(x)

~:. dx) J:co F(t) dt+ f:/(x)F;(x) dx

= C f:co F(x) dx -

(19)

f:cof'(x) Fz(x) dx,

where C is a constant. The last integral in (19) vanishes, since f'(x)=o. Hence,j(x)=C. THEOREM 9. If g(y) is a generalised function and yg(y) =

0,

then

g(y) is a constant times o(y). PROOF. This theorem follows immediately from theorem 8 by taking Fourier transforms (using the second part of theorem 7 and example 7). It can aL,> be proved independently as follows. If G(y) is a good function, then 1I1 G1(y) = G(y)-G(o)e(20)

y is a good function. Hence f:co g(y) G(y) dy= G(o) f:co g(y) e- II ' dy

+ f:coyg(y) G1(y)dy= CG(o),

(21)

where C is a constant, and the last integral vanishes because yg(y)=o. Henceg(y)=Co(y).

2.3. Ordinary functions as generalised functions DEFINITION 7. Iff(x) is a function of x in the ordinary sense, such that (I + X2)-N f(x) is absolutely integrable from - 00 to 00 for some N, then the generalised function f(x) is defined by a sequence f..(x) such that for any good function F(x)

l~J:/..(x)F(x)dx= f:/(x)F(X)dx.

(22)

22

FOURIER ANALYSIS AND GENERALISED FUNCTIONS

GENERALISED ,FUNCTIONS AND FOURIER TRANSFORMS 23

The integral on the right is the integral in the ordinary sense, which exists as the integral of the product of (I + x2)-Nf(x), which is absolutely integrable, and (I +X2)N F(x), which is a good function. When the generalised functionf(x) has been defined, this integral has a' meaning also in the theory of generalised functions, and equation (22) states that these two meanings are the same. NOTE.

PROOF OF CONSISTENCY.

It must be shown that such a sequence

exists. We take·

Secondly,

U:oofn(X) F(x) dx- f:oof(X)F(x) dx I =

If~l S(y) dy {f:oof(t) e-('In' F(t-~) dt- f:oof(t) F(t) dt} I

~ max Ifoo 1111<1

f(t) e-I'/n' {F(t

-00

fn(x) = f:,/(t) S{n(t-x)}n e-I'/n' dt,

- f:oo f(t) F(t)(I -

where the' smudge function' S(y) is any good function which is zero for Iy

I;;. 1 and positive for Iy I < I and satisfies f

1

S(y) dy = I.

-1

For example, S(y) may be taken as

We must now prove that the fn(x) are good functions, and that equation (22) is satisfied. First,

If~)(x) I=If:",f(t)( -n)P scp){n(t-x)}ne-t'/n'dt/ I

I+ I)2}N

f over a small interval

where A and B are constants and the facts that F(x) is a good function and that (I +t2)-Nf(t) is absolutely integrable have again been used. This completes the proof of consistency. Definition 7 increases enormously the range of generalised functions available to us. Not only can all ordinary functions f(x) with (I +x2)-Nf(x) absolutely integrable from -00 to ex> be used as generalised functions, but one can obtain from them new generalised functions by differentiation in accordance with definition 6.

1

The discontinuous function sgn x, which is for x < 0, is a generalised function, and

1

for

dsgnx/dx=28(x).

(25)

PROOF. sgn x satisfies the condition of definition 7 (with N and, for any good function F(x),

foo

/Xl-I
dt

(26)

EXAMPLE 10.

where we have used the fact that where the integrand is non-zero

x+n- 1). The e- l '/'" factor makes it •good' at infinity.

I

t2

as n-+oo,

x> 0 and -

X f:", (I + t 2)-N /f(t) I dt

= 0(1 Xi)-M as I x 1-+00 for all M,

dt

~ ~f:oo If(t) I(I +~2)Ndt+~ f:oo If(t) I(I ~2)Ndt

for Iy I < I and zero for Iy 1;;. I; note that on this definition aU the derivatives of S(y) exist even at y = ± I (they are all zero there).

I

e-('/n')

~ f:oo If(t) I {~I~:~<11 F'(x) I} dt+ f:oo If(t)F(t) I I ;2

-+0

~ nP+lmax sP)(y) e-tlz/-l),/n' {I +(/ x

_l.)n -F(t)} dt

-00

(x _ n-1

'

= I)

dsgnx F(x)dx= -foo sgnxF'(x)dx dx -00

= - f: F'(x)dx+ f:", F'(x)dx=2F(0). (27)

FOURIER ANALYSIS AND GENERALISED FUNCTIONS

GENERALISED FUNCTIONS AND FOURIER TRANSFORMS 25

THEOREM 10. If f{x) is an ordinary differentiable function such that both f(x) and f'(x) satisfy the condition of definition 7, then the derivative of the generalisedfunction fonned from f(x) is the generalised function formed from f'(x).

2.4. Equality of a generalised function and an ordinary function in an interval DEFINITION 8. Ifh(x) is an ordinaryfunction and f(x) a generalised

24

PROOF. This theorem, which shows that the notation f'(x) can be used without risk of confusion, follows from the fact that both definitions of it satisfy

I:,/'(x)F(X)dx= - I:oof(x)F'(x)dX

(28)

for any good function F(x). With the second definition, equation (28) assumes thatf(x)F(x)~o as x~+oo or x~-oo. However, the product must tend to some finite limit in each case, since

I:""f(x)F'(x)dx+ I:""f'(X)F(x)dx exists, whence both limits must be zero since I:oof(x)F(x) dx exists. THEOREM II. If f(x) is an ordinary function which is absolutely integrable from -00 to 00, so that its F.T. g(y) in the ordinary sense exists, then the F.T. of the generalised function f(x) is the generalised function g(y). PROOF. The generalised function g(y) exists because g(y) satisfies the condition of definition 7 with N = I ; the integral

Ioo

-00

d ~ I

+y

I""_"" f(x) e- "lxv dx 2

remains convergent when each term is replaced by its modulus. But, by the note following theorem 5, the ordinary function g(y) satisfies I:oo g(y) G(y) dy = I:""f(x) F( - x) dx for any good function F(x) with F.T. G(y). Hence, by definition 7, the generalised functiong(y) also satisfies (30). Hence, by theorem 6, it is the F.T. of the generalised function f(x). This theorem again eliminates possibilities of confusion, this time between different uses of the expression 'Fourier transform'.

function, and

foo _J(x)F(x)dx=

fb h(x)F(x)dx

(3 1 )

a

for every good function F(x) which is zero outside a < x < b (here, a and b may be finite or infinite, and we assume the existence of the right-hand side of (31) as an ordinary integral for all such F(x), thus imposing a restriction on the function hex) in a < x < b, although it need not even be defined elsewhere), then we write f(x)=h(x)

for

(3 2 )

a<x
PROOF OF CONSISTENCY. The definition is consistent with the maxim that everything is equal to itself, since if h(x) satisfies the condition of definition 7, then the generalised function hex) equals (in the sense of definition 8) the ordinary function h(x) in any interval. EXAMPLE

II.

o(x) = 0 for

0

< x < 00 and for

-00

< x < o.

PROOF. If F(x) vanishes outside either of these two intervals, then F( 0) = 0, and so by equation (10)

f:oo o(x)F(x)dx=o. THEOREM 12. If h(x) and its derivative h'(x) are ordinary functions both satisfying the restriction imposed on hex) in definitwn 8, andf(x) is a generalised function which equals h(x) in a < x < b, then

f'(x)=h'(x)

in

a<x
PROOF. If F(x) is a good function which is zero outside a < x < b, then

I:oof'(x)F(X)dx= - f:J(x)F'(X)dx = -

I: h(x)F'(x)dx= I: h'(x) F(x) dx,

(33)

which proves the theorem. The assumption in the last integration by parts that h(x)F(x)-+o as x-+a (and, similarly, as x-+b) is

26 FOURIER ANALYSIS AND GENERALISED FUNCTIONS

GENERALISED FUNCTIONS AND FOURIER TRANSFORMS 27

proved exactly like the corresponding assumption in the proof of theorem 10. The product must tend to some finite limit for any good function F(x) vanishing outside a < x < b, and if a = - 00 the

i

limit is zero because f: h(x)F(x)dx exists. If a>

r

-00,

however, it

is zero because otherwise h(x)F(x)/(x-a) would not tend to a finite limit, although F(x)/(x - a) is itself a good function vanishing outside a < x < b. EXAMPLE 12. Any repeated derivative 8
EXAMPLE 15. Iff(x) is an even (or odd) ordinary function satisfying the condition of definition 7, then the generalised function f(x) is even (or odd). PROOF. This follows Immediately from definition 9. THEOREM 13. If the generalised function f(x) is even (or odd, respectively), then its derivative f'(x) is odd (or even), its F.T. g(y) is even (or odd), while ¢(x)f(x) is even (or odd) when the fairly good function ¢(x) is even, and odd (or even) when ¢(x) is odd. PROOF. This follows immediately from theorem 6 and the corresponding results for good functions. EXAMPLE 16. 8
(34)

which proves the result. Note that, by theorem 9, the various functions f(x) with xf(x)=g(x) all differ by constant multiples of 8(x), which does not invalidate the result since 8(x) =0 in such an interval.

2.5. Even and odd generalised functions DEFINITION 9. The generalised function f(x) is said to be even (or odd, respectively) tf f:/(x)F(X)dx=o for all odd (or even) good

f:/(x)F(x)dx= ± f:/(X)F(-X)dX =

for all odd good functions F(x).

±

f

b

h(x)F(-x)dx= ±

a

f-a h(-x)F(x)dx. -b

which proves the theorem.

2.6. Limits of generalised functions DEFINITION 10. If ft(x) is a generalised function of x for each value of the parameter t, and f(x) is another generalised function, such that,for any good function F(x),

EXAMPLE 14. 8(x) is even. 0

(35)

with the upper sign if f( x) is even and the lower if f( x) is odd.

functions F(x). PROOF. F(o) =

-b<x< -a,

PROOF. If F(x) is zero outside -b < x < -a, then F( -x) is zero outside a < x < b, and also F(x) +F( - x) is odd (or even). Hence

f:",f(X)F(X)dx= f:", xf(x).r1F(x)dx = f:", g(x)x-1F(x) dx= f: h(x).r1F(x)dx,

I I

then we say

!~f:",ft(x) F(x) dx= f:/(X) F(x) dx,

(3 6)

limft(x)=f(x).

(37)

t->-c

28

GENERALISED FUNCTIONS AND FOURIER TRANSFORMS 29

FOURIER ANALYSIS AND GENERALISED FUNCTIONS

Here, c may be finite or infinite, and t may tend to c, through all real values or (when c=oo) through integer values only. THEOREM IS. Under the conditions of definition

DEFINITION II. Ifft(x) is ageneralisedfunction of xfor each value of the parameter t, we define

~ ft(x) =

10,

limJ;(x) =f'(x),

lim.ft(ax+ b) = f(ax+b),}

' ... c

t ... c

lim ¢(x)ft(x) = ¢(x) f(x),

at

(3 8)

equals

limgh)=g(y), ' ... c

limf'" f;(x)F(x)dx= -limf'" ft(x)F'(x)dx -co

f:/(x)F'(x)dx= f:",f'(x)F(x)dx, (40)

lim e I x 1£-1= 28(x). .... 0

PROOF. This is obtained by differentiating the result lim I x jesgnx=sgnx

exists and

I.

Prove that

ml

x n 8<ml(x)=(-I)n_-'-8<m- nl(x) (m-n)!

(m;?;n)

I

:;;; ie f:a> pog Ix

II (I + I X 1£) IF(x) Idx=O(e)

'

0

(m
(44) Prove that the general solution of j
and using example 10 and theorem IS. To prove (41), let F(x) be any good function. Then x 1£ - I) sgn xF(x) dx

NOTE. Similarly, the F.T. of (a/at) JI(X) is (a/at) gt(Y)' Theorem 15 also shows that we can differentiate or take Fourier transforms of series, tenn by tenn. This fact will be continually used in chapter 5.

EXERCISE 2. If ¢(x) is any fairly good function, prove that

.... 0

If:", (I

ft {o~ft(X)}

lim f;,(x) - f;(x) = ~ lim ft,(x) - ft(x) t, ... t t 1 -t dXt.-+-t t1 -t

EXERCISE

whence'by definition 10 the first result follows; and similarly with the others. EXAMPLE 17.

(43)

PROOF. By theorem IS,

PROOF. The proof of this remarkable theorem offers no difficulty, following closely the lines of the proof of consistency of definition 6. For example, if F(x) is any good function,

= -

i ft(x),

t=O

o~ {iJl(x)}.

where gt(Y) and g(y) are the F.r.' s offt(x) and f(x).

t-+-c

lim n ... a>

provided in each case that the limit function exists.

for any fairly good function ¢(x), and

-co

~ ft(x) =

t-O

EXAMPLE 18. If (a/at) ft(x) exists, then

t ... c

t-+c

lim k(x) - ft(x) , t.... , t 1 -t

n-O

(4 2 )

as e-7-O, where the inequality Itanhz I:;;; Iz I has been applied to z=telog 1x I· Two specially useful kinds of limiting operation are now given their usual special names and symbols.

(45)

More generally, from the results of example 8 and exercise I, or otherwise, prove that m ml ¢(x)8<m)(x)= ~ (- I)n I ¢
n.(m-n)/

EXERCISE 3. If f(x) is a generalised function and g(y) its F.T., find the F.T. of xnf(x). EXERCISE 4. Prove that lim sinnx =8(x). n-+-a>

(47)

1TX

NOTE. This is the Fourier transform of a much simpler result.

PARTICULAR GENERALISED FUNCTIONS

30

CHAPTER

3

DEFINITIONS, PROPERTIES AND FOURIER TRANSFORMS OF PARTICULAR GENERALISED FUNCTIONS 3.1. Non-integral powers In this chapter a number of particular generalised functions are defined and studied, some for their intrinsic interest and widespread utility, and others solely for their application to the technique of asymptotic estimation of Fourier transforms described in chapter 4. We begin by defining non-integral powers, and more precisely (since non-integral powers of negative numbers have without further particularisation no precise meaning) the functions Ixl"', Ixl"'sgnx and x«H(x)=Hlxl"'+lxl"'sgnx), (I) where H(x), equal to I for x> 0 and 0 for x < 0, is Heaviside's unit function. The first of expressions (I) is an even function, the second odd, and the third (the mean of the other two) vanishes for x < o. These expressions are generalised functions of x by definition 7 when IX > - I. The differentiation rules d: 1 x I"'=IX I x Id"'-lsgnx,

d: I x l"'sgnx=IX 1 x

1"'-1,) (2)

dx x«H(x) =ax"'-IH(x), follow from theorem I 0, provided that IX> 0 (so that IX - I as well as IX exceeds - I). It is convenient to use these equations for IX < 0, repeatedly if necessary, to define the appropriate generalised functions for non-integral IX less than - I. DEFINITION 12. If IX < - I and is not an integer, then we define three new generalised functions as follows:

I x I"'} Ixl"'sgnx x«H(x)

(IX+I)(IX+~)

{ I x I"'+n(sgnx)n } ... (IX+n)d:: Ixl",+n(sgnx)n+l, x«+nH(x)

where n is an integer such that IX + n > -

I.

31

PROOF OF CONSISTENCY. The value of n chosen is immaterial, by equations (2). Definition 12 now extends the validity of these equations to all non-integer IX. The relation (I) between the three functions, known in the first place for IX > - I, is similarly extended, by repeated differentiation, to the three new generalised functioI\s defined by equation (3). By theorem 13, I x I'" is an even, and I x I'" sgn x an odd, generalised function. Each of the three new generalised functions of definition 12 is equal to the ordinary function of the same designation in the interva's 0 < x < CO and - co < x < 0, by repeated application of theorem 12. (Note that the restriction on hex) imposed in definition 8 is satisfied by all these functions in the intervals stated, because a good function F(x) which is zero outside either interval must tend to zero faster than any power of I x I as X40.) Hence, as the ordinary functions are undefined at x = 0, no conflict between the new and established usages is possible. One can use definition 12 to interpret 'improper' integrals like

f:

x«F(x) dx, in which IX is non-integral and < -

I

and F(x) is a

good function,- as

f:""

{x«H(x) -x«H(x-a)}F(x) dx,

where the second term in curly brackets is to be interpreted as a generalised function by definition 7. Integratio'n by parts can be used to express (4) as an ordinary integral. We need the fact that, by theorem 10, iff(x) is an ordinary function (here x«) differentiable for x ~ a, then d d dx {f(x)H(x-a)} = dx[{f(x) - f(a)}H(x-a)] +f(a) 8(x-a) =f'(x)H(x-a) +f(a) 8(x-a),

(5)

which is easy to remember because of its similarity to the ordinary rule for differentiating a product. Applying this repeatedly to the (3) • More generally, it is sufficient that F(x) be differentiable any number of times in some interval which includes (0, a). This is because any such function necessarily coincides with some good function in (0, a).

32

PARTICULAR GENERALISED FUNCTIONS

FOURIER ANALYSIS AND GENERALISED FUNCTIONS

plane which is equal to the positive number x" for r.eal positive x; in other words, the many-valued function x" must be made precise by taking I arg x I < t1T· Hence, if a is any complex number with positive real part,

second term (and using equation (3) for the first), we obtain

x"H(x) -x"H(x-a) I dn ;----,--;--,-------;-----;: - [x"+n{H(x) - H(x - a)}] (a+ I)(a+2) ... (a+n)dx n aa+1 aa+2 +-o(x-a)+( )( )o'(x-a)+ ... a+1 a+1 a+2

+

aa+n o(n-l)(x-a) (a+ I)(a+2) ... (a+n) ,

I: (6)

X

0

1

X ( )

dx

which holds for a> -

II: (7)

Equation (7) is what the ordinary formula for repeated integration by parts would give formally, if all the contributions from the lower limit (which involve 00:+1,00:+2, ... , oo:+n, and so are' infinite' if n is the least integer with a + n > - I) were omitted. For this reason the interpretation (7) was given the name 'finite part' by Hadamard, who showed that the finite part obeys many of the ordinary rules of integration. Its inclusion within the framework of generalised-function theory makes this conclusion readily intelligible. The rest of this section is devoted to obtaining the Fourier transforms of the functions of definition 12. We need some properties of the factorial function a!, called rca + I) in the older books, and defined for a> - 1 as

I:

x"e-Xdx=al,

z"e-"dz=a!a-"'-t,

(9)

limx"e-xt H(x) = x"H(x),

)n-l aa+n pn-l>(a) (a+ l)(a+2) ... (a+n) . -

I:

(10)

t~O

aa+l aa+2 +-F(a)-( a+1 a+1 )(a+2 )F'(a)+ ...

+ (.

x"e-=dx=a-a-l

by the substitution ax=z; and in (9) we must have I arga 1< !7T, in order that arg z = arg a shall satisfy the same condition. To obtain the F.T. of x"H(x) for a > - I, we use the limit property

which on substitution in (4) gives the interpretation

(-I)n fa x"+nFCn) foax"F() dx= (a+ I)(a+2) ... (a+n)

33

(8)

where x" is of course positive. The straight path of integration in (8) may be rotated about the origin through any angle < t1T, provided that x" is taken as a function regular in the right-hand half

I

because for any good function F(x)

(I -e-xt)x"F(X)dx!.;;;A

I: ~:-xt I

dX=2A.j(1Tt)-+0 (II)

as t-+o, where A = max I x"+iF(x) I. Now, by theorem II, and equation (9) above, the F.T. of x" e-xt H(x) is

I

et>

o

dn (a-n)l(t+27Tiy)n-O:-l x" e-(I+2"III)zdx=a! (t+ 21Tiy)-a-l='. . dyn ( - 21T1)n (12)

But, by theorem IS, the F.T. of x"H(x) is the limit of (12) as t-+o, and this is the nth derivative of the limit of the quotient, namely, dn (a-n)! (21T Iy I)n-"'-l e!>rl(n-a:-l)sgn II

dyn

( - 21Ti)n =

{e-*"I(o:+I)Sgnll}a! (21T Iy I)-a:-l.

(13)

In (13) the limit of the quotient is given for n - a - I > 0, when the limiting result is easy to prove, using the rule about the argument of a in (9). The nth derivative of this limit is then evaluated from definition 12, which' defines as repeated derivatives the generalised functions Iy 1-",-1 and Iy 1-a:-1sgny, of which, by de Moivre's theorem, expression (13) is a linear combination. Now, by theorem 7, the F.T. of f'(x) is 21Tiyg(y) if the F.T. of f(x) is g(y). Applied as a check to (13), this says that the F.T. of axa-1H(x) is {e-i "lasgnll}al(21T Iy I)-a:, (14) 3

LPA

PARTICULAR GENERALISED FUNCTIONS

34 FOURIER ANALYSIS AND GENERALISED FUNCTIONS since iy = et "ISgn II Iy

I. Equation (14) shows that the expression (13) for the F.T. of XT-H(x) is valid for a - I if it is valid for a, and so extends it by induction to all values of a, which is necessary because the proof given, using the integral expression for the factorial function, applies only when a> - I. Since 1x

I"=XT-H(x)+( -x)" H( -X)'}

Ix '''sgnx=XT-H(x)-( -x)" H( -x),

(IS)

{2 cos t11(a+ I)} a! (211 Iy 1)-,,-1

(16)

(which is even, in agreement with theorem 13), and that the F.T. of I x '''sgnx is {- 2i sin t11(a+ I )}a! (211 Iy 1)-,,-1 sgn y

NOTE. These equations are of course true in the ordinary sense for a > - I. For non-integral a < - I, they may be taken as defining the generalised functions on the left-hand sides; the derivatives with respect to a (in the sense of definition I I) then exist by repeated application of the result of example 18. This example shows also that the ordinary rules of differentiation apply to these functions; for example,

0 0

d

and by theorem 7 the F.T. of ( -x)" H( -x) is obtained by changing the sign of yin (13), we deduce that the F.T. of Ix I" is

J

0

= Ix 1"'-lsgnx+a Ix 1"'-1 log I x I sgnx,

o

(17)

oa [{2 cos t11(a+ I )}a! (211 I y 1)-"'-1] = {2 cos 111(a + I)}a! (211 Iy 1)-«-1 x {-10g(211 Iy D+~(a)-t11tant11(a+ I)},

~

dx

EXERCISE 6. Prove that the equation xf(x) = 1X f(x) = I X 1"'-1 sgn x. Are there any other solutions?

d

where

=t 11 +t.

I'" is satisfied

by

~(a)= dalog(a!).

NOTE. The standard formula

(22)

Similarly, that of I x I'" log 1x 1sgnx is

x { -log(211 I.y

I) + ~(a) + 111 cot !11(a + I)}, (23)

and that of XT- log xH(x) is {e-h1("'+llsgn II} a! (211 Iy 1)-«-1 {-10g(211 Iy I) + ~(a) - t11i sgn y}.

at ( -a- I)! = -11cosec11a

(24)

will be required.

3.3. Integral powers

3.2. Non-integral powers multiplied by logarithms

Throughout the rest of this chapter, n signifies any integer ~ 0 and m any integer> o. By the second and third parts of theorem 7, if f(x) has F.T. g(y) then

DEFINITION 13.

XT-IogxH(x) = oa {XT-H(x)}.

(21)

{- 2isin !11(a + I)} a! (211 Iy 1)-"'-1 sgn Y

EXERCISE 7. Check that Fourier's inversion theorem (see theorem 7) is satisfied both by I x I'" and by 1x I"'sgnx.

Ix I"'log Ix I= oal Oxl"'sgnx), o x I"', Ix 1"'10: Ix Isgnx= oa O }

(20)

as one would obtain for positive a by direct differentiation. The F.T. of Ix I'" log I x I, by the note following example 18, and by equation (16), is

EXERCISE 5. Show that l

0 0

ax(l x I"'log Ix 1)= ox oa Ix 1"= oa ox Ix 1"= oa (a 1x 1"'-1 sgn x)

(which, similarly, is odd). Conversely, if (16) and (17) had been derived first, (13) would follow from them by equation (I).

OX·(I+X)

35

(19)

xnf{x)

has F.T. (-211i)-1l-g
(This is the solution to exercise 3.)

(25)

36

Hence, by example 7, the F.T. of x n is (-2m)-n8Cn)(y). This, as theorem IS requires, is actually the limit as a --+ n of expression (16) when n is even, namely

d

n

=

]

(21T)"'+1 dyn Iy 11+"'--1l n 2 cos t1T(a + I)} I d n-a} "'~ n-a (21T)n+1 dyn <£~ Iy 11+",-n

(26)

where theorem IS and example 17 have been used. Similarly, it is the limit as a--+n of (17) when n is odd. We next define negative integral powers of x; the only problem is how to define X-I, the others being deduced from it by differentiation. xf(x) = I; and

14. .r1 is the odd generalised function satisfying dm- I I x--m= (m-I)! dxm-I(x- ).

lim Iy l-a-I=y-n-I. "'-+n

These properties are satisfactory features of definition 14· Definition 14 is also closely connected with Cauchy's' principal value' of an integral whose integrand has an inverse-first-power singularity. In the present theory, if F(x) is a good function and

°

a < < b, the integral

f:""

°

( - 21Ti)-n(1Ti)-1 d~n (y-I) = 2(nl)(21Tiy)--1l-I.

(29)

{x-l_x-IH(x-b)-x-IH(a-x)}F(x)dx.

(3 2 )

d = dx[1og! x I{I -H(x-b)-H(a-x)}]

The equation xf(x) = I has such a solution, since f(x) = d log I x I/dx is an odd generalised function by theorem 13 and example IS, and it satisfies By theorem 9, the general solution is f(x) + Co(x), but this is odd only for C=o, by example 14. Hence the definition specifies X-I uniquely, and theorem 12 shows that it equals the ordinary function X-I in < x < 00 and in - 00 < x < 0. Note also that x--m has a similar property and (by theorem 13) is even when m is even and odd when m is odd. Now, if sgnx has F.T. g(y), then by example 10, theorem 7 and example 7, 2myg(y) =2. But g(y) is odd by theorem 13, and so (by definition 14) it is (1Ti)-ly-l. Hence also, by equation (25), the F.T. of xnsgnx is

.rIF(x) dx must be interpreted as

X-I-x-IH(x- b) -x-1H(a-x)

PROOF OF CONSISTENCY.

(28)

f:

But, by equation (5),

(27)

I.

I"', and so (29) must be compared

with the limit of (16), and we infer that in this case

( _ I )m-I

xf(x) = d(x log 1 x I)/dx -log I x I=

(30)

"'-+110

{r

=mn (2~n+1 :;n{20(y)}=( -21Ti)--1l8Cn)(y),

DEFINITION

lim Iy l-a-Isgny=y-n-I. "'-..n

When n is odd, x"'sgnx= lim I x I

X

{r

<£-..n

paring the limit of (17) as a--+n with (29) (and using theorem IS), we see that

;~n[{2cost1T(a+ I)} (-a) (-a+ I~!.. (-a+n- I) I

37 n Now, when n is even, x sgn x = lim I x I'" sgn x, and so, by comPARTICULAR GENERALISED FUNCTIONS

FOURIER ANALYSIS AND GENERALISED FUNCTIONS

+ (log b) 8(x-b)-(log I a i) 8(x-a), which substituted in (32) gives the interpretation

f:

x-IF(x)dx= -

I:

(log I x I) F'(x) dx+ F(b) logb-F(a) log I a I· (34)

This may be compared with the' Cauchy principal value', defined as lim (f-' +fb) x-IF(x) dx= lim {- (f-' +fb) (log Ix I) F'(x) dx

e~O

a

6

e-i'-O

a

e

+F(b) logb-F(e) loge+ F( -e)loge-F(a) log I a I},

(35)

which is seen to be identical with (34)· In the same way, integrals involving other inverse-integralpower singularities can be interpreted by definition 14, and this

PARTICULAR GENERALISED FUNCTIONS

38- FOURIER ANALYSIS AND GENERALISED FUNCTIONS interpretation also has been anticipated by a number of writers. We obtain as above (but with repeated integration by parts)

fb

x-F(x) dx=

-

_I_fb

(m-I)!

a

x-Ip<m-I)(x) dx

I I b -'1nP(b) _a -'1nP(a)

m-I

a

DEFINITION IS. The symbol I x \-1 will stand for any generalised function f(x) such that xf(x) = sgn x. The symboL x-'1n sgn x wiLL stand for (- I).""...I/(m- I)! times the (m-l)th derivative of any of these functions. Thus (by theorem and example 19) Ix can be written as

9

b2-'1nP'(b) -al--mP'(a) b- Ip<m-2)(b) _a- Ip<m-2)(a) (m-I)(m-Z) - ... (m-l)(m-z) ... 1 ,(3 6)

where the Cauchy principal value itself has been left uninterpreted because most readers will be familiar with these integrals. The interpretations (36) and (34) are valid (see the footnote to equation (4)) if F(x) is any function differentiable any number of times in an interval including (a, b). We have treated x n, xnsgnx and x-; more serious difficulties are presented by x-'1n sgn x. These difficulties are already fully present in the case m= I, that is, for the function Ixl-I. The limit of the generalised function I x 1.-1 as e--*o does not exist, as example 17 clearly shows. Again, all the solutions of the equation xf(x)=sgnx

are even, so that the method of definition 14 cannot be used to pick out one of them. Particular solutions can be specified, but they do not obey the manipulation rules which one expects of a function called Ixl-I. EXAMPLE 19. If f(x)=d(log I x Isgnx)/dx, then xf(x) = sgn x, but f(ax) Ia I-If(x). f(ax) = d{(log I x

I+ log Ia j)sgna sgnx} adx

I d =101 dx(log I x I s~nx+~og Ia Isgnx)

1

and X- sgn x as ( l)m-I dm (:_ I)! dx'" {(log Ix I + C)sgnx}

( 1)711-1 dm = ) , ::;-:- (log Ix I sgnx) + C8<m-I)(x), (m-l . ~m where C is an arbitrary constant in each expression (not the same in each). Study of examplC? 19 shows that, with C arbitrary in this sense, definition 15 gives Iax 1-1 = I a 1-1 Ixl-I. Again, it is only with C arbitrary in this sense that the relation x(x-'1n sgn x) = x-<.""...I)sgn x holds. EXAMPLE zO. lim {I x 1·-I- zc I8(x)}= I xl-I. ...... 0

PROOF. An equation like this means that the limit exists and It is obtained by differentiating equals one of the values of I x

1-1.

I \e

the result

lim( x ......0

EXAMPLE

Z1.

) -I

sgnx=loglxlsgnx.

(40 )

e

The F.T. of log Ix I is

-t Iy 1-1.

PROOF. Since log I x I= lim (I -I x I-<:)/e, its F.T. (by theorem IS ...... 0

and equation (16» is lim {8(y) - (z sin t1Te)( -e)! (Z1T Iy j).-I}/e ...... 0

= - t lim ([ I +e{log (Z1T) -1fr(0)} + 0(e2)] Iy 1.-1 _ zc18(y)} ...... 0

= ral {f(x) + zlog Ia 18(x)}. = -

The only satisfactory definition is one which admits the indeterminacy, in the same way as does the definition of the •indefinite integral'.

1-1

d d dx(log I x I sgnx)+ C8(x) = dx {(log I x 1 + C)sgnx},

*

PROOF. The first part is proved as in equation (z8); also

39

t lim {I y 1<-1_ ze-I8(y)}-{log (21T) -1fr(o)} 8(y),

(4 1 )

...... 0

by example 17. Expression (41) is (-t) times one of the values of Iy (though a different one from that in example zo).

1-1

40

FOURIER ANALYSIS AND GENERALISED FUNCTIONS

PARTICULAR GENERALISED FUNCTIONS

Conversely, the F.T. of Ix 1-1 is - 2(log Iy I + C),

(4 2 )

where the arbitrary constant C is present because Ix 1-1 contains an arbitrary multiple of o(x). It follows from (42) by definition 15 and theorem 7 that the F.T. of .r""sgnx is

-2

( - 21Tiy)m-l (m- 1)1 (log Iy I+ C).

Note finally that, by theorem 12 and example 12, all the determinations of .r"" sgn x are equal in the intervals - co < x < 0 and o < x < co, and equal to the ordinary function x-m sgn x in these intervals.

3+ Integral powers multiplied by logarithms The generalised function x n log I x I exists by definition 7, and its F.T., by example 21 and equation (25), is

.

n!

-m(2my . )n+1 sgny, a result which can be checked against equation (43) and Fourier's inversion theorem. The generalised function x-m log Ix I requires definition, but like x-7n itself it presents no difficulty. In fact, if m is even, Ix I" log Ix I tends to a limit as a-+-m, as is proved by the fact that its F.T. (expression (21)) can be written as

1T

( - a - I )'. cos t 1Ta (21TlyJ)-"-I{-log(21TlyJ)+tfr(-a-l) -1T cot 1Ta + t1T cot t1Ta} (45) (where equation (18) and its logarithmic derivative have been used to put (45) into a convenient form), and that this tends to a limit

as a -+ - m. Similarly, we may infer that I x I" log Ix I sgn x tends to a limit a-+-m if m is odd, from the fact that its F.T. (23) tends

41

to a limit, which again can be thrown into the form (46). These facts make the following definition appropriate. DEFINITION 16. The generalised function .r"" log I x I is the function whose F.T. is (46), so that it is the limit as a-+-m of Ix I" log Ix I if m is even and of Ix I" log Ix Isgn x if m is odd. Fourier's inversion theorem applied to definition 16 gives us that the F.T. of xnlog I x Isgnx is

n!

-2 (21Tly .)n,.'-1 {log(21T Iy D-tfr(n)}.

(47)

In connexion with (46) and (47) the reader may like to be reminded that I I I tfr(n) = -y+I+-+-+ ... +-, (48) 2 3 n where y= -tfr(o) =0'577 2 is Euler's constant. Finally, we come to the function x-m log I x Isgn x, which like x-m sgn x presents more serious difficulties, and involves a certain indeterminacy. It is most expeditiously approached from the special case n=o of (47); the F.T. of log I x Isgnx is I I d - - . {log(21T Iy D+y}= - - . -d {log(21T Iy D+y}2· my 2m !y

(49)

If now f( x) is taken as the function whose f'. T. is {log (21T Iy J) + y}2, it follows from (49) and (25) that

x.f(x) = log Ix Isgnx.

(50)

However, by theorem 9, the general solution of equation (50) is

f(x) + Co(x), and there is no way of selecting one solution as more suitable than the others (for example, all are even) and, indeed, if C is not left arbitrary, the ordinary rules of manipulation cannot be applied to this function and those 'derived from it by differentiation. DEFINITION 17. The symbol I x 1-1 log I x I will stand for any generalised function f(x) which satisfies equation (50). The symbol x-m log I x Isgn x will stand for

(-';--:....,~fCm-l)(x)+ - I )m-l ( I I ... + -I -) .r""sgnx 1+-+-+ (m-I)! 2 3 m-I '

where f(x) is any generalised function which satisfies (50).

(51)

42

FOURIER ANALYSIS AND GENERALISED FUNCTIONS

43

s

NOTE. (51) is the equation relating the corresponding ordinary functions. Under definition

G +

+

17,

the F.T. of Ix

1-1 log I x I is

{log(21T Iy l)+y}2+C,

~ 00

..9

(52)

+..,

where C is arbitrary, and that of x-m log Ix Isgn x can be thrown into the form

«~~;;l[{log(21T Iy I) -ifr(m- 1)}2+ C],

c

. I

(53)

~ N I

where C again is arbitrary (and not in general the same in each formula).

'"

~

I

.~

E

-0

f'f7 i:: N

I~

'" I

3.5. Summary of Fourier transform results The complete set of Fourier transform results for the elementary functions possessing algebraic or algebraico-logarithmic smgularities at x = 0, which have been derived in the foregoing sections, are collected for easy reference in table I. By use of this table it is possible, as will be seen in chapter 4, to write down the asymptotic behaviour of the F.T. of a function as Iy I~oo by inspecting its singularities and adding up the F.T.'S of the elementary functions which have the same singularities. As these singularities are not always at the origin, the formula which expresses the F.T. of f(ax+b) in terms of that of f(x) (theorem 7) must frequently be used in conjunction with table 1 and accordingly has been written under it. The kind of singularity occurring in most applications consists of some linear combination of those in table I. Very occasionally, however, terms involving higher powers of log I x I are present. The reader should be able to derive the F.T. of a function involving (log I x 1)2 or higher powers by the methods of this chapter (for example, in § 3.2, further differentiations with respect to IX would be necessary); one set of results involving (log Ix 1)2 can in fact be obtained directly by applying Fourier's inversion theorem to the result (53). Table 1 can also be used to find directly the Fourier transform of any rational function. By a familiar theorem in algebra any rational function can be expressed 'in partial fractions', or more precisely as a linear combination of integral powers x" and negative integral

....

G +

c+

I

b..,

I

~

I::

~

I

II

'1

+

0i: I

..,

E -

~

.... I::

~~

I::

.!:!.

.. i::

00

I'"

..

.s

f7 .. E

I

0;;'"

8 ::=.. x

j E

~

I

1'-"'

~

I E

~ I

I

E -

+

~

....

~T .. i::

I'"

~

I::

8'" .!:!. O~

00

..9

"II

TII

44

PARTICULAR GENERALISED FUNCTIONS

FOURIER ANALYSIS AND GENERALISED FUNCTIONS

powers (x-c)-m for different real and complex values of c. The of x n , and of (x-c)-m for real c, can be read off from table I. For complex c we need the following additional result.

whence, by table

F.T.

EXAMPLE 22.

g(y) =

The F.T. of {X-(CI +icz)}-m for cz+o is

I

and example

22,

- 2~i o'(y) + HeZ1rili + e-Z7rill ) ( - msgn y) -l{2m){ -H(y)e-Z7rIl +H( _y)e Z7rll},

( -2my)m-1 Z1r 217iH(-c y)sgnc e- !l/(C1+!c.>. z z (m-I)!

which on simplification takes the form given.

The result (12), with cx=m-I and t=217Ic z l, says that the F.T. of (55)

EXERCISE

8. Show that

EXERCISE

9. Find the

PROOF.

is g(y) = (y - i ICz D-m, whence by Fourier's inversion theorem the F.T.

is

of

( . )-m = (sgnc z)m g( ) X -IC X sgnc z

z

EXERCISE 10.

( _217iy)m-1 (sgncz)m j( -ysgncz) = 217isgncz (m- I)T eZ7rC ,1I H( -ycz)· (57)

Hence, by theorem 7, the F.T. of {X-(CI +icz)}-m is e-Z7ri e,lI times (57), as (54) in fact states. It is worth noting that expression (54) is 2H( -czy) times what is got by inferring (incorrectly) the F.T. of {X-(CI +icz)}-m from that of x-m by means of the result quoted at the bottom of table I (which is valid only for real a and b, and therefore applicable only for cz=o). It follows that the F.T. of (x-cl)-m is not the limit of that of {X-(CI +icz)}-m as cz.-+o either from above or below, but that (since H(-czy)+H(czy) = I) it is half the sum of the two limits. (This point emerges clearly also from a contour-integration approach to the evaluation of Fourier integrals.) We end this section with an example on finding the F.T. of a rational function. EXAMPLE

23. The F.T. ofj(x) =x5j(x'- I) is '0'( ) g(y) =!.-L + !1Ti(e-Z1r 1111 - cos 217Y) sgn y. 217

PROOF.

(58)

In partial fractions,

t

t

!

.1.

X+I

X-I

X+I

X-I

j(x)=x+--+----.-~,

(59)

F.T.

of

(I -x)-ilog(I -x)H(1 -x).

(56)

45

Find the

F.T.

of

(60)

ASYMPTOTIC ESTIMATION OF FOURIER TRANSFORMS

47

by a simple substitution, whence

Ig(y) I = I~J~oo (!(X)

-f(x+ 2~)} e-21rlxV dxj ~;J~oo If(X)-f(x+ 2~) Idx,

CHAPTER 4 THE ASYMPTOTIC ESTIMATION OF FOURIER TRANSFORMS

4.1. The Riemann-Lebesgue lemma An asymptotic expression for a function is an expression as the sum of a simpler function and of a remainder which tends to zero at infinity, or (more generally) which tends to zero after multiplication by some power. When we do not know the Fourier transform of a given function, it is convenient to possess at least an asymptotic expression for it. In this chapter we develop a method which leads quickly to such an asymptotic expression for most functions occurring in applications. The method involves writing the given function, say f(x), as the sum of a simpler function F(x), whose F.T. G(y) we know, and of a remainder fR(X), whose F.T. gR(Y) tends to zero, or (more generally) is such that the F.T. (21Tiy)NgR(Y) of its Nth derivative fRN)(x) tends to zero. Then the F.T. off(x), say g(y), satisfies

as Iy I-+co. To develop such a method, we need a simple technique for recognising functions whose Fourier transforms must tend to zero as Iy I-+co. The following theorem is the classical result which does this for ordinary functions.

If f(x) is an ordinary fumtion absolutely integrable from - co to co, and g(y) is its F.T., theng(y)-+o as Iy I-+co. lemma)~

PROOF. This theorem is proved in standard books on analysis; we reInind the reader that one writes

g(y)=Joo f(x) e- 21r1xv dx= -co

-Joo f(X+!-) e2y -co

EXAMPLE 24. If f(x) is continuous except at x=xl , X=X 2 ,

..• ,

x=x1l1 , and if

f(x)=O(J x-xm IPm) as X-+x m, where fJm> -I for m= I toM, and f(x) = 0(1 x IPo) as

Ixl-+oo,

thenf(x) is absolutely integrable from F.T. g(y)-+o as Iy 1-+00.

where fJo< -I, -00

to

00

(4)

(5)

and therefore its

PROOF is immediate from the integrability properties of the comparison functions.

g(y) = G(y) + gR(y) = G(y) + 0(1 Y I-N)

THEOREM 16 (The Riemann-Lebesgue

which tends to 0 as Iy I-+co by a fundamental theorem of integration. To use theorem 16 one must be able to recognise absolute integrability in commonly occurring functions. The most useful test is as follows.

EXAMPLE 25. The function

f(x) = I.0- I 1-1sgnx

(6)

satisfies the conditions of example 24, with Xl = - I, x 2 = 0, X s = I, fJl= -t, fJ2=0, fJs= -t and fJo= -2. Hence its F.T. g(y)-+o as Iyl-+oo.

4. 2 • Generalisations of the Riemann-Lebesgue lemma We need results similar to theorem 16 for generalised functions. We begin with some rather trivial definitions. DEFINITION 18. If g(y) is a generalised function, then any

statement like 21rIXV

dx

(2)

g(y)-+o, g(y) = O{h(y)},

or g(y)=o{h(y)},

(7)

48

FOURIER ANALYSIS AND GENERALISED FUNCTIONS

as y-+c (or as Iy 1-+(0), means that g(y) is equal in some interval including y = c (or in some interval I y 1 > R, respectively) to an ordinary function gl(y) satisfying the stated condition.

EXAMPLE 26. 8(y) +y-l-+ 0 as Iy 1-+ 00. DEFINITION 19. If f(x) is a generalised function which equals an ordinary function fl(x) in the interval a < x < b, andfl(x) is absolutely integrable in the interval (a, b), then we say that f(x) is absolutely integrable in (a, b). EXAMPLE 27. 8(x) is absolutely integrable in (0,00) and (-00,0), but not in ( - 00, (0). THEOREM 17. If a generalised function f(x) is absolutely integrable in (-00,00) andg(y) is its F.T., theng(y)-+o as Iy 1-+00. PROOF. This, by definitions 18 and 19, is just a rewritten version of the Riemann-Lebesgue lemma. Now, the condition in theorem 17 may be split up into two conditions-absolute integrability in every finite interval ( - R, R), which holds for the function of example 24 subject to equation (4) alone; and absolute integrability up to infinity, which in that example requires also (5). The first condition (absolute integrability in every finite interval) cannot easily be relaxed if we are to haveg(y)-+o as Iy 1-+00. Thus, functions satisfying (4) only with 13m ~ - I do not normally have g(y)-+o, as table I shows. The reader should check that all the functions in table I with the ;rm factor, or the factor I x I'" for C% < - I, have Fourier transforms which are non-zero or infinite at infinity, while the result about the F.T. off(ax+b) shows that, if the non-integrable singularity is at X= -bja instead of x=o, the behaviour of the F.T. as Iy 1-+00 is still not a subsidence, but rather a finite or infinite oscillation. Again, the delta function and its derivatives have F.T.'S which are non-zero or infinite at infinity, and so generalised functions which miss being absolutely integrable because of singularities of this type will not have their F.T. -+0. However, table I also shows that the second condition (absolute integrability up to infinity, as implied by equation (5) for example) is by no means generally necessary. The reader should check that all the functions in table I which are absolutely integrable in every finite interval (namely, those with the x" factor, or the factor I x I'"

ASYMPTOTIC ESTIMATION OF FOURIER TRANSFORMS 49 for Cl: > - I) have Fourier transforms which -+ 0 as 1y 1-+ 00, even though none of these functions is absolutely integrable up to infinity. It would be wrong to conclude from this that absolutely any generalised function which is absolutely integrable in every finite interval has its F.T. tending to 0 as 1y 1-+00. EXAMPLE 28. TheF.T. ofel"" is e-I "'v'(I+i).y'(!1T), which does not -+0 as Iy 1-+00. PROOF. The F.T. of the ordinary function e(l-f)"'·, of which ei ",' is easily seen to be the limit as e-+o, is

J:II) eC

i -€)",'-Z"I2:11 dx = e"'v·/(I-€)

=e"'I1'/(I-€)

J:II) eC

l -€){",-"III/(I-€»)2 dx

J

(e:i)'

(8)

which in the limit as e-+o is the function stated. However, the function of example 28 is somewhat exceptional in that it oscillates with a frequency which itself increases to infinity as I x 1-+00. Most functions occurring in applications do not do this, and the following definition and theorem represent an attempt to include most of them in a general statement without making the latter too complicated to prove. DEFINITION 20. The generalised function f(x) is said to be 'well behaved at infinity' if jor some R the function f(x) - F(x) is absolutely integrable in the intervals (-00, -R) and (R,oo), where F(x) is some linear combination of the functions elkzlxl',

elk"'lxl'sgnx,

elkzlxl'loglxl,

elkzlxl'loglxlsgnx, (9)

for different values of 13 and k.

NOTE. Obviously no values of 13 < -

I

need be present in F(x).

THEOREM 18. If the generalised function f(x) is well behaved at infinity and absolutely integrable in every finite interval, and g(y) is its F.T., theng(y)-+o as Iy 1-+00. PROOF. It is convenient to divide up the function F(x) of definition 20 into the terms l\(x) with 13> - I and the terms Fz(x) with 13= - 1. If .Fs(x) = 0, the proof of theorem 18 is trivial, for l\(x) is absolutely 4

LFA

SO

FOURIER ANALYSIS AND GENERALISED FUNCTIONS

ASYMPTOTIC ESTIMA nON OF FOURIER TRANSFORMS

integrable in every finite interval (it satisfies equation (4)) and thereforef(x)-l\(x) is; but the latter is given to be also absolutely integrable in (-00, -R) and (R,oo), and hence finally in (-00,00). Hence, by theorem 17, if GI(y) is the F.T. of l\(x),

g(y)-GI(y)-+o as :Iy 1-+00.

(10)

But GI(y)-+o by table I, and sog(y)-+o as Iy 1-+00. This method fails when J;(x) =1= 0, since F2(x) is not absolutely integrable in any interval including the origin. However,

F2(X) Fa(x)= {F;(R)(R+x):~( -R)(R-x)

CI x I> R),} (I x I
(II)

is absolutely integrable in every finite interval (in fact, it is a continuous function), and hence we can conclude as before that f(x)-l\(x)-Fa(x) is absolutely integrable in (-00,00). Therefore, if Ga(y) is the F.T. of Fa(x),

g(y)-GI(y)-Ga(y)-+o as ,. '.

Iyl-+oo.

(12)

where Ixlv-lN-1 (v+n-t)! F(x) = .)(27T) n~o n!(v-n-t)!(2i lxi)n x {( - l)nei(l x l-!v7T-t>r)+e-l(Ix!-tv7T-i

I x 1-+00

EXAMPLE 29. Ifg(y) is the F.T. of f(x) = I x Iv J.,(I x D, where J.,(x) is the Bessel function of the first kind, then g(y)-+o as Iy 1-+00 if v> -t. PROOF. f(x) is continuous except at x=o, where it is 0(1 X 12v), so it satisfies equation (4) if v > - t. Hence it is absolutely integrable in every finite interval. It is not absolutely integrable up to infinity, but, by the asymptotic expansion for J." f(x) = F(x) + 0(/ x Iv-i-N)

as

Ix 1-+00,

(13)

satisfies equation (5) and so is absolutely integrable up to infinity. Hence, by theorem 18,g(y)-+0 as Iy 1-+00. We now check this conclusion by calculating g(y). Note first that, if v> -I, lim {e-<,x'f(x)}=f(x), since for any good function

. F(x) we have

II:",

.~o

(I-e-6IX1 ) IxlvJ.,(1 xDF(X)dxl

~e if v> -

I.

I:",

I x Iv+! I J.,(I x i)F(x) I dx=O(e)

(14)

Now, the F.T. of e-6IX' f(x) is·

2v(V-t)! 2V (V-t)! + {I + (e+ 211iy)2}v+l.)11 {I + (e - 211iy)2}v+l ')11 .

(IS)

Hence g(y), the limit by theorem IS of (IS) as e-+o, is 2

and hence is absolutely integrable from - 00 to 00 (the derivative F~k(X) has only simple discontinuities at X= ±R) and so its F.T. (211iy-ik) Gak(y)-+o as Iy 1-+00, whence Gak(y)-+o and, on adding up these results for the different values of h, GaCy)-+o. Hence finally, by (I2),g(y)-+0 as Iyl-+oo.

7T)}

is a linear combination of functions of the type (9). It follows thatf(x) is well behaved at infinity, since if N> v+t thenf(x) - F(x)

But GI(y)-+o. Also, if Faix) signifies the contribution to Fa(x) of terms carrying the el "'" factor, then F~k(X) - ihFak(x) is 0(1 x 1-2 10g I x D as

51

v

+!(V-!)!

(I-4112y2)"+l.)11

(

I) lyl<211'

V

2 +l(v-i)!sinv11 -(4112y 2_r)"+l.J11

(I y I>211' ~) (16)

°

This checks that g(y) does tend to as Iy 1-+00 when v> - t; and the fact that g(y) does not tend to 0 for - I < V ~ - t reconfirms the importance of the condition that f(x) be absolutely integrable in every finite interval.

4.3. The asymptotic expression for the Fourier transform of a function with a finite number of singularities DEFINITION 21. A generalised function f(x) is said to have a finite number of singularities x = XII X 2, ... , X 111 if, in each one of the intervals -oo<x<xl , Xl <x<x2 , ... , xM-I <x <x1l1' XM<X
ASYMPTOTIC ESTIMATION OF FOURIER TRANSFORMS 53

52 FOURIER ANALYSIS AND GENERALISED FUNCTIONS

transform theory the transforms Gm(y) would exist only if all the Fm(x) were composed solely of functions of the type (17) with - 1<(3<0. The method of theorem 19 will now be illustrated by a number of examples. After studying these, the reader should practise the method on several of the exercises at the end of the chapter. It is instructive to begin with an example to which we already know the answer. EXAMPLE 31. Find an asymptotic expression for the F.T. of

f(X) is equal to an ordinary function differentiable any number of times at every point of the interval. EXAMPLE 30. o"(X) + Ix4- 5X2+41-! has the singularities - 2, - I, 0, I and 2. Most ordinary or generalised functions which occur in applications have only a finite number of singularities; for these, the method of the present section is very effective. The principal exceptions are periodic functions, which are treated separately in chapter 5. x=

j(x) = 1x Iv J,,(I x I).

THEOREM 19. If the generalised function f(x) has a finite number of singularities x = Xl' x 2, ... , XM' and if (for each m from I to M) f(x) - F.,.(x) has absolutely integrable Nth derivative in an interval including Xm, where F.,.(x) is a linear combination offunctions ofthe type

Ix-x.,. IP, 1X-Xm jPsgn(x-xm ), IX-Xm IPlog 1X-Xm I,} IX-Xm IPlog I X-Xm Isgn(x-xm )

SOLUTION. The behaviour ofj(x) near its only singularity x=o is given by the series j(x) = I x Iv (t I XpV+2n( ~ I)n. n=O n.(v+n).

£

Hence, if

(17)

and o(Pl(x-xm), for different values of (3 and p, and if j
g(y)= I: Gm(y)+o(IYI-N) as m-I

lyl-rCXJ,

where Gm(y), the F.T. of Fm(x), can be obtained from table

2V 1

(20)

is the leading term of this series,f(x)-FI(x) is 0(\ x 12v+2) as X-rO, and its Nth derivative is absolutely integrable in an interval including the origin x = 0 if N is the least integer ;;;. 2V + 2. Also, j( x) and its derivatives are well behaved at infinity (see example 29), and so, by theorem 19 and table I,

(18) I.

M

PROOF. Letf(x)- I: Fm(x)=fR(x) have F.T. gRey). ThenJkNl(x)

g(y)=GI(y)+o(!yl

m=1

has F.T. (21Tiy)NgRey), and to prove (18) we must show that this Now,JkNl(x) is absolutely integrable in an interval including Xm but no other singularity, because fNJ(x) - FJ:>(x) is, and so are FiNl(x), ... , F~I(X), F~~I(X), ... , Ft,gl(x). This being a correct conclusion for m = I to M, it follows that fRNl(x) is absolutely integrable in every finite interval; also, it is well behaved at infinity, since f(NJ(x) is given to be, and each component in each of the F~Nl(x) obviously is. Hence, by theorem 18, the F.T. (21Tiy)NgRey) offlNl(x) tends to zero as Iy l-r ex:>, as stated in equation (18). The result of theorem 19 is most often useful when f(x) is an ordinary function. Note, however, that even in this case the statement (18) of the result, let alone its proof, would be meaningless outside generalised-function theory, since in ordinary Fourier-

x

I F,(x)=I v! 2 V

-rO as Iy l-rex:>·

-N

)=

{2COSt1T(2V+ I)}(2V)! I j-N V!2v(21TlyI)2V+l +o( Y )

__ 2

-

•

>

,

+l( V-21)".smV17 +0(1 I-N) (21T Iy 1)2V+1..j17 Y , V

(21)

where g(y) and GI(y) are the F.T.'S of f(x) and FI(x) , and the 'duplication formula' (2V)!=V!(V-t)!2 2v 17-t (22) has been used to throw the result into a form which can be immediately checked from the exact form (16) of g(y). Note that, by including more terms of the series (19) in FI(x), one could make higher derivatives (the (N + 2)th, the (N + 4)th, and so on) of j(x)-FI(x) absolutely integrable in an interval including x=o, and so reduce the error in the equation g(y) = GI(y)

54

ASYMPTOTIC ESTIMATION OF FOURIER TRANSFORMS

FOURIER ANALYSIS AND GENERALISED FUNCTIONS

for large Iy I successively to 0(1 y I-N-2), 0(1 y I-N-4), and so on, depending on how many terms were included. In this way one would build up an 'asymptotic expansion' of g(y), which in the particular case here discussed would be simply the binomial expansion of 2V+I(V- i)! sin V11 (4112y2 - q+l.,j11 in descending powers of y, as the reader may check. EXAMPLE 32. Find an asymptotic expression for the F.T. of f(x) = I x II x+ I III x- I I-t with an error o(ly 1-2). SOLUTION. The singularities of fix) are X= - 1,0, + 1. Expressingf(x) near each singularity as a sum of terms (17), with an error whose second derivative is absolutely integrable in an interval including the singularity, we have

f(x)=

IX+Ill , .,j2 +O(lx+I Ii), f(X)=IX1+0(IXI2),}

f(X) = ,

.,j2Il+Slx~11 X-I 2 2

t

(23)

EXAMPLE 33.

+ e-27711l{.,j2 .,j(211) +_S_ ( - isgn y ).,j(!11)} + o(ly 1-2) (211 Iy I)i 2.,j2 (211 IY I)I

I

g(y)= with an error 0(1 y SOLUTION.

1-2).

I

l

0

e-277l2:11 cosh x (I-xi)l dx,

(2S)

This g(y) is the F.T. of

(26)

which has the singularities x=o and 1. Near x=o,f(x) -~(x) has absolutely integrable second derivative, being in fact 0(1 x 12), if Fl(x)=.H(x). Nearx=l,

(_1_ e277111 + sisgny e- 277111) Iy I-I 411 .,j2 811 .,j2

- 2112y- 2+ 0 (ly 1-2).

Find an asymptotic expression for

f(X) =(1 -xi)-l(coshx)H(x)H(1 -x),

=e277111 -2.. -,J(t11) +_2_ .,j2 (211 Iy 1)1 (211iy}2

= (.,j2) e-277lll ly I-t_

large Iy I. One would build up in this way g(y) for large Iy I as the sum of three asymptotic series, one in simple powers of Iy I, one in powers with the factor e277111 outside and one in powers with the factor e-277ill outside. The' leading term' ing(y), that is, the one asymptotically biggest for large Iy I, is that in Iy I-i, which arises from the term inf(x) which is of order I x- 11-1 as x-.+ 1. This illustrates a general principle, obviously implied by theorem 19 and table I, that the 'worst' singularity of a function always contributes the leading term to the asymptotic expression for its Fourier transform. (Here, 'worst' is used in the sense that the singularity x = X m , where f(x) is of order 1X-Xm IP, is 'worst' if f3 is algebraically least.) Note that the precise order of magnitude of the error term in (24) (instead of the rather vague information that y2 times it tends to zero) can be deduced from the orders of the error terms in (23); the worst of these is of the 0(1 X-Xm Ii) form, so its F.T. is 0(1 Y I-!)· We can increase the precision of equation (24), therefore, by replacing the o(ly 1-2) by O(ly I-i).

sgn(x-I)+O(lx-Ili),

as x-.+ - I, 0 and I respectively. Taking the right-hand sides of (23) (without the error terms) as Fl(x), J<;(x) and J<;(x) respectively, we see that the conditions of theorem 19 with N = 2 are satisfied, and therefore (in the notation of that theorem) g(y) = Gl(y) + Gly)+ Ga(y)+o(ly 1-2)

55

f(x) = cosh 1 +(x- I) sinh I H(I -x)+ 0(1 -x)i (I -X)l{2-!(1 -x)} =F2(x) + 0(1 x- 1 Ii),

(24)

As in example 3 I, if higher terms in the expansions of f(x) near each of its singularities were retained in the Fm(x), then the error term in the expression for g(y) could be reduced in magnitude for

(27)

where

F2(x) =(! cosh 1)(1 -x)-lH(1 -x) + (}cosh 1 -tsinh 1)(1 -x)tH(1 -x).

(28)

56

ASYMPTOTIC ESTIMATION OF FOURIER TRANSFORMS

FOURIER ANALYSIS AND GENERALISED FUNCTIONS

with an error o( Iy 1-1), where K o is the modified Bessel function of the second kind of order zero.

Applying theorem 19 with N = 2, we deduce by table I that I

g(y) =

-.+

21T1y

{ eVrI ago l/ .j1T e-2,r1l/ (t cosh I) ----,-2.,.:(21T Iy Dt

+ (icosh I -tsinh I)

SOLUTION.

e~lsgol/ (t.j1T)}

-

(21T Iy DI

which has singularities x= -

34. If F(x) and all its derivatives exist as ordinary functions for x~o, and are well behaved at infinity, derive the asymptotic expansion EXAMPLE

F(o) reo) FIV(o) F(x) sm 21TXy dx ~ - (-)3 + (--5 - .... 21Ty 21Ty 21Ty)

er.>.

o

f(x)=F(1 x Dsgnx.

of

(3 1 )

Now f(x) has only one singularity, x=o, near whichf(x)-FI(x) has absolutely integrable (2P)th derivative if

r(o) }'(2P-2)(0)} FI(x)= {F(0)+--x 2+ ... + X2p - 2 sgnx. 2! (2p - 2)!

(3 2 )

Hence the conditions of theorem 19 with N = 2P are satisfied, whence, using table. I, p-l }'(2n l(0) (2n)! g(y) = l~o (2n)! 2(2my)2n+I+ o(lyl-2 P )

(33)

as Iy I~ co. The fact that this is true for all p is what is meant by the' asymptotic expansion' formula (30) for tig(y). Similarly, under the same conditions, we have

f

er.>

o

F'(o)

F'R(o)

Fv(o)

2~

2~

2~

F(X)COS21TX)'dx~ - - ( )2+(-)'-(-)6+ ....

EXAMPLE

f:

Ko(x) COS 21TX)'dx

and +

I,

where

D,

}

f(x)=H -log(t I x i)-y}+O(lx 1210g I x i), f(x)=tKo(I)H(I -x)+O(1 x- I i), respectively. Hence, by theorem 19 with N = I and table I,

e2l11 l/ I ( sgn y) e- 217I l/ g(y)=tKo(I)-. -- - - +tKo(I) - - . +o(ly 1-1) 21T1y 2 2y - 21T1y I

-2

=Ko(I)--+-I-,+O(IYI ), 21Ty 4 Y where the precise form of the error term is 0(1 y 1-2) because the 'worst' error term in (37) is of the 0(1 x - X m i) form. EXERCISE 1 I. Find an asymptotic expression for the F.T. of e-ixi with an error o( 1y 1-3), and check it against the exact expression obtained by direct integration. EXERCiSE 12.

Derive the asymptotic expansion of er.> e-217ixl1 dx -er.> 1 + I x Is .

f

Find an asymptotic expression for the F.T. of Ix'- 5x2+41-tsgnx with an error o(ly 1-1), and state the precise order of magnitude of the error. EXERCISE 13.

EXERCISE 14.

Derive the asymptotic expansion of

f: EXERCISE 15.

(I -X)-ICOS21TX)'dx.

Find an asymptotic expression for I logx --)-. sin 21TX)' dx o (I-X

I

35. Find an asymptotic expression for

g(y) =

1,0

f(x)=tKoCx)H(x+ 1)+0(1 x+ I

sin21Ty

SOLUTION. The 'half-range' Fourier sine integral (30), bY§I'4 (see especially equation (18)), signifies i-ig(y), where g(y) is the F.T.

Thisg(y) is the F.T. of

f(x)=tKo(l x i)H(x+ I)H(I -x)

+o(ly 1-2)

as Iy I~co. From the detailed expressions for the errors in FI(x) at x = 0 and in F2(x) at x = I, we can say that the precise order of magnitude of the error in (29) is O(ly I-i).

f

57

with an error 0(1 y the error.

1-2), and state the precise order of magnitude of

FOURIER SERIES

CHAPTER

5

'5' c 1 n=O

FOURIER SERIES

s·x.

Convergence and uniqueness of trigonometrical series as series of generalised functions THEOREM zo. The tngonometrical series

'" 2;

C

n=-co

n

G(vldv

'J '"

(5) where the right-hand side of (5) converges even when each term is replaced by its modulus because G(y)=O(ly I-M) for all M. Hence, by theorem IS, the series obtained by differentiating (4)

'" C 8(y - n/zl), converges to the generalised term by term, namely 2; n

-'"

eim,zll

converges to a generalised function fix) if (and only if) Cn = 0(1 n IN) for some Nasi n 1-+00, in which case the F.T. off(x) is

'" c 8(y-n/zl). g(y)= 2; n n=-oo

(z)

Further, the sumf(x) of (I) can be zero only if all the Cn are zero. From this follows the uniqueness theorem, that two different tngonometrit:al series (I) cannot converge to the same function (since then their difference would converrre to ",,,,.0 hUT 1."",,, nnn_",,,.·,,.

-S/2L

-4/21

-3/2t

-2/2i

-1/2L

T

elli 1/2L

y

2/2L

3/2L

4/2L

5/21

II ~

" . ~.

-;~

.'

.

'0'

Ult: tit:nt:ti ~:i\J, ootalnea oy takIng 1'0urler transtorms term bv term converlZes to~ the F.T. of ff:r\ H~n{'p III d,11U

umy

11

tiraph of the 'step function' gl{Y) of eouation

1"lg·4· J: ,11I>L,

function

assume mal: cn UU n r ') ror some ~ defined (see fill". ..I.) as 0

"

'5'c l>i\Y J

.

.,..

N.

" 1 ) C 8=0

Then the step U~:~

(0 ~ y ~ I/Z!},

{rhZ,;::, v,;::, (r..J.;v;;n {

..

r/zl,;::,v,;::, .fr

I",

_,

\' '"

~I

£11

~

" J ' O\J

\'1

'.

'o"v

wnose 1".'1'. IS glYJ. Note also that, 1t ((x) were zero thenPrVl must be zero and so, hv ~ ~ rtf.,\ IX

_

•

•

'J

\':>/~'

...~

'II,

l11Lll>L u<;; ,£,<;lU.

ISlInplt: prum 01 convergence ana UlliQueness under the rather unrestrictive condition r .nfl n IN) ," ..:l1~. ~'"

,.\1,,11.

J. IHti

..

uenlliuon 7, and so can be reg-arded as a generalised function. Further, we can write ....<;

.

Ul<;;

8=1

(~),

OI

'" cnH(y-n/zl)- 2; '" cnH( -y-n/zl), gl(y)= 2; 5=1

~=1

(4)

prom 01 necessity ot the COndlt1on (as stated In the theorem) is so easy, however, that it may as well be given. If there were no N for which Cn = O( In F'l), then there must be al1 increasing sequence of n's, say nl> n2 , fla, ... , such that Ic.... 1> In"I" for each r. Now let

60

61

FOURIER ANALYSIS AND GENERALISED FUNCTION"S

FOURIER SERIES

G(y) be any good function such that G(nlzl) = 0 if n is not a member of this sequence but G(nrlzl) = c;,1. (For example, one might take

is of no use where generalised functions are concerned, as these cannot be integrated between finite limits.

'" G(y) = Zc;.IT(z0'-n r),

(6)

r~l

EXAMPLE

37. No suitable definition of such integration is pos-

sible which will give a meaning to

where T(x) vanishes for I x I;;:: I but is say e-x2f(l-x')

J:

8'(x) dx.

However, the idea of 'integration over a period' can be reproduced by the use of a special kind of function.

for I x I < I and in particular is I when x=o. Then G(y) is a good function by the inequality satisfied by the cTlr .) We may then infer that the series (z) does not converge according to definitions 10 and II, because

THEOREM Zl. A 'unitary function' U(x) can be found, which is a good function vanishing for Ix I;;: I and such that

'" Z n--co

U(x+n) = I

(10)

for all x. The Foun"er transform V(y) of any such function has V( 0) = I, but V(m) = 0 if m is an integer other than zero.

has all its terms zero, except those with n in the sequence n I , n 2 , " •• which are all I, and hence it increases without limit as N -+00. This completes the proof of theorem zo. DEFINITION zz. A generalised function of the form (z) is called a 'row of deltas' of spacing liz!. A generalised function f(x) is said to be periodic with period zl iff(x)=f(x+ zl). EXAMPLE

36.

elm,x/l

is periodic with period zl. Hence, by theorem

15, the f(x) of theorem zo is periodic. Theorem zo therefore states that the F.T. of a row of deltas of spacing I/zl is a periodic function of period z!. (For the converse result, see § 5.3.)

5.2. Determination of the coefficients in a trigonometrical series We now consider the problem: if we know that f(x)=

'" ~

, , _ _ 0)

,.-fl

c.. e1

zl

JI f( x) e-1m=/l dx, _I

U(x)+U(X-I)=I

for

O~X~I,

(II)

and that all derivatives of U(x) should vanish at X= ± I (so that they are continuous with zero, their value for I x I> I). One may take 1 U(X)=J exp { - I ()}dtffl exp { - I ()}dt, Ixl tl-t 0 tI-t for instance (see fig. 5). The exponential ensures that all the derivatives of U(x) vanish (with U(x) itself) at X= ± I. Condition (II) is easily proved by making the substitution t = I - s in the integral. Lastly, if m is any integer, V(m) = J:", e-2>rlrnx U(x) dx= ,.-~",J:~:e- 2>rlrnx U(x) dx

(8)

for some C,., how can these coefficients be determined? The deeper problem, to prove the existence of such an expansion for any periodic function, is postponed to § 5.3. Note that the classical solution of the present problem (equation (I z) of chapter I), Cm = -I

PROOF. Many such functions U(x) can be found. For any x, at most two terms of the series (10) differ from zero (those with I x + n I < I). Therefore, it is necessary only that

=

,.A",f.

=Jt

-t

e-2>rlrnx U(x+n)dx

e- 2>rlrnx dx ={1 0

(m=o),} (m9=o),

(13)

which completes the proof of theorem Zl. The idea of integrating a periodic function f(x) over a period can now be replaced by the idea of integrating f(x) U(xlzl) from - 00

62

63

FOURIER SERIES

FOURIER ANALYSIS AND GENERALISED FUNCTIONS

(8) holds, or (what is the same thing) then g(y), the

:!: U(xj21+ n) I; but the integration is permissible in the theory n==-CO

"" U(2ly - n) = first step towards this, since Z

of generalised functions since U is a good function.

F.T.

of Jf(:i),

I.

n=-oo

THEOREM 23. If f(x) is a periodic generalised function with period 21 and F.T. g(y), and if

U(x)

HI

cn = ;IJ:""f(x) U(~) e-1nl""/ldx=J:<7> g(y) V(n- 21y) dy, (16) 0075

'Where the equality of the t'Wo forms of Cn follo'Ws from theorems 6 and 7, then g(y) U(2ly-n)=c'n 8(y-nj21).

0·5

PROOF.

We are given that

0.25

f(x)-f(x+ 21) =0.

(18)

Hence, taking Fourier transforms, -1-0

-0-5

0

0.5

(x-1)

x

which means that

g(Y)(I -e41rllll)=o,

Fig. 5. Graph of the unitary function U(x) of equation (1Z), illustrating the property U(x)+U(X-1)=1 foro';;x';;1.

n

-------------C~m-_'MilJ:,J(x)U(x/21) e

unctIOn

(14·)

(21)

6-3' Existence or Fourier-series representation for any periodic genelalised function All the main objects of a theory of Fourier series listed in § 1.3 have now been achieved, except that of proving that, if f(x) is any periodic generalised function, and the Cm are defined by (1+), then

which proves the theorem. To complete the proof that KCy) satisfies equation (1) it ooed only be proved (after equation (17» that

IIT11lWPdx,

(22)

<7>

g(y)= Z g(y) U(zly-n) n--c:o

FOuRIElt SERIES

65 because the left-handseries is absolutely convergent. Butfor m M r , say, co co r (29) k k tlm... > k (W e, 8-1 .. -N.+l .-1

64 FOURIER ANALYSIS AND GENERALISED FUNCTIONS fox any generalised functWng(y). TIns appeals almost ob9ious by theorem 21, but the proof regUlres SOme care because our generalised functions are such a very unrestricted class of objects. A lemma on convergence of series is first needed. co

by (26) and (27)' Hence for the increasing sequence M 1 , M 2 , M a, ..•

..-0

the series k Xn am... increases without limit, in contradiction to the ..-I hypothesis. This proves the theorem.

k x..am... is absolutely

THEOREM 24. If the am... are such that

CX>

convergent and tends to afinite limit as m ~ OJ for any sequence x.. which co

is O(n) as

n~OJ, then k

lim Om,nconverges to the sum lim

i: am ..'

m-+«> n=-O

n=O m-+(O

t

THEOREM 25. If g(y) is any generalised function, and U(x) is a unitary function, then

PROOF. If the conclusion were false, then an infinite sequence of N's such that

\

~

lim am,.. - lim

11,.=0 m-..oo

£ am... I>e

co

g(y) = k g(y) U(2Iy-n).

(24)

PROOF. By definitions 10 and G(y) is any good function, then

would exist for some e > 0. This means that lim k"" am,.. > e, m_ co ..-N+l

(25)

To prove this it is necessary to go back to the definition of the generalised function by a regular sequence of good functions :".(y). Writing

am,..

(",1'.\

+ 1 -".,.,.

for all m ~ Ms. Now choose N 8+1> N s as a member of the sequence satisfying (25) such that 00

k

..-N.+.+i ,

n I a.'l{•. n I < e.

n-O

n~.,.. is

n=-()

(27)

00

absolutely convergent.

k x"am... = k k am,..' 8-1 .. -N.+l n-l

¢(y)

""....,..00

(28)

lim :L urn,.., 116-+0)

11<-0

(33)

~

,,-a

x..{U(zly n) + U(2ly + n)}

(34)

is a fairly good function if x.. = O( nN) as n -+ co for some N a result which we use for N = I. (The proof is trivial, since at most t~o terms of the series can be non-zero for any y.) Hence ¢(y) G(y) is a good

: /

~

co

co

llO

(3 2 )

'where on the right hand side equation (10) has been used. Now, WIth the same gloss regarding the term in curly brackets,

If naw, for each n, x.. is the number of "V8 which are less than n, then x.,. - O(n) as n~OJ, and so co

n) + U(21y + n)} G(y) dy,

co

:L lim um...

co

This is possible since I:

fcoco gm(Y) {U(21y

except for n o when the term in curly brackets is taken as only -v[Zly) lDstead Of 2U(2ly), we can wnte equation (31) m thelorm

<Xl

~

equation (30) means that, if

(3 1)

~;;;~;~:~:~;i~~~~i.:~:::h~~:::7:~:;'::~: bY (25)) such that Ie . ..- N. n

II,

N fco ~~ n!:N _cog(y) U(20'-n)G(y)dy= fco _cog(y)G(y)dy.

or that it is < -e. There must therefore be an infinite subsequence of these N's satisfying only one of these alternatives; the proof proceeds along siInilar lines whichever it is, but for definiteness suppose it is (25)' Let the first of these ..:v's be N 1 , and then define the sequences N 8 , M 8 by mductlon as fonows. If N 1 , ... ,N8 and

~

(3°)

.,..=--00

m-+oo '8=0

s

LFA

FOURIER ANALYSIS AND GENERALISED FUNCTIONS

FOURIER SERIES

function, by theorem I, and, by the definition of a regular sequence (definition 3),

THEOREM 27. Under the conditions of theorem 26, cn is necessarily 0(1 n IN) for some N as In 1-+00. Hmce

66

fi() X is absolutely convergent and tends to a finite limit as m-+oo. Hence, by theorem 24, we have equation (33) and hence equation (31) and hence the theorem. After this digression on convergence matters the main theorem follows at once. THEOREM 26. If f(x) is any periodic generalised function with period 21 and F.T. g(y), then ~

n--oo

c" =2..I~ f(x) 21 _~

where

(36)

U(::.-) e-imrx/ldx. 21

PROOF. The second of equations (36) follows from theorems 25 and 23, and the first follows from it by theorem 15. NOTE. If in additionf(x) is absolutely integrable (as assumed in the classical theory of Fourier series) then we have simply

c" = I21

II

-I

f(x) e-i"nx/ldx,

since for such anf(x) we can write I

~

c,,=- },:

f(2m+ll1

21m=_~ (2m-Ill I

<0

=-},:

21m=_~

II

-I

fl(X) =

dN+2fl(x)

cixN+2

(-1 (I (N + 2)! + n-~~ + n~1 n in1T ~)

XN+2

Co

) '

)N+2

C

(39)

einnx/I

is a continuous function. PROOF. By theorem 20, the series (36) which have just been proved to converge could not do so unless en = 0(1 n IN) for some N. The first of equations (39) is obtained by term-by-term differentiation (permissible by theorem 15). The fact that fl(X) is continuous follows from the fact that the series for it is absolutely and uni~

~

f(x)= },: c"el"=/1 and g(y)= },: c"o(y-nj21),

where

67

f(X)U(xj21)e-l"nXlldx

f(x) U(xj21+m)e- 1nnX/ldx,

and then interchange the order of summation and integration. Therefore, the classical Fourier-series theory is included in our more general result. There is a simple corollary of theorem 26, of little practical importance, but of some general interest.

formly convergent, by comparison with the series},: n- 2 •

n-l

The fact that a periodic generalised function must necessarily be a repeated derivative of some continuous function is interesting as showing that there is a limit to the seriousness of the singularities that these functions can have. We have now derived all the general properties which were noted in § 1.3 as necessary for a satisfactory theory of Fourier series; there exists a unique Fourier-series representation of any periodic function, which converges to the function, whose coefficients can be determined, and which can be differentiated term by term. We conclude with a number of more special results.

5.4. ExaIllples. Poisson's sUIIlIllation forIllula This section is mainly concerned with consequences of the following result. EXAMPLE 38. The generalised function

f(x) =

~

},:

o(x- 2ml)

m--oo

exists by theorem 20 and is periodic with period 21 by theorem IS. Its nth Fourier coefficient is

5'2

68

FOURIER SERIES

FOURIER ANALYSIS AND GENERALISED FUNCTIONS CD

and so by theorem 26 CD

I

f(x) =-

but I + ~ (0) cos mTX/ l. In the theory of generalised functions, n-l however, we have seen that any series can be differentiated term by term. To reconcile the apparent contradiction, note that the sum of (47) is not x, but rather is a periodic function which coincides with x in the period ( -I, I). Thus, if

e1n""'II,

~.

21 n--CD

an equation whose real form I

CD

~ m--CD

I

CD

n1TX

8(x-2ml)=-+- ~ cos21 I n-l I

f(x)=x

x-2ml {(2m-I)/<x«2m+I)/}.

(-l<x
is worth noting; and the F.T. of f(x) is I g(y)=2/

n)

CD ~

__ CD

n

(48) then by theorem 26 and the note following it f(x) is equal to the series (47), and by differentiation

8 ( y-21 .

In words, a row of equal deltas has as its F.T. a row of equal deltas. EXAMPLE 39. From (43), by differentiation, 1T

CD

~

n-l

n1TX

CD

8'(x-2m/) = -2

~ nsin-.

I n-l

m--CD

I

NOTE. This is a trigonometrical series on which the old-fashioned 'summability' methods of handling divergent series (which preceded the introduction of generalised functions) broke down. They gave its sum as zero for x 4= 2ml (which is correct, as the generalised function is equal to 0 in any interval not including these points), but also gave it as zero for x = 2ml, on the ground that every term of the series vanished at these points. Thus they missed the true character of the singularity at x = 2ml; and (what was perhaps even worse) uniqueness was absent in these theories, because of the existence oftrigonometrical series whose' sum' was everywhere zero. EXAMPLE 40. In the classical theory of Fourier series one calculates the nth Fourier coefficient of the function x in the range

( -I, I) as

I C

n

giving

= -

II

21

-I

21

CD

-

1T

~

n-l

il x e-11l1,,,,/1 dx = - ( - I)n

n1T'

(-

1)11.-1

n

•

n1TX

CD

f'(X)=2 ~ (-I)n-lcos-I-·

CD

= I -21

~

8{x-(2m- I)/}.

(5°)

m--ClO

Thisf'(x) is not the function

I.

In fact, by equation (43) above, it is

CD

n1TX

n-l

l

-2 ~ ( - I)nCOS-,

in perfect agreement with (49). We learn something useful by applying Parseval's formula to the result of example 38. THEOREM 28 (Poisson's summation formula). If F(x) is a good function and G(y) its P.T., then

(4 6)

(52)

n1TX

PROOF. This is the equation

SIO-

l

as the full-range Fourier series of x in (-1,/). Now, it is usually stated that such a series cannot be differentiated term by term; and, CD

in fact, the Fourier series of I in ( -I, I) is not 2 ~ ( n-l

I )_1 cos n1Tx/ I

f:CDf(X)F( -x) dx= f:CD g(y)G(y) dy

of theorem 6, withf andg as in example 38 (equations (40) and (44)), and 21 replaced by A.

70

FOURIER

ANAl YSIS

A.ND G:IlNBR1'"LISED

FUNCTIONS

FOURIER SERIES

EXAMPLE 41. If F(x)=e-x", then G(y)=7Tfe-"2y' and theorem 28 gives m -w

'-

1 2•

n--co

This equation is obvious only for it=.j1T. The left-hand side converges very rapidly for it> .j1T, and so does the right-hand side for it <.j1T-in both cases, faster than

PROOF. We may write

cosnz n-l 1 +n a:>

2k

I + 2e-1T + 2e-4.. + 2e-lI1r + ...

= 1 +0'0864278+0'0000070+(10-12)+ ....

(55)

The equation can therefore be used to compute either side very rapidly for any positive it. EXAMPLE 42. If F(x)=e-X·-2..1."., then G(y)=1Tfe-1Tl(YH)' and theorem 28 gives 1 +2

In the period 0 < Z < 21T we have

EXAMPLE 43.

nz)

2 it

where

e1mz

a:>

--+ 2 1=

k

--= 2

m--ao 1

+m

a:>

k F(m),

m--a:>

(58)

Ixz

F(x)=~ and its F.T. is G(y)=1Te-l2.-y-s1 I+X

(as obtained by the method of § 3.5). Hence, by Poisson's summation formula, a:>

a:>

k F(m)= k G(n)

m--oo

n=-co

.j1T e-"'''' (1a+2 :> 21T ka:> e-m·"·cos(21Tmitz)=_ k e-n·..·j"·cosh-,

m-l

it

n-l

(56) which is Jacobi's transformation in the theory of theta functions. Similar remarks about convergence apply (the cosh in the series on the right WOlsellS its convergence only shghtl ,smce the ex resston IS a enodic function -1 take 13' 1< i- J - 1 for computational put poses), and the equation is thelefole the key to the computatIon of the theta functions and, through them. of the elliptic functions We may note here that Poisson's sUlIlmation formula (denved ill theorem 28 for good functIons) can be extended to functions which are not good by approximating to them by a sequence of

where the geometric series have been summed for the case 0 < z < 21T. Equation (57) now follows from (58) and (60) by rearrangement.

The following theorem enables us to use the method of chaplel 4 to find the asymptotic behaviour as In I ~co of the Founer coeHlclents Cn of a gwen function [(x). THEOREM 29. I]j(x) zs a perzodzc generalised function with period 2l, then C(y), the F T af (2l)-lf(x) U(x/21), is a continuous /u1tction uner coe cien Cn 0 x.

ing e lffilt. or e ormula to be valid in the limit, it is sufficient for F{x) in theorem 28 to be replaced by a function continuous and of bounded vaIiation in (-co, co) and such that the uifinite

--~::-~~'~t~;E;;~:i~;;:~ small it, as in the above examples. Occasionally the formula leads to a simple analytical form for the sum of a trigonometrical series.

PROOF.

By theorem

IS

the F.T. of (61)

IS This is an absolutely and uniformly convergent series of continuous functions in any finite interval of y, since by theorem 27 Cn = 0(/ n IN) for some N but V(20J -n) = 0(1 n I-N-2) as In' ~co.

72

FOURIER SERIES

FOURIER ANALYSIS AND GENERALISED FUNCTIONS

Hence C(y) is continuous. Also, C(m/21) =Cm since, by theorem V(m-n)=o except when m=n, and V(o) = I.

21,

NOTE. It is easily shown that C(y) is in fact a fairly good function, but this result is not needed below. Now, it was noted in connexion with definition 21 that a periodic functionf(x) could not have a finite number of singularities (unless the number were zero). However, theorem 29 shows that, provided only f(x) U(x/21) has a finite number of singularities, then the method of chapter 4 can be applied to determine the asymptotic behaviour of C(y) and hence of the cn's. Since U(x/21) vanishes for I x I> 21, the condition is simply thatf(x) have a finite number of singularities in anyone period.

73 Xl> -1(1 -e). In this case, (21)-1 f(x) U(x/21) is a generalised function with only the singularities X = Xl' X 2 , ' , ., XM, and is equal to (21)-lf(x) in an interval -/(1 -e) < x <1(1 + te) including all of them. It and all its derivatives are 'well behaved at infinity' (they are zero for I X I ~ l(I + e», and so its F.T. C(y), by theorem 19, satisfies I M (65) C(y)=-l I: Gm(y)+o(IYI-N) as Iyl-+oo, 2 ",-1

whence equation (64) follows by theorem 29. U(x)

10()

DEFINITION 23. The periodic generalised function f(x) , with period 21, is said to have afinite number ofsingularities x = Xl' X2 ' ' ." XM in the period -1 < x ~ l if, for some e> 0, f(x) is equal, in each one of the intervals

0-75

o-s

-l<x<xl , Xl <X<Xa' ... , XM-l <X<XM, xM<x
I X-Xm IP, IX-X", IPsgn(x-x",), IX-X", IPlog IX-X", I,} I X-Xm IPlog I x-x", Isgn(x-xm)

(63)

and 8
~

2 m-l

where Gm(y),

th~ F.T.

Gm(!!.l)+o
l-

N ) as

Inl-+oo,

ofFm(x) , can be obtained from table

(64) 1.

PROOF. The result follows most directly if we take the unitary function U(x) in theorem 29 to be one which equals I when -!(I -e) ~X ~ HI + ie) and 0 when X ~ -HI -ie) or x~ HI +e). Here, e is that of definition 23, assumed chosen so small that

0'25

-10() -0·5 0 0'5 1-0 Fig. 6. The unitary function U(x) of equations (66) and (67). in the case e=O'2. Note that U(x) + U(x- 1)= I for o~x~ I, and also that, wherever in the period -I < x ~ 1 the singularities Xl> X,... '. X", are (the crosses indicate possible values of x",/zl), we have U(x",/21) = I for III = I to M for sufficiently small e.

To complete the proof, we need only produce a unitary function with the stated property. This U(x) (see fig. 6), besides being 0 for X ~ - t + Ie and x ~ t + t e, and I for - t + ie ~ X ~ t + le, is

f

6

6

(2%+1-t )/(t

o

)

exp { _ _I_} dt/fl exp { _ _I_} dt t( I - t) 0 t( I - t)

(66)

for -t+le<x< -t+ie, and

I

I

(20:-1-t6)/(te)

exp{ _ _ I }dt/fl exp{ _ _ I }dt

t(I -t)

0

t(J.-t)

(67)

for !+le<x< i+te. Theorem 30, like theorem 19, is most often useful whenf(x) is an ordinary function, but again the theorem would be difficult to

74

FOURIER

ANALYSIS AND GENERALISED FUNCTIONS

FOURIER SERIES

the apparatus of generalised functions. The theorem will now be illustrated by some examples.

state without

. in an asymp 0 c expressIOn, WI error 0 for the nth Fourier coefficient c,. of the periodic function

.f(x) = e1coaa:I".

b

n

Icosxl3 cos6 x Icosxl9 ji() x =1+--,-+-,-+--,-+ ..., I.

where

2.

•

=21C

n

(68)

SOLUTION. In the period -11 < x:::;; 11 the singularities of f(x) are wherecosx=o, thatisatx= ±!11. Whenx-+±I11,

75

as x ... 0 (w here for OlIce the error in the expression quoted 1S zero!) and as x-+ 1. Hence, by theorem 30 with N = 2, and table I,

=-1 -i

(~11) "2"

1 (11n/I).

= 2( -

I )n-1-,jI 1Tn

1

21 11in

-I~e-n1Ti-+o(n-2)

}

+ -,j(tI ) + 0(n-3)

(74)

11nt

as n -+ cx:>, where the precise form of the error term follows from the detailed expression for the error in (73).

3.

_ I _ I I _ I Icosxl=lx+!11I-31lx+!1113+51 x+t11 5_ ....

(70)

Hence 3 f(x) = I + I x+: t11 1 -!-1 X+:!11 15 +M-1 x+: t11 17

+t(x +: t11)6- t(x+: t11)8+ 0<1 x+: t11 19 )

(71)

as X-+±!-11. Hence, by theorem 30 with N=9 and 1=11, and table I,

NOTE. This example exhibits a common feature, where Fourierseries representations of continuous functions in a limited range are concerned-namely, that the 'worst' singularity of that periodic function, which coincides with the given function in the range, is a simple discontinuity at one of the end-points. By theorem 30 and table I, the nth Fourier coefficient for large n is then necessarily of order n-l . But there are, of course, exceptions to this rule, that is, functions for which no such discontinuity occurs at either endpoint (see, for example, exercise 18). XERCISE 16. Sum the serie EXERCISE

I

.

Find an

2

error

n

where the precise form of the error term follO\n from the detailed expression for the error in (71).

ExAMPLE 45. Find an asymptotic expression for the coefficlents bn m the Fourier sine series for .lx in the range 0 < x < l, with an error O(n-2) as n-+oo. SOLUTION. By § 1.3, this Fourier sine series is simply the full~nge F;rier series of the odd function I x It sgn X III ( -1, 1); and coe dents b.. in the sine series are zi times the Fourier co-efficients En of the periodic function f(x) which equals 1x ,i'SgllX in the period -1 < x ii!i; 1. Now j( x) has SlllguIarittes at x = 0 and 1in this period, and

e

f(x) = I x li sgnx, .f(x) = -Itsgn(x-l)+ tz-t(x-I) +0(1 x-Iii),

(73)

0(1 n I 2), and state the preC1se order of magnitude of the error.

ExERCISE 18. Fmd an asymptotic expreSSlOn for the coefficients an in the Fourier cosine series for xlogx in the range 0 <ex <e I,

with an error the error.

0(11-3),

and state the precise order of magmtude of

77

INDEX Absolutely integrable, 2, 21-4, 46-56 in eve"Y finite interval, 48-56 Arbitrary constant in definitions of particular generalised functions, 3
Asymptotic estimation, I, 10 of Fourier transforms, 46-57 of Fourier coefficients, 71-5 Asymptotic expansion, 50, 54-7 Bessel functions, 50-I, 53-'7 Boundary conditions, 4-5, 9 Cauchy principal value, 14, 37-8 Classical theory, of Fourier series, 7, 60, 66 of Fourier transforms, 8-<), 24 Constant, arbitrary, in definitions of particular generalised functions, 39-43 Constant generalised function, 17, 20 Contour integration, 2, 33, 44 Convergence of Fourier series, 6-'7, 58~0

Delta function, ',9-'3,23,25--<),3', 36-4 ,48 ' of, 17 definition derivatives of, 11-13, '9, 26-'7, 29, 32, 36, 39,43-5, 52, 61, 68 F.T. of, 9, 19-20 row of, I, 58-69 Differential equation, I, 3-10 Differentiation, of generalised function, 18, 24-5 of series term by term, 3-4, 7, 29. 59,68-<) with respect to a parameter, 29, 34-5 Dipole, 13 Dirac, P. A. M., I, 9-10 Distributions, I Elliptic function, 70 Equivalent regular sequences, 17 Error term, 46, 52 precise order of magnitude of, 55-7, 74-5 Even function, 5-'7,'0.26-'7,3',38,4'

Factorial function, 32-6, 5', 53-4 Fairly good function, IS, 18--20, 28-<), 65,7 2 Fourier coefficients, 3-'7, 60-'75 asymptotic estimation of, 71-5 Fourier cosine integral, 10, 56-'7 Fourier cosine series,S, 7, 68-'7 1, 75 Fourier inversion theorem, for good functions, 16 for generalised functions, 20 for particular generalised functions 34, 40-4 Fourier series, I, 3-'7, 58-'75 classical theory of, 7, 60, 66 convergence of, 6-'7, 58~0 differentiation of, 3-4, 7, 29, 67-<) existence of, for any periodic generalised function, 62-6 half-range, quarter-range, 4-'7, 74-5 summability of, 14, 68 uniqueness of representation by, 4, 7, 58-60, 68 Fourier sine integral, 9, 56-7 Fourier sine series,s, 7, 68, 74-5 Fourier transform (F.T.), 1,8-10 asymptotic estimation of,

I~

10,

46-57 classical theory of, 8-<), 24 definitions of, 8, IS, 18 half-range, 9-10, 56-'7 limit of, 28-<), 36, 40-1 of generalised function, 18-21, 24, 27 of derivative of generalised function, 20,23 of particular generalised functions, 30-45 of good function, 15-16 of ordinary function, 24 of periodic function, 58-68 of rational function, 42-5 summary of results on, 42-5 table of, 43 Frequency, 8, 49 Generalised function, I, Ie-I4 constant, 17, 20 definition of, 17 derivative of, 18, 24- 5

INDEX

Generalised functiol1 ~CQm.J derivative of with respect to a parameter, 29, 34-5 equal to ordinary function in an interval, 25-{) F.T. of, IB-2I, 24, 27 F.T. of derivative of, 20, 33 general properties, 16-29 limit of, 27-9 limit of derivative of, 28-9, 33, 39 limit of F.T. of, 28-9, 36, 40-1 ordinary function as, 21-4 particular, 30-45, 53-'7 periodic, 60"-75 series of, 29, 58-75 well behaved at infinity, 4~57, 73 Good function, 13-16 definition of, 15 fairly, 15, 18-20, 28-9, 65, 72 F.T. of, 15-16 sequence of, 10-14, 16-18, 21-3, 65-6,70 Hadamard finite part, 14, 31-2, 34 Half-range, Fourier integral, ~IO, 5 6 -'7 Fourier series, 4-7, 74-5 Hardy, G. H., 7 Heaviside unit function H(x), 30-45, 55-8,69 Historical account of subject, 13 Improper integral, 31-2, 37-8 Indefinite integral, 14, 38 Integral powers, 35-40 multiplied by logarithms, 40-2 Integration by parts, 11,15,19,31-2,37 Interchange of order, of limit processes and differentiation, 28-9, 35 of limit processes and Fourier transfonnation, 28-9, 36, 40-1 of summation and/or integration 2, 16, 22-4, 59. 64-6 Jacobi's transfonnation, 70 Laplace's equation, 3-4 Laurent's theorem. 4 Lebesgue theory of integration, 2 Limit, of.generalised functions. 27-9 of derivatives of generalised functions. 28-9, 33. 39 of F. To'S of generalised functions, 28-9, 36, 40-1

Logarithmic singularities, 34-5. 40-3, 52, 5 6 -7. 75 Mikus/nski, J. G .• 13 Non-integral powers, 30-4 multiplied by logarithms, 34-5

o and 0 notations, 2, 47 Odd function, 5-7. 9. 26-'7, 31, 3 6 Ordinary function, as generalised function, 21-4 derivative of, 23-5 equal to generalised function in an interval. 25-6 F.T. of. 24 Oscillations, 3, 8. 49 Parseval's theorem, 16, 19, 69 Partial fractions, 42-5 Particular generalised functions, 3045, 53-'7 arbitrary constant in definition of. 3~43

Periodic function, 3-'7 equal to given function in an interval, 5-6,8.69 generalised, 60-']5 Point charge, 12 Poisson summation formula, 67-71 Polar coordinates, 3-4 Powers, integral, 35-40 non-integral, 30-4 Psi function ifr(a.), 35, 39-43 Quarter-range Fourier series, 4-6 Rational function, F.T. of. 42-5 Regular sequence. 16-18. 21-3, 65-6 Remainder, 46, 52 Riemann-Lebesgue lemma. 46-51 Rogosinski, W. W .• 7 Row of delta functions, I. 60, 67-9 Schwartz, L., I. 14 Sequence of goc.d functions, 10-14, 16-18, 21-3, 65-6. 70 Series of generalised functions, 29, 58-'75 Sgn function, 23, 28, 30-45, 47, 49. 52. 54. 5 6 -'], 7 z • 74 Singularity, 42, 51-7, 72-5 defined, 5', 72 'worst', 55, 57, 75 Smudge function, 22

Step function, 58-9 String. plucked. 5-6 Summability of Fourier series, 14, 68 Summary of Fourier transform results, 42-45 Temple, G., ,,8. '3-14 Test function. 13-14 Theta function, 70

79

Titchmarsh, E. C .• 8 Unitary function, 61-7, 71-3 Watson, G. N., 51 Waveform, 3 Wave number. 8 Well behaved at infinity, 4~57, 73 'Worst' singularity, 55, 57,75