Methods of the Theory of Generalized Functions (Analytical Methods and Special Functions)

It was soon pointed out by mathematicians that from the purely mathematical point of view the definition is meaningless. It was of course clear to Dirac himself that the J'-function is not a function in the classical meaning and, what is important, it operates as an operator (more precisely, as a functional) that relates, via formula (*), to each continuous function cp a number
PREFACE

xiii

The theory of generalized functions can be especially effectively applied to the concept of generalized solutions of linear differential, integral, integra-differential, convolutional (more generally, pseudo-differential) equations. Generalized solutions appear when studying integral relations of the type of local balance, and the taking account of these solutions leads to generalized settings of boundary-value problems of the mathematical physics. The exact definition of a generalized solution is based on the notion of a generalized derivative of a (generalized) function. A notion of a generalized solution for one or another problem of mathematical physics has been introduced by many mathematicians and physicists, beginning with Maxwell's, Kirchhoff's and Heaviside's works of XIX century. The history of this question is presented in detail in the book by Liitzen (see Liitzen (71]). In connection with the concept of a generalized solution it is relevant to cite here the following phrase from the fundamental work of L. Euler "Integralrechnung" (Euler (32, p. 199]). After obtaining the general solution of the wave equation

u(x, t)

=::

f(x - at)

+ g(x + at),

he writes: "In this way, the highly astute man (D'Alembert - author's note) has obtained the complete integral, however he did not pay attention to the fact that as the functions introduced (f and 9 - author's note) one can take not only any continuous functions (analytically expressed - author's note) but also functions completely deprived of the property of continuity." It follows from this citation that Euler came close to the notion of the generalized solution of the wave equation. In German the phrase cited is the following: "Auf diese Weise hat der hochst scharfsinnige Mann das vollstandige Integral erhalten, hat aber nicht bemerkt, daB statt dieser eingefiihrten Funktionen, nich allein die continuirlichen Funktionen jeder Art, sondern aush jene Funktionen, by welchen durchaus das Gesetz der Continuitiit mangelt, genom men werden konnen." The present monograph is an expanded version of a course of lectures that the author has been delivering to students, postgraduates, and associates of the Moscow Physics and Technology Institute and the Steklov Mathematical Institute. This book appeared as a result of revision and complementation of the second edition of my monograph "Generalized Functions in Mathematical Physics", Nauka, Moscow, 1979 (see Vladimirov (120]). Section 14 is added which is devoted to Tauberian and Abelian theorems for generalized functions from the algebra 8' (f). The sections considering convolution (Sections 4.1 and 6.5) and multiplication (Section 1.10) of generalized functions as well as divisors of identity in the algebra H(C) (Section 12.5) and to the estimates of the growth of holomorphic functions with nonnegative imaginary part in tubular regions over acute regular cones (Section 17.3) are essentially revised and extended. New sections are devoted to the study of the generalized functions P( 1l'v Ix la-I) (Section 2.7) and homogeneous generalized functions (Section 5.7) and their Fourier transform (Section 6.6) as well as the Mellin transform of generalized functions (Section 6.7) and the sweeping principle (Section 15.8). The numerous changes and additions are made, inaccuracies and misprints are corrected. In the large, the international terminology used in the theory of generalized functions is accepted in the book. The question only arose, which can be answered ambiguously, how to translate into English the main object of our investigation ~

xiv

PREFACE

an element of the space V': "distribution" or "generalized function"? In French, beginning with L. Schwartz, the term "distribution" took root; at the same time in Russian the term "generalized function" is well customized (see I. M. Gel'fand et al), however the latter is often used for more general objects. In English literature the term "distribution" is used more frequently, although "generalized function" is used sometimes as well (see, for instance, Lighthill [67]). As a result of this analysis, we chose the term "generalized function" the more so as this term was used when translating the first version of this book into English (Vladimirov [120]). I take this opportunity to thank all my associates for their constructive criticism that has helped me to improve the presentation. In particular I wish to thank my pupils and collaborators Yu.N. Drozhzhinov, B.I. Zavialov, V.V. Zharinov and R.Kh. Galeev. I express a special gratitude to Prof. A.P. Pruclnikov for his active interest and support of the publication of this book and to Prof. Jacob Korevaar for his support and a series of valuable remarks. v. S. Vladimirov

SYMBOLS AND DEFINITIONS We denote the points of an n-dimensional real space lR n by x y, ~, ... ; x (Xl, X2, ... I x n ). The points of an n-dimensional complex space en are given as z, (, ... ; Z = (Zll Z2,"" zn) = x + iy; x ~z is the real part of Z and y = ~z is the imaginary part of z, Z x - iy is the complex conjugate of z. In the usual manner we introduce in ~n and the scalar products 0.1.

I

=

=

=

en

+ '" + xn~nl Zl(1 + '" + zn(n

(X,~) = x16

(z, () =

and the norms (lengths)

Ixi = j(x, x) = (xi + Izi = ~ = (lzll2 +

+ +X~)1/2,

+ IZn 12)1/2.

0.2. Open sets in lR n are denoted by V, V', ... ; {}V is the boundary of V, or {)V = V \ V. We will say that the set A is compact in the open set V (or is strictly contained in V) if A is bounded and its closure A lies in V; we then write A <E O. The following designations are used: U(xo; R) is an open ball of radius R with centre at the point Xo; S(xo; R) aU(xo; R) is a sphere of radius R with centre at the point Xo; UR U(O; R), SR 5(0; R). Below, all point sets A, B, V, ... are assumed measurable. We use ~(A, B) to denote the distance between the sets A and B in lR n , that

= =

=

IS

~(A, B)

=

inf

:cEA, yEB

Ix - yl.

A' ;,,---...-

I

1/

\

/ I ,

\

I \

,

I

I

~ ,-.-_.."1

I

IE I~

I

b)

a) Figure 1

SYMBOLS AND DEFINITIONS

2

tf(x ' )+(I-t)f(x") _'r

8(x) 11-----

o

x

tx'+(I-t)x" b)

X'

a)

x" x

Figure 2

=

We use A£ to denote the e-neighbourhood of a set A, A£ A+Ue (Fig. la). If 0 is an open set, then ()£ designates the set of those points of () which are separated from 80 by a distance greater than c (Fig. 1b);

OE

= (x: x EO,

~ (x,

80)

> E].

We use int A to denote the set of interior points of the set A. The characteristic function of a set A is the function BA (x) which is equal to 1 when x E A and is equal to 0 when x f/. A. The characteristic function B[o,co)(x) of the semiaxis x > 0 is called the Heavlside unit function and is denoted O(x) (Fig 2a):

B{x) =0, O(x) = 1,

x O.

=

We write On(x) 8(xd ... B(x n ). The set A is said to be convex if for any points x' and x" in A. the line segment joining them, tx' + (1 - t)x", 0 < t < 1, lies entirely in A. We will use ch A to denote the convex hull of a set A. A real function f(x) < +00 is said to be convex on the set A if for any points x' and x" in A such that the line segment tx/ + (1 - t)x" joining them lies entirely in A the following inequality holds (Fig. 2b):

f(tx '

+ (1 -

t)x") < tf(x') + (1- t)f(x").

The function j(x) is said to be concave if the function - f(x) is convex. 0.3.

The Lebesgue integral of a function

J

f

over an open set () is given as

/ f{x) dx == / f(x} dx.

f(x) dx,

o

En

The collection of all (complex-valued, measurable) functions for which the norm

r

p

11/110(0)

= { [[ If(x)IP dx

ess sup If(x)L

xEO

,

1~ p

p

< 00,

= 00,

f specified on

(1

SYMBOLS AND DEFINITIONS

3

=

is finite will be denoted as .cP(O), 1 < p < 00; we write 1111 IIllc2(lltn), .cP(R n ) = £P. If f E .cP(O') for any 0' @ 0, then f is said to be p-locally integrable in 0 (for p = 1, we say it is locally integrable in 0). The collection of p-Iocally integrable functions in 0 is denoted .cfoc(O) , .cfoc(lRn) = .cfoc' A measurable function is said to have compact support in 0 if it vanishes almost everywhere outside a certain 0' @ O. The set of all functions in .cP(O) that have compact support in 0 is denoted .cb(O).

=

0.4. Let 0' (all 0'2, ... , an) be a multi-index, that is to say its components 0'.1 are nonnegative integers. We have the following symbolism:

(;) - (;;) (;:) ... (;:) -

lal = al + a2 + ... +

fJ!(QQ~ fJ)!' Q'n'

= (Q'l) Q'2, .•. , an) will be used to denote a multiindex with components of any sign: Q'j > O. <

It may sometimes happen that

Q'

0.5. We use Ck(O) to denote the set of all functions f(x)that are continuous in 0 together wi th all derivati ves aCt. f (x), 10'1 ::; k; coo (0) is the collection of all functions infinitely differentiable in 0. The set of all functions f(x) in Ck(O) for which all derivatives ex f(x), lal ::; k, admit continuous extension onto 0 will be denoted by Ck(O). We introduce the norm in Ck(O) for k < 00 via the formula

a

IIfll C k(O)

= sup loex f(x) I· xED

1Ct.I~k

We also write C(O) = CO(O), C(O) = CO(O). The support of a function f (x) continuous in 0 is the closure in 0, of those points where f(x) i- 0; the support of f is denoted by suppf. Ifsuppj @ 0, then f has compact support in 0 (compare with Sec. 0.3). We denote the collection of functions, with compact support in 0, of the class Ck(O) by C~(O); Co(O) = C8(0). Finally, the set of all functions of the class Ck(O) that vanish on 80 together with all derivatives up to order k inclusive will be denoted by C~(O); Co(O) = C8(0). We write Ck(I~n) = Ck; C~(I~n) = C~; C~ (I~n) C~, Co = cg (C~ is the set of functions in C k that vanish at infinity together with all their derivatives up to order k inclusive).

=

SYMBOLS AND DEFINITIONS

4

Symbolism: (a, b) is a bilinear form (linear in a and b separately); (a, b) is a sesquilinear form (linear in a and antilinear in b): 0.6.

(aal

+ {3a2

J

,Xb l

+ pb 2) ::::: a.\(aI' bI ) + ajL(all b2) + (3.\(a21 bI ) + (3Jt(a2, b2); (Tn

=

2rrn/2

f

ds = f(n/2)

Iz l=l is the surface area of a unit sphere in llt n ; AT is the transpose of the matrix A. We shall also use the alternative notation:

V' :::::

{j :::::

6. =

grad, 2 8 o = -2

ox o

-

0; + o~ + ... + {j~J

2

2

V' ::::: 00 - 6..

We denote the uniform convergence of a sequence of functions, {rpn (x) } to a function
iPn(x) if A

= JRn, then instead of xE1

XE1

n ~ 00;


we write

*

::b-.

* *

The sections are numbered in a single sequence. Each section is made up of subsections, the numbers of which are included in any reference to a section. Formulae are numbered separately in each subsection; they contain the number of the formula and of the subsection. When reference is made to a formula in a different section, the number of that section is also given. Symbol 0 denotes that the proof is complete.

CHAPTER 1

GENERALIZED FUNCTIONS AND THEIR PROPERTIES The exposition of the theory of generalized functions given in this chapter is tailored to the needs of theoretical and mathematical physics.

1. Test and Generalized Functions 1.1. Introduction. A generalized function is a generalization of the classical notion of a function. On the one hand, this generalization permits expressing in a mathematically proper form such idealized concepts as the density of a material point the density of a point charge or dipole, the spatial density of a simple or double layer, the intensity of an instantaneous point source, the magnitude of an instantaneous force applied to a point, and so forth. On the other hand, the notion of a generalized function can reflect the fact that in reality one cannot measure the value of a physical quantity at a point but can only measure the mean values within sufficiently small neighbourhoods of the point and then proclaim the limit of the sequence of those mean values as the value of the physical quantity at the given point. This can be explained by attempting to determine the density set up by a material point of mass 1. Assume that the point is the origin of coordinates. In order to determine the density, we distribute (or, as one often says, smear) the unit mass uniformly inside a sphere of radius E centred at O. We then obtain the mean density Ie (x) that is equal (see Fig. 3) to l

o

Ie (x)

_l_

iflxl<E,

= { 0~1rE3

if

E

2e

X

Figure 3

Ixl > E.

We are interested in the density at c = +00. To begin with, for the desired density (which we denote by J(x)) we take the point limit of the sequence of mean densities IE: (x) as E --t +0, that is, the function

a(x) =

{+oo o

if x = 0, if x i- O.

(1.1)

Of the density it is natural to require that the integral of the density over the entire space yield the total mass of substance, or

J

J{x) dx = 1.

(1.2)

=

But for the function <5 (x) defined by (1.1), J IS (x) dx o. This means that the function does not restore the mass (it does not satisfy the requirement (1.2)) and 5

1, GENERALIZED FUNCTIONS AND THEIR PROPERTIES

6

therefore cannot be taken as the desired mass. Thus the point limit of a sequence of mean densities 1£ (X) is unsuitable for our purposes. What is the way out? Let us now find a somewhat different limit of the sequence of mean densities 10: (X) the so-called weak limit. It will readily be seen that for any continuous function if'( x) J

lim

£-++0

f

10: (x)
(1.3)

Formula (1.3) states that the weak limit of a sequence of functions fo:(x), c --+ +0, is a functional if'(0) (and not a function!) that relates to every continuous function
Ie ( X )

weak

----+ J (X ) ,

and we understand by this the limiting relation (1.3). The value of the functional J on the function cp (the number \0(0)) will be denoted thus:

(J,
=
(1.4)

This equality yields the exact meaning of the formula (*) (see Preface). The role of the "integral" J J(x)if'(x) dx is played here by the quantity (J, cp), which is the value of the functional J on the function cpo Let us now check to see that the functional J restores the total mass'. Indeed, as we have just said, the role of the "integral" J J(x) dx is handled by the quantity (J,1), which, by virtue of (1.4), is equal to the value of the function identically equal to 1 at the point x = 0, that is, (J, 1) = 1. Also, generally, if masses Ilk are concentrated at distinct points Xk, k ::::: 1, ... 1 N, then the density that corresponds to such a mass distribution should be regarded as equal to

L

P-kJ(X - Xk).

(1.5)

l
The expression (1.5) is a linear functional that associates with each continuous function
L

Ilk/f(Xk).

l
Thus, the density corresponding to a point distribution of masses cannot be described within the framework of the classical concept of a function; to describe it requires resorting to entities of a more general mathematical nature, linear (continuous) functionals. 1.2. The space of test functions V(O). In the case of the delta-function we have already seen that it is determined by means of continuous functions as a linear (continuous) functional on those functions. Continuous functions are said to be test functions for the delta-function. It is this viewpoint that serves as the basis for defining an arbitrary generalized function as a continuous linear functional on a collection of sufficiently "good" so-called test functions. Clearly, the smaller the set of test functions, the more continuous linear functionals there are on it. On the other hand, the supply of test functions should be sufficiently large. In this

1. TEST AND GENERALIZED FUNCTIONS

7

subsection we introduce the important space of test functions 1)(0) for any open set 0 C ~n. Included in the set of test functions 1)(0) are all functions with compact support infinitely differentiable in 0; 1)(0) = CO'(O) (see Sec. 0.5). We define convergence in V(O) as follows. A sequence of functions i{)1, (0)) if there exists a set 0' @ 0 such that supp i{)k C 0' and for every a 8 Ct i{)k(X)

XE?

aQ'
k --t

00.

We then write:
V

= v(~n), V(a, b) = V((a, b)).

Clearly, if 0 1 C02, then also V(OI) C 1'(0 2 ), and from the convergence in V(Od there follows convergence in V( ( 2 ), An exampIe of a nonzero test function is the "cap" in Fig. 4: w~(x)

c

=

{ 0,

e

e - .. 2~:",F

Ixl ::; c,

-2£

J

Ixl > E.

we(x)dx

= 1,

that is,

X

Figure 4

In what follows, the function We will play the part of an averaging function: and so we shall regard the constant Ce as such that

f

2£

-£

f

Gee n

e-

1-~E12 d~ = 1.

1~1<1

The following lemma yields other examples of test functions. LEMMA.

For any set A and any number

£

>

0 there is a function T/e E

COO

such that

T/e (x) = 1,

x E A( ;

o:s T/e (x) < 1, PROOF.

T/e (x) = 0,

x

tt. A 3e ;

laCtT/e(x)1 S; Ker£-Ia l .

Let 0A2.. be a characteristic function of the set A 2e (see Sec. 0.2). Then

the function

T/e(X) =

f

OA2,,(Y)W e (x-y)dy=

f

we(x-y)dy,

A2..

where

We

D

is the "cap", has the required properties.

Let 0 be an open set. Then for any 0' TJ E V(O) such that 7J{x) I, x E (J', 0 S; 7J(x) ::; 1. COROLLARY.

=

@

0 there is a function

This follows from the lemma when A = t)l and £ = ~.6.(0', 80) > O. Let Ok, k = 1, 2 be a countable system of open sets. We say that this system forms a locally finite cover of the open set 0 if 0 = Uk> 1 Ok Ok @ 0, and any compact [{ ~ 0 intersects only a finite number of sets {O~}. l ... I

J

1. GENERALIZED FUNCTIONS AND THEIR PROPERTIES

8

I (partition of unity). Let {Ok} be a locally finite cover of O. Then there is a system of functions {ek} such that THEOREM

ekEV(Ok),

O<ek(x)
Lek(x)=l,

xEO.

k>1

For each x E 0 there is only a finite number of nonzero summands ek (x); the set of functions {ek} is termed the partition of unity corresponding to the given locally finite cover {Ok} of the open set O. REMARK.

We shall prove that there is another locally finite cover {O~} of the set 0 such that O~ @ Ok. Construct O~ and set PROOF.

Then ]{ 1 C 0 1 @ 0 and ]{ 1 is closed in O. Hence]{ 1 @ 0 1 , for 0i we take an open set such that ]{1 € O~ @ 0 1 . Then the sets O~, O 2 , . .. form a locally finite cover of O. In similar fashion we construct an open set O 2 @ 02 etc. We thus obtain the required cover {O~}. By the corollary to the lemma, there exist functions such that I

x E O~,

Putting

ek(x)

TJk (x)

= L k>1 TJk (X )'

we obtain the required partition of unity. This completes the proof.

D

We have thus seen that there are various functions in V(O). We will now see that there are a sufficiently large number of such functions. Let I be a locally integrable function in 0, I E £foc( 0). The convolution of I and the "cap" , w E'

IE (x)

=

f

I(Y)WF; (x - y) dy

=

!

WF; (y)/(x - y) dy

(wherever it is defined) is called the mean function of I· Let I E £P(O), 1 ::; p < 00 (f(x) is regarded as zero outside 0). Then fE: E Coo and the following inequality holds: (2.1) Indeed, the fact that IE: E Coo follows from the properties of the function f and from the definition of a mean function. When 1 < P < 00 the inequality (2.1)

1. TEST AND GENERALIZED FUNCTIONS

9

follows from the Holder inequality:

! :s ! ! =f f
11/€II~p(o) =

lie: (x)I P dx

o o

o

J/

=

f(y)we: (x - y) dy

0

If(Y)lPw,(" - y) dy

[j w,(" -

y)

P

dx

dyr1

dx

0

I/(y)IPwe(x - y) dydx

0

l/(y) IP dy =

Ilfll~p(o)'

o

The case of p

= 00 is considered in a similar fashion.

0

=

II. Let I E .cA(O) and I(x) 0 almost everywhere outside K ~ 0. Then for all c < Ll(K,80) the mean function I€ E V(O) and (see Sees. 0.3 and 0.5 fOT the symbolism) THEOREM

if f E Co(O),

in C(O) c ---t +0

{

:s

in .cP(O) (1 p < 00) almost everywhere in 0

if IE .cg(O), if I E .c~(O).

If c < ~(f{,aO), then Ie: (x) is with compact support in 0, and since f€ E Coo(O), it follows that IE: E V(O). Let / E Co(O). Then from the estimate PROOF.

IIe(x) - f(x)1 = fU(Y) - f(x)]we:(x - y) dy

< max If(x) - f(y)1 \x-yl~e:

=

f

wE(x - y) dy

max I!(x) - f(y) I.

x E 0,

Ix-yl~£

and from the uniform continuity of the function f follows the uniform convergence, on 0, of fE(X) to f(x) as c ---t +0. Let / E .cg(O), 1 p < 00. Take an arbitrary 0 > O. There is a function 9 E Co(O) such that

:s

11/ -

e5

9110(0)

< 3'

From what has been proved, there will be an co such that

o

119 - 9e: 11.0(0) <"3

for all

c

< co·

From this, using the inequality (2.1), we obtain, for all c

IIf -

f€ 11.0(0)

< co,

:s III - 9110(0) + 119 - 9€ II.c p(o) + II(g < 2111 -

20

9110(0)

f)e: II.c p(o)

t5

+ IIg - ge II.cp(o) < "3 + :3 = o.

And this means that fE ---t f,e ---t +0 in .cp (0). Now if f E .cgo(O), we can indicate a sequence of functions taken from Co(O) that converges to f(x) almost everywhere in O. From this fact and from what has

10

1,

GENERALIZED FUNCTIONS AND THEIR PROPERTIES

been proved, it follows that ft{x) ---+ f(x), theorem is proved.

or 0

€

---+ +0, almost everywhere in O. The 0

< 00.

COROLLARY

1. V(O) is dense in .cP(O), 1
COROLLARY

2. V(O) is dense in C~(O) (in the norm Ck(O)) i/O is bounded

= ffi.n.

1.3. The space of generalized functions V'(O). A generalized function specified on an open set () is any continuous linear functional on the space of test functions V(O). We will write the value of the functional (generalized function) f on the test function cp as (I, cp). By analogy with ordinary functions, we sometimes formally write f(x) instead of I, and regard x as the argument of the test functions on which the functional f operates. We now give an explanation of the definition of a generalized function.

(1) A generalized function f is a functional on 1)(0), that is, with each 1'(0) there is associated a (complex-valued) number (f,Ip).

Ip

E

(2) A generalized function f is a linear functional on V(O), that is, if t.p and 1/J belong to V(O) and A and IJ are complex numbers, then

(/,)..1{) + p"p) = A(I, cp)

+ J.l(f, ?jJ).

(3) A generalized function f is a continuous functional on V(O), that is, if rpk --+ !p, k --+ 00 in V(O), then k --+

00.

The generalized functions f and 9 specified in 0 are said to be equal in 0 if they are equal as functionals on V(O), that is, iffor any rp in V(O), (f,
=

=

Suppose / E V'(O). We define a generalized function complex conjugate of I, as follows:

(1, rp) = (/, c,o),

J in V'(O),

cpE'V(CJ).

The generalized functions

SRi = f + f, 2

are respectively the real part and the imaginary part of f so that

I =

'iRf

+ i<;J/,

If ~f = 0, then f is said to be a real generalized function. EXAMPLE.

The delta-function is real.

which is the

1. TEST AND GENERALIZED FUNCTIONS

11

Let us define convergence in V' (0). A sequence of generalized functions /1, h, ... in V' (0) converges to a generalized function I E V' (0) if for any test function 'P E V(O) (!k,'P) -+ (I,r.p), k --t 00. We then write

!k

---t

I,

k ....-+ 00

V' (0).

in

This convergence is termed as a weak convergence. The linear set V' (0) together with the convergence with which it is equipped is called the space V' (0) oj generalized junctions. In symbols: V' =v,(~n), V'(a,b) =V'(a,6)). Quite obviously, if 01 C02, then V'(02) C V'(Od, and from convergence in V'(02) follows convergence in V'(Ot}. For this reason, for any generalized function / in 'D'(O) there is a (unique) restriction to any open set 0' C 0 such that f E V'(O'). Linear functionals on V(O) need not necessarily be continuous on V(O). However, not a single discontinuous linear functional has been constructed explicitly on V(0); one can only prove their existence theoretically by using the axiom of choice. REMARK.

For a linear functional f on V(O) to belong to V'(O), that is, jor it to be a generalized function in 0, it is necessary and sufficient that for any open K(O') and m m(O') such that set 0' @ 0 there exist numbers K THEOREM.

=

=

'P E V(O').

(3.1)

Sufficiency is obvious. We prove necessity. Suppose f E V'(O) and 0' @ O. If the inequality (3.1) does not hold, there will be a sequence 'PkJ k 1J 2, ... , of functions in V( 0') such that PROOF.

=

(3.2)

But the sequence

=A

VJk

'Pk kll'Pkllck(O')

since supp 1/Jk C 0'

@

0, and for k

l a13 1Pk(X)1

Therefore,

(I, ¢k)

= afJ

-+ 0, k -+

1(!,1Pk)l==

-t

k --t

0,

2 1.81

In

1>(0),

we have

< .~ -+ 0,


VkII'Pkllck{o') -

00.

00

V

k -+

00.

k

On the other hand, by virtue of (3.2) we get

1(!,Y'k)1 ?Vk-+oo, Vkll
k....-+oo.

D

This contradiction proves the theorem.

Suppose f E D'(O). If it is possible in the inequality (3.1) to choose the integer m as independent of 0', then we say that the generalized function! is of finite order; the smallest such m is termed the order of I in O. For example) the order of the delta-function is 0; the order of the generalized function

(!,l

in (0,00) is infinite.

1. GENERALlZED FUNCTlONS AND THElR PROPERTIES

12

The theorem just proved signifies that if we introduce in the space V( 0) a topology of an inductive limit (union) of an increasing sequence of countableDormed spaces Co(Ok), where 0 1 @ O 2 @ ... , Uk>1 Ok 0, with norms REMARK.

=

If' E C~ (0 k) ,

v = 0 I 1, ... ,

then V'(O) becomes the conjugate space ofV(O) (see Bourbaki [11], and Dieudonne and Schwartz [17]). Here, the inequality (3.1) is preserved for all functions 'P in -=' G[f(O ) (see Corollary 2 to Theorem II of Sec. 1.2). 1.4. The completeness of the space of generalized functions V' (0). The property of the completeness of the space V'(O) is extremely important.

Let there be a sequence of generalized functions It, h, ... in V' (0) such that for every function r.p E V(0), the numerical sequence (fk, 'P) converges as k --+ 00. Then the functional f on V(O) defined by (I,
The linearity of the limiting functional I is obvious. Let us prove its continuity on V( 0). Let 'Pv --+ 0, v --+ 00 in V(O); we have to prove that (I, Ipv) --+ 0, lJ --+ 00. Assuming the contrary and passing, if necessary, to a subsequence, we may assume that for all v = 1,2, ... the inequality 1(/, !Pk)1 > 2a holds true for some a > O. Since (I, 'Pv) = 1imk-+= (/k,
Given a sequence offunctionals II, 12, ... taken from a weakly bounded set M' C V'(O), that is, l(f,Ip)1 < Cr.p, f E M' for all

Assume the lemma is not true. Then, passing, if necessary, to a subsequence, we can say that I(!k, 'Pk)1 ~ c> O. The convergence of 'Pk to 0 in V(O) means that sUPP 'Pk C 0' @ 0 and for every a PROOF.

k --+

00.

Therefore, by again passing, if necessary, to a subsequence, we can assume that

lad ~ k = 0,1, .... Set

1/Jk

= 2k ipk; then supp VJk

C

0'

@

0 and

10'1

~ k

= 0, 1, ... ,

(4.1 )

so that k --+

00.

(4.2)

Now construct the subsequences {fk,,} and {1/)k,,} in the following manner. Choose hI and 1/Jk l so that 1(!k l1 vJk l )1 ~ 2. Suppose Aj and 1/Jkjl j = 1, ... , v-I, have already been constructed; construct Ik" and VJk v ' Since 1/Jk --+ 0, k --+ 00 in

1. TEST AND GENERALIZED FUNCTIONS

V(O), it follows that (A; ,1fJk) ---t 0, k ---t a number N such that for all k ~ N,

1(I"jI1/l,,)1 S

00,

j

1

13

= 1, ... , V -I, and therefore there is

j == 1, ... , v - 1.

2v -.i'

(4.3)

Now note that I(!k, 1/Ik;) 1 < Ckj' j = 1, ... , v - 1. Finally, by virtue of (4.2) choose a number k v > N such that

L:

1(!k",tPkJI >

Ckj

+v+

(4.4)

1.

1~.i:$11-1

Thus, by (4.3)-(4.4), the (generalized) functions fk" and 1fJk" thus constructed are such that 1 (4.5) IUk; ,1/IkJI ::; 211 - j ' j 1, ... , v-I,

=

L

IUk" , ?PkJI ~

I(/kj' 1Pkj)! + v + 1.

(4.6)

l:$i:$11-1

Set

tP =

Lj>ltPkj. By virtue of (4.1) this series converges in V(O) and, hence,

1/J E V(O) and

(fk.,,7J;)

= (!kv, tPh.,) + '2:')11.."

lh j ).

(4.7)

j :::1

jt-v

Whence, taking into account the inequalities (4.5) and (4.6), we get

1(!k."tP)I~I(Jk."tPkJI-

00

L

1(lk."tPkJI-

1

> '"" -.- 1I + 1 - ~ 2311

L

1(!kv,.,pkJI

j~v+l

1:$j:$v-l

= v.

j~1I+1

That is, (!k1l,1/;) ---t 00, v ---t 00. But this contradicts the boundedness of the sequence (fk,.,p), k ---t 00 (!k EM'). D If a set M' C 1)' (0) is weakly bounded, then fOT any 0 ' ~ 0 there exist numbers K and m such that the inequality (3.1) is preserved for all

The proof is analogous to the proof of the theorem of Sec. 1.3 with the lemma invoked as well. 1.5. The support of a generalized function. Generally speaking, generalized functions do not have values at separate points. Nevertheless, one may speak of the vanishing of a generalized function in an open set. We say that a generalized function f in V'(O) vanishes in an open set 0 ' COif its restriction to 0 ' (see Sec. 1.3) is zero functional in V'(O'), that is, (I,
=

o.

14

1. GENERALIZED FUNCTIONS AND THEIR PROPERTIES

Uv> 1 O~

:::: (}. By the Hein~Borel lemma, the compact 0; is covered by a finite

nu~ber of neighbourhoods U (y): U (yI), ... , U (YNJ; the compact 0; \ O~ is covered likewise by a finite number of such neighbourhoods: U(YN 1 +I), ... , U(YNl+N~); and so forth. Setting Ok :::: U(Yk) n O 2, k :::: 1, ... , Nil Ok :::: U(Yk) n (O~ \ ~), k = Nl + 1, ... , N l + N 2 • and so on, we obtain the required cover {Ok}. Let {ek} be the partition of unity that corresponds to the constructed cover {Ok} of the set 0 (see Sec. 1.2). Then for any 'P in V(O), supp(
(I, <,0)

::::

(fl L

ek'P) ::::

k>l

I)f, 'Pek) :::: O. k>l

Thus, we proved that the following lemma holds true. If a generalized function in 1"(0) vanishes in some nighbourhood of every point of an open set 0, then it also vanishes in the whole set (}. LEMMA.

Suppose f E V'(O). The union of all neighbourhoods where f :::: 0 forms an open set Of which is called the zero set of the generalized function f. By the lemma, f :::: in Of; furthermore Of is the largest open set in which f vanishes. The support of a generalized function f is the complement of Of to 0; the support of f is symbolized as sUPP!, so that supp! = 0 \ Of; Stipp! is a closed set in O. If supp! @ 0, then I is said to be with compact support in O. From the foregoing follow the assertions: (a) If the supports of f E V'(O) and'P E 1'(0) do not have any points zn common, then (I, 'P) :::: 0. (b) To have x E supp I, it is necessary and sufficient that! should not vanish in any neighbourhood of the point x. Let A be a closed set in O. We denote by V' (0, A) the collection of generalized functions in 1"(0), whose supports are contained in A, and with convergence: fk ---+ 0, k ---+ 00 in V'(O, A) if /k ---+ 0, k ---+ 00 in V'(O) and supp!k c A. We write D'(A) = V'(R n , A). A similar meaning is also attributed to other spaces of generalized functions, for example, S'(A), £;(A) and so on (see Sees. 5 and 7 below). The lemma that was proved in this subsection admits of a generalization. In Sec. 1.3 we saw that any generalized function f in V' (0) induces, in each 0' cO, its own local element f E '[)'(O'). The converse is also true: it is possible to "join together" a single generalized function out of any collection of compatible local elements. To be more precise, the following theorem holds.

° I

THEOREM ("of piecewise sewing"). Suppose that for each point yEO there exist a neighbourhood U (y) @ 0 and in it there is specified a generalized function !y, wherein !Yl (x) :::: f!J2(x) if x E U(yt} n U(Y2) :I 0. Then there is a unique generalized function f in V' (0) that coincides with fy in U (y) for all yEO. PROOF. Again, as in the proof of the lemma, we construct, with respect to the cover {U(y), YEO}, a locally finite cover {Ok}, Ok C U(Yk), of the set 0 and the appropriate partition of uni ty {ek}. Set (/, l


(5.1)

1. TEST AND GENERALIZED FUNCTIONS

15

Since the number of summands in the right member of (5.1) is always finite and does not depend on

=

(I, l -

k>l -

That is, I = fy in U(y). The uniqueness of the generalized function! thus constructed follows from the lemma. D 1.6. Regular generalized functions. The simplest example of a generalized function is a functional generated by a function f (x) locally integrable in 0:

(I, Ifi) =

!

f(x)lfi(x) dx,

(6.1)

From the properties of the linearity of an integral and from the theorem on passage to the limit under the integral sign it follows that the functional in the right member of (6.1) is linear and continuous on V(O), that is,.! E V'(O). Generalized functions determined, via (6.1), by functions locally integrable in o are termed regular generalized functions. All other generalized functions are said to be singular. LEMMA (Du Bois Reymond). For a junction f(x) that is locally integrable in o to vanish almost everywhere in 0, it is necessary and sufficient that the regular generalized junction f generated by it vanish in O. PROOF.

Necessity is obvious. We prove sufficiency. Let

f

f(x)
(6.2)

ep E '0(0).

Take an arbi trary 0' <s 0; let 00 , be a characteristic function of 0'. By Theorem II of Sec. 1.2, there exists a sequence of functions
=

f

If(x)1 dx

=

!

=

kl~~ {f f(X)
f(x )e- i arg!(x)oCY (x) dx

0'

!

= kl~~ f(x)

[e-iargj(X) -

[e-; acgJ(x)Oo' (x) -
epk(X)] dx

= 0,

0'

so that f( x) = 0 almost everywhere in 0'. Due to the arbitrariness of the set 0' @ 0, we conclude that f(x) = 0 almost everywhere in O. D From the lemma just proved it follows that any regular generalized function in o is defined by a unique function (unique up to the values on a set of measure zero) that is locally integrable in O. Consequently there is a one-to-one correspondence between functions locally integrable in 0 and regular generalized functions in O.

1. GENERALIZED FUNCTIONS AND THEIR PROPERTIES

16

For this reason we will henceforth identify a function f(x) locally integrable in 0 with the generalized function from V' (0) that is generated by it via (6.1). In this sense, "ordinary" functions (that is, functions locally integrable in 0) are (regular) generalized functions taken from V' (0). From the Du Bois Reymond lemma it follows likewise that both definitions of the support of a continuous function in 0 that were given in Sec. 0.5 and Sec. 1.5 coincide. Also note that if the sequence fk(X), k 1,2, ... , of functions locally integrable in 0 converges uniformly to the function f(x) on each compact I< @ 0, then, also , !k --+ f, k --+ CX) in 1)' ( 0) . Suppose f E V' (0) and 0 1 CO. We will say that a generalized function / belongs to a class Ck(Od if, in 0 1 , it coincides with the function /1 of the class Ck(Od, that is, for any if> E V(Ot},

=

!

(I, rp) :::: 1f, besides,

II

/1 (x)
E C k (VI) I then we will say that

f belongs to the class

C k (OJ).

1. 7. Measures. A more general class of generalized functions that contains the regular generalized functions is generated by measures. A measure on a Borel

set A is a completely additive (complex-valued) function of sets JL (E)

::::

!

Jl( dx),

E

which function is specified and finite on all bounded Borel subsets E of the set A, IJl(E) I < 00. For details of measure theory and integration see Kolmogorov and Fomin [58]. The measure J1.(E) on A can be uniquely represented in terms of fOUf nonnegative measures /lj(E) > 0 on A via the formula J.l Jll - J.l2 + i(1l3 - J.l4); here,

=

! E

Jl-(dx)

=

!

J.tl(dx) -

E

!

J.l2(dx)

+

E

if J.t3(dx) - i! Jl-4(dx). E

(7.1)

E

The measure J.t(E) on an open set 0 determines a generalized function J.t in 0 by the formula

(Il,
=

f

if>(x)JJ(dx) ,

V(O),

(7.2)

where the integral is the Lebesgue-Stieltjes integral. From the properties of this integral it follows that we do indeed have Jl E V' (0). For measures Jl on 0 that are absolutely continuous with respect to the Lebesgue measure, that is, p(dx) = f{x) dX where f E .cfoc(O), formula (7.2) defines regular generalized functions f (see Sec. 1.6). REMARK.

I

For a measure Jl-(E) on V to be a zero measure, it is necessary and sufficient Jor the generalized function J1. defined by it to vanish in O. LEMMA.

1. TEST AND GENERALIZED FUNCTIONS

17

The proof is based on the following assertion: for the measure J-l( E) on 0 to be a zero measure, it is necessary and sufficient that PROOF.

f


= 0,

Co(O)

(7.3)

o whence immediately follows necessity. We now prove sufficiency. Assuming that (7.3) holds for all

/

=!

o

lim
k-+oo

0'

=

lim / 'Pk(x)Jl(dx)

k-+oo

=

0,

0

D

which is what we set out to prove. The lemma is proved.

From the lemma it follows that there exists a one-ta-one correspondence between measures on CJ and the generalized functions generated by them via formula (7.2). For this reason, we will in future identify the measure Jl(E) on 0 and the generalized function Jl in V' (0) generated by that measure. I. For a generalized function f in D'(O) to be a measure on 0, it is necessary and sufficient that its order in 0 be equal to O. THEOREM

PROOF.

and any
@

Necessity. Let f E V' (0) be a measure J.l on O. Then for any 0' V(O') we have

l(f,
@

0

::;/IJ.l(dx)lm~I
whence we conclude that the order of f in 0 is 0 (see Sec. 1.3). Sufficiency. Let the order of f in 0 be 0, that iS l for all 0' @ 0

V(O/).

(7.4)

Let Ok, k = 1,2, ... I be a strictly increasing sequence of open sets that exhausts 0: Ok @ Ok+1, Uk Ok O. Since the set V(Ok) is dense in CO(Ok) in the norm C(Ok) (see Corollary 2 to Theorem II of Sec. 1.2), it follows from inequality (7.4) that the functional f admits of a (linear) continuous extension onto Co (0 k ). By the Riesz-Radon theorem there is a measure Ilk on Ok such that

=

I

From this it follows that the measures /-lk and J-lk+1 coincide on Ok; therefore there exists a single measure J.l on 0 that coincides with the measure Ilk in Ch and it coincides with the generalized function f in CJ. The proof of the theorem is D complete. A generalized function f in V' (0) is said to be nonnegative in 0 if (I, for all

II. For a generalized function f in V'(O) to be a nonnegative meait is necessary and su.fficient that it be nonnegative in O.

THEOREM

su.re on 0

I

°

1. GENERALIZED FUNCTIONS AND THEIR PROPERTIES

18

Necessity is obvious. We prove sufficiency. Suppose f E V' (0) is nonnegative in O. Suppose I{) E V(O'), O'@ O. By the corollary to the lemma of Sec. 1.2 there is a function 1] E V(O), 1](x) 1, x E 0'. For this reason, PROOF.

=

x EO.

Whence, using the nonnegative nature of the functional f in 0, we get

or

1(/,

This inequality shows that the order of the generalized function f is O. By Theorem I, f is a (nonnegative) measure on 0, and the proof is complete. 0 The simplest example of a measure (and, what is more, a measure of a singular generalized function) is the delta-function of Dirac (see Sec. 1.1), which operates via the rule

(tS,

Clearly, .5 E V', d"(x) :::: 0, X =1= 0 so that suppd" = {O}. We will now prove that o(x) is a singular generalized function. Suppose, on the contrary, that there is a function f E .ctoc{IR n) such that for any

= (.5,
J f(x)t.p(x)dx Since

Ixl 2 rp E V,

(7.5)

it follows from (7.5) that

J

2

f(x) Ix 1
= Ix I
for all

wc(x)

--t

o(x),

€ -7

+0

in V'.

(7.6)

The sequence wE(x), c --t +0 is depicted in Fig 4. Indeed, by the definition of convergence in V' the relation (7.6) is equivalent to lim Jw£(x)
£--++0

=

V.

This equation follows from the estimate

J w£ (x)
:s J w~ (x) l

< max l
=

Ixls~

and from the continuity of the function <po

o

1. TEST AND GENERALIZED FUNCTIONS

19

The surface d-function is a generalization of the point d-function. Let S be a piecewise smooth surface in lR n and let J-t be a continuous function on S. We introduce the generalized function J-tds that operates via the rule (J-tds, rp)

=

!

J-t(x)
s Clearly J.ld E V'; J.le5s (x) = 0, x rt S so that supp J-tds C S; J-tds is a singular measure if J-t 1- O. The generalized function J-tos (x) is termed a simple layer on the surface 5. It describes the spatial density of masses or charges concentrated on the surface 5 with surface density J.l. (Here, the density of the simple layer is defined as a weak limit of the densities that correspond to a discrete distribution on the surface 5,

L J-t(Xk)~Skd(X -

Xk),

k

when the surface S is unrestricted refinemented; compare Sec. 1.1.) Locally integrable functions and e5-functions describe the density distribution of masses, charges, forces and the like (see Sec. 1.1). For this reason, generalized functions are also termed distribution,.; (see Schwartz [89, 90]). If, for example, a generalized function 1 is the density of masses or charges, then the expression (1,1) is the total mass or total charge (on the assumption that 1 is meaningful 'on a function identically equal to 1 because 1 is not with compact support!). In particular, (6,1) = 1; (/,1) = f I(x} dx if 1 E £1. REMARK.

1.8. Sochozki formulae. We now introduce another important singular generalized function P ~ that operates in accordance with the formula

The functional P~ is a linear functional. Its continuity on T> the equation

(p~,;?)

= VP

J;?~x) J +

= T>(~n) follows from

dx

R

VP


XX'P' (x') dx

-R R

<

J

1
-R

~

2Rmaxl
ip

E V( -R, R).

(8.1)

Here, Xl is some point in the interval (- R, R). Thus P ~ E V'. The generalized function p1.x coincides with the function 1.x for x f. 0 (in the meaning of Sec. 1.6). It is called the finite part or principal value of the integral of

20

1. GENERALIZED FUNCTIONS AND THEIR PROPERTIES

the function ~. Let us now set up the equality lim £--++0

Indeed, if
· 11m e--++O

!

! i.p(x~+ x

+

dx == -i7rip(0)

l£

VP!

It'(x) dx. X

(8.2)

= 0 for Ixl > R, then

!

R

i.p( x) dx == I"1m X + ic €--++o

x - 'l£ i.p (x) d X x 2 + £2

-R

!

R

== i.p(O) lim

€--++o

!

-R

R

x - i£ 2 2 dx x +£

+ lim

£--++0

!

-R

x - i£ 2 2 [
R

== -2i
+

== -imp(O)

c

VP!


-R

If'(x) dx. x

The relation (8.2) means that there is a limit to the sequence x~i£ as [ --+ +0 in D' . which limit we denote by X~iO; and this limit is equal to -im5(x) +p~. Thus

x

1.

+ zO

==-i1r6(x)+P~. x

(8.3)

Similarly 1 x

'0 == irr6(x)

-l

1

+ P-. x

(8.3')

The formulae (8.3) and (8.3') were actually first obtained in "integral" form of the type (8.2) in 1873 by the Russian mathematician Julian Sochozki (see Sochozki [98]). At the present time these formulae are widely used in quantum physics. We will now prove that the order of P ~ in ~ 1 is equal to 1. Indeed, from (8.1) it ~ In x follows that its order in ~ 1 does not exceed 1. If its order in ~ 1 were equal to 0) then by Theorem I of Sec. 1.7, P~ would be a measure on ~ 1. But then the integral vp f 'P~x) dx would be defined on all x continuous functions that are with compact o support in ~ 1, whieh, as we know, is not true (for example, it is not defined on functions equal to ~~~ in the neighbourhood of 0; Fig. 5). We note in passing that the order of P ~ in {x =f. O} is equal to 0 because P~ coincides with the locally integrable function ~ Figure 5 when x =f. O. The generalized function P ~ is a continuation of the regular generalized function ~ from the set {x #- O} onto the whole axis ~1.

1. TEST AND GENERALIZED FUNCTIONS

21

The generalized function P ~ is called the regularization of the function ~ xi- O. Similarly, the regularization of the functions lxl a and sgn xlxl- a . x i- 0 for ?Ra > 1 are defined (see Sec. 2.7). For instance, the generalized function P f N = 1,2, ... , are defined by the formula

I

W'

f

) = Pf_1_ ( ]xjN I If'

If'(x) - SN-dx; If') d

Ixl N

x

f

+

Ixl<1

ip(x) d

Ixl N

(8.4)

If' E V,

xI

Ixl>1

where SN (x; If') is the Taylor polynomial of the function If' at 0 of degree N If'(k) (0) k

N

SN(X;
1

=L

k!

X.

Ic=O

The question now is: does any locally integrable function in 0 i- ~1 admit a continuation onto the whole space ~ n as a generalized function from V' (~1 )? The answer is negative, as will be seen from the following example:

# 0).

e1/ x E V' (x

f

If there exists a function have

E V' (IR. 1) that coincides with e 1 / x for x

(f,lf') = Let If'o E V, If'o(x) = 0 for x

f

e1/xlf'(x) dx,

< 1 and

f

x

i=

i- 0).

If' E V(x

0, we would

(8.5)

> 2, If'o(x) > 0 and

= 1.

If'o(x) dx

Then If'k(X)

= e- k / 2 klf'o(kx) -+ 0,

f

e1/xipdx)

dx

k --+

in

00

f =f ek(~-~)lf'o(Y) >f =

V,

= e:-~kipo(kx) dx 2

dy

1

2

If'o(Y) dy

1,

1


i- 0),

but this contradicts (8.5): 1

~

f

el/xlf'k(X) dx

= (f,lf'k) --+ 0,

k --+ co.

1.9. Change of variables in generalized functions. Let x = Ay + b be a nonsingular linear transformation of 0 onto 0 If' E V(OI) we have

f 0

1

f(Ay

+ b)
= Ide~ AI

f 0

1

f 1•

E £foc(O) and Then for any

f(x)
22

1. GENERALIZED FUNCTIONS AND THEIR PROPERTIES

a(x) x

Figure 6

This equality is taken for the definition of the generalized function f(Ay any f(x) in V'(O):

( f(Ay

+ b),
= ( f(x),


+ b)

for

(9.1)

Since the operation
(j(cy),~) = Ic~n If A

= I, then

(I, ~W) .

(a shift equal to b)

(f(y+b),
(a) J( -x) (b) (J(x - xo),
= ~(x);

Let a Eel. We define the generalized function J(a(x)) via the formula

6(a(x)) = lim wE(a(x)) E--t+O

in

D'(c, d)

(9.2)

where WE is the "cap". Suppose the function a(x) has isolated and simple zeros which we denote by Xk, k = 1,2, ... (Fig. 6). Then J(a(x)) exists in V'(~l) and is given by the sum

~(a(x)) ='" J(x -

L.:

x k ).

la'(xk) I

(9.3)

By virtue of the theorem of piecewise sewing (see Sec. 1.5), it suffices to prove formula (9.3) locally, in a sufficiently small neighbourhood of each point. Let

1. TEST AND GENERALIZED FUNCTIONS

23

we have the following chain of equalities:

f

Xk+Ek

( &(a(x)),
= £-4+0 lim

WE

[a(x )]
Xk-E:k


= ("::,~~i)' ~) Now if 'P E V (a:, {3), where the interval (a:,.8) does not contain a single zero of Xk, then

I

f3

( 6 (a(x)), tp)

= £-4+0 lim

WE

[a(x )]
= O.

Q'

The local elements li~D::)l in (Xk - tk, Xk agreement. The proof of (9.3) is complete. EXAMPLES.

(a) J(x 2

-

a2 ) =

1

2a

+ ck)

[J(x - a)

and 0 in (oJ.8) are clearly

In

+ J(x + a)];

00

(b) J(sin x) =

L

J(x - k1T").

k=-oo

1.10. Multiplication of generalized functions. Suppose

f

E L]loc(O) and

a E COO(O). Then for any

(aI,
=

I

a(x)/(x)tp(x) dx

= (/, alP)·

This equality is then taken for the definition of the product of the generalized function I in V' (0) by the function a that is infinitely differentiable in 0:

(ai, 'P) = (I, alP),

'P E'D(O).

(10.1)

Since the operation 'P --+ a
=0

\ Oaf C 0 \ (Oa U OJ)

=(O\Oa)n(O\Oj) = supp a n supp f. If I E V' (0), then we have the equality

I = TIl

(10.2)

1. GENERALIZED FUNCTIONS AND THEIR PROPERTIES

24

where TJ is any function of the class Coo, that function being equal to 1 in the neighbourhood of the support of /. Indeed, for any r.p E V(O), the supports of / and (1 - TJ}cp have no points in common, and for this reason (see Sec. 1.5)

(/, (1 - TJ)r.p) = 0 = (/(1 -

7]),

cp).

o

which is equivalent to (10.2). EXAMPLES.

= a(O)o(x) since for all cp E V (ao,
(b) xP~ = 1 since

(Xp~, I") = (p~, xI") = II"(x) dx = (1,1"), The following question arises: Is it possible, in the case of generalized functions, to define the multiplication of any generalized functions and so that the multiplication is associative and commutative and agrees with the above-defined multiplication by an infinitely differentiable function? L. Schwartz demonstrated that no such multiplication can be defined. Indeed, if it existed, then, using examples (a) and (b), we would have the following contradictory chain of equalities:

o = OP~x = (xo(x))p~x = (o(x)x)p.!.x = o(x) (xp.!.) = 6(x). x In order to define a product of two generalized functions / and g, it is necessary that they have (to put it crudely) the following properties: / must be just as "irregular" in the neighbourhood of an (arbitrary) point as 9 is "regular" in that neighbourhood, and conversely. As for the genera.l definition of the product of generalized functions, see Sec. 4.7 and Sec. 6.5. Here we give the definition of the product of the generalized function / E V' (lR n) with the characteristic function 00 (x) of an open set 0 C lR n. The problem consists in construction of the generalized function 00 / from V' which is equal to /(x) for x E 0 and equal to zero for x E IR n \ O. The product 00 f exists if / is regular in a neighbourhood U of the boundary 8(} and it can be represented by the formula

(Bo/,
= (/,00 (1 -

'1)'1')

+

1

/(x)'1(x)

(10.3)

UnO

where 7] is any COO-function, supp 7] C U and 7] = 1 in a neighbourhood V' C V of the boundary &0. Indeed, noting that 00 (1 - 7]) E Coo, we conclude that the right-hand side of equality (10.3) determines a linear continuous functional on V, i.e., Oof EV', and it does not depend on the auxiliary function 7] with the properties indicated. The other properties of the generalized function 00 f also follow from representation (10.3). 0 EXAMPLE.

O(x _ xo)6(x)

= {O(x), 0,

if if

Xo

XQ

> 0, < O.

2, DIFFERENTIATION OF GENERALIZED FUNCTIONS

25

2. Differentiation of Generalized Functions

0',

2.1. Derivatives of generalized functions. Let f E Ck(O). Then for all 10'1 :S k, and rp E V(O) we have the following integration-by-parts formula:

({l~f,
!

aOf(x)
= (_1)1

0

1! f(x)aO''P(x) dx

= (_I)la l (j,8 a lp). We take this equation for the definition of a (generalized) derivative aa f of the generalized function f in V' (0):

(1.1 )

'P E V(O).

Since the operation lp -+ (-l)lolaalp is linear and continuous from V(O) into D(O), the functional aO' f defined by the right-hand side of (1.1) is a generalized function in V'(O). In particular 1 when f = 6, then (1.1) takes the form 'P E 'D.

It follows from this definition that if a generalized function f in 'D'(O) belongs to the class C k (Od in 0 1 C 0 (see Sec. 1.6), then its classical and generalized derivatives oaf, lal:S k, coincide in 0 1 . The following properties of the operation of differentiation of generalized functions holds true: 2.1.1. The operation of differentiation f -+ (Yx f is linear and continuous from V'(O) into V'(O). Linearity is obvious. We will prove continuity. Suppose fk --+ 0, k --+ 00 in 1)'(0). Then for all

(o°!k, cp) and this signifies that For example,

an fk

= (_1)10'1 (Jk, aarp) -+ 0, --+ 0, k --+

00

k -+

00

D

in 'D'(O). €

-+ +0

In

V'.

(1.2)

The relation (1.2) follows from the relation (7.6) of Sec. 1. The sequence E -+ +0, is depicted in Fig. 7. In particular, if the series

w~(x),

L uk(x) = S(x), k>l

converges uniformly on every compact f{ @ 0) then it may be differentia.ted term by term any number of times, and the resulting series will converge in D'(O),

LaO'Uk(X) = aO'S(x). k>l

True enough, the sequence of the partial sums of this series converges to V' (0) (see Sec. 1.6).

aO' S( x)

in

26

1. GENERALIZED FUNCTIONS AND THEIR PROPERTIES

2€

Figure 7

2.1.2. Any generalized function f E 'D'(O) (in particular, any function locally integrable in 0) is infinitely differentiable (in the generalized sense). Indeed] since

f

E 1)'(OL it follows that

:1

E V'(O); in turn,

J

a~. ,

(:1) J

E

D'(O) and so forth. 2.1.3. ation:

0 The result of differentiation does not depend on the order of differenti-

( 1.3) Indeed] (8a+~ f,rp) = (-l)lol+I~I(J,ao+t3rp)

= (-1) I

0

I (8 13 f, 8°
= (aO (a{3 f) I
= (-1)1131(8 f, afJ
= (a/3 (8° f) t.p) I

I

whence follow the equalities (1.3). 0 2.1.4. If f E 'D'(O) and a E COO(O), then the Leibniz formula holds true for the differentiation of a product af, (1.4)

2. DIFFERENTIATION OF GENERALIZED FUNCTIONS

27

Indeed, if ~ E V(O), then

=_

o(aJ) ) ( 8 Xl lIP

(fa'88

IP )

Xl

(f,a~) OXI

= = _

(f' o(a
= -

= (:;, ,ay»

aX!

y»

+ (;:, I,Y»

8f 8a ) = ( a OXl + ax} f, ~ , whence follows (1.4) for 2.1.5.

0'

o

= (1,0, ... ,0). supp 00. f

c

supp f.

(1.5 )

Indeed, if f E V'(O), then for all

= (_I)l o l(/,8
=0 o

so that Oaaf J OJ, whence follows the inclusion (1.5).

2.2. The antiderivative (primitive) of a generalized function. Every function f(x) continuous in an interval (a, b) has in (a, b) a unique (up to an additive constant) anti derivative j(-l)(x),

f /(~) x

f(-l)(X) =

d€ + C,

/(-1)' (x)

= f(x).

The last equality is what we will start with to define the antiderivative or primitive of an arbitrary generalized function / (of one variable). Suppose f E V'(a,b). The generalized function j(-l) in V'(a,b) is termed the antiderivative (or primitive) of the generalized function / in (a, b) if /(-1)' = /, that is, (/(-1) I
= -(I, lp),

t.pEV(a,b).

(2.1)

The equality (2.1) shows that the function j(-l) is not specified on all test functions taken from V(a J b), but only on their first derivatives. Our problem is to extend that functional onto the whole space D(a, b), and in a manner so that the extended functional j(-l) is linear and continuous on D(a, b), and to determine the degree of arbitrariness in such an extension. First assume that /(-1) (the antiderivative of j) exists in V'(a, b). Construct it. Let lp E V(a, b). (We assume the function

~(x) =

1/J'(x) + w,(x - xo)

f lp(~) d~

(2.2)

28

1. GENERALIZED FUNCTIONS AND THEIR PROPERTIES

a'

b' =! b"

!

Xo

a xo-f,=a/l

Xo

+E

b

x

Figure 8

where

WE

is the "cap" when e

!

< min(xQ -

a, b - xo) (see Sec. 1.2) and

00

=

"'(x)

[
1 ) -

w,(x' - xo)

!


(2.3)

-00

We will prove that .,p E V(a, b). Indeed, .,p E Coo and .,p(x) 0 for x a" = min(a',xQ - c) > a ifsupp\O C [a',b'] C (a,b). Furthermore, for X> 6" max(b', Xo + c) < b,

!

00

.,p(x)

=

!

00


-00

f

<

=

00

we(x' - xo) dx'

-00


-00

Thus, supp'IjJ C [a",b"] C (a,b) (Fig. 8). Hence 'IjJ E V(a,b). Applying the functional I( -1) to (2.2), we obtain

(/(-1), rp)

= (/(-1), .,p') + (/(-1) ,We(X -

That is to say, taking into account (2.1),

(f(-I), lp)

= -(f,.,p) + C

!

Xo))

!

rp(f,) dE,.

rp(E,) dE,

(2.4)

where C = (/(-l),we(x - xo)). Thus, if f(-I) exists, then it is expressed by (2.4), where 'I/J is defined by (2.3). Let us now prove the converse: given an arbitrary constant C, the functional /(-1) defined by equalities (2.4) and (2.3) defines the antiderivative of fin (a,b). Indeed, the functional f(-I) is clearly linear. Let us prove that it is continuous on V(a,b). Let 'Pk -t 0, k -t 00 in V(a,b), that is, SUpPr,ok C [a',b'] C (a,b) and rpia ) (x) ~ 0, k --+ 00. Then, by what has already been proved,

!

x

",,(x)

=

[
!
dx'

=a

-00

outside

[a", b"] C [a, b]

and, obviously, .,plQ)(x) =S. 0, k --7 00, that is, 'ljJk --7 0, k -t Therefore, by virtue of the continu.ity of f on V(a, b), we have k --+

00

in V(a, b).

00,

which is what was affirmed. Consequently, /(-1) E V'(a, b). It remains to verify that f(-I) is the antiderivative of f in (a, b). Indeed, substituting tp' for tp in (2.3) and noting that f tp'(E,) dE, = 0, we get 1/J = r,o, and then from (2.4) there follows the equality (2.1), which is what we set out to prove. We have thus proved the following theorem.

2. DIFFERENTIATION OF GENERALIZED FUNCTIONS

29

Every generalized junction f in 'D'(a, b) has in (a, b) an antiderivalive f( -I), and every antiderivative of it is expressed by the formula (2.4), where t/J is defined by (2.3) and C is an arbitrary constant. THEOREM.

This theorem states that a solution of the differential equation

f,

u' -

f

E 1)' (a, b)

(2.5)

exists in V' (a, b) and its general solution is of the form u = /(-1) + C, where f(-I) is some antiderivative of I in (a, b) and C is an arbitrary constant. In particular, if f E C(a, b), then any solution in V'(a, b) of the equation (2.5) is a classical solution. For example, the general solution of the equation u' a in V'(a, b) is the arbitrary constant. The definition of the antiderivative I( -n) of order n in (a, b) of the generalized function f E V'(a, b), I(-n)(n) = f, is similar. Applying this theorem to a recurrent chain for f(-k) (the antiderivatives of f of order k),

1

=

1(-1)'

= I,

1(-2)'

= f(-I),

... ,

I(-n)'

0 Figure 9

= /(-n+I),

we conclude that j(-n) exists in 'D'(a, b) and is unique up to an arbitrary additive polynomial of degree n - 1. 2.3. Examples. 2.3.1. Let us compute the density of charges corresponding to the dipole of the moment +1 located at the point x 0 and oriented in a given direction 1 (iI, ... ,In), III 1 (Fig. 9). Approximately corresponding to this dipole is the charge density (see Sees. 1.1 and 1.7)

=

=

1

1

C

£

=

-o(x - cl) - -o(x), Passing to the limi t here as

£

1) = -[
-o(x - £1) - -o(x),

C

8cp(O) aI

> O.

-+ +0 in V' (IR. n),

(1

----+

£

=

0,

£

= -


!

we conclude that the desired density is equal to

- ao~;) =_(1,86 (x)) . Let us now verify that the total charge of the dipole is 0:

(-~1 ,1) and that its moment is equal to 1:

= (J,

~:) =

(J, 0) = 0,

30

1. GENERALIZED FUNCTIONS AND THEIR PROPERTIES

8c)J;)

2.3.2. A generalization of is a double layer on a surface. Let 5 be a piecewise smooth two-sided surface, n the normal to 5 (Fig. 10) and va continuous function on S. We introduce the generalized function (vc5 s ), which operates via the rule

:n

8 ) ( -&n(vc5s ),cp =

f

v(x)

8
an

r.p E V.

dS,

s Clearly

-~(VOS)E'I),.

SUPp[-:n(vosl]

cS.

The generalized function - aOn (v8s) is called a dou.ble layer on the surface S. It describes the spatial density of charges corresponding to the distribution of dipoles on the surface S with surface moment density v( x), the dipoles oriented in the given direction of the normal n on S. (Here, the density of the double layer is defined as the weak limit of the densities corresponding to the discrete arrangement of dipoles on the surface 5,

s Figure 10

- Lk a8

[V(Xk).6.Skc5(x - Xk)] I

Xk

E 5,

nk

by an unbounded refinement of the surface S; compare Sec. 1.7.)

f(x)

a

b

x

Figure 11

2.3.3. Let a function f(x) be piecewise continuously differentiable in (a, b) and let {Xk} be points in (a, b) at which it or its derivative has discontinuities of the first kind (Fig. 11). Then

J' = f~1 (x) + I: [fJx

k

8 (x - Xk)

(3.1)

k

where f~l(x) is the classical derivative of the function f(x), equal to f'(x) when x =f Xk, and is not defined at the points {Xk}; [J]Xk is ajump of the function f(x) at the point Xk 1

2. DIFFERENTIATION OF GENERALIZED FUNCTIONS

31

Indeed, for any 'P E V(a, b) we have (!', rp) = - (I, rp') Xk+1

J

=- L k

f(x)r.p'(x) dx

Xk

f f~l(x)'P(x)

Xk+l

= L: k

f

dx - L[f(Xk+l - O)r.p(xk+d - f(Xk

+ O)r.p(Xk)]

k

Xk

= f~l(X)'P(X) dx + L:[f(xk + 0) -

f(xk - O)]r.p(Xk)

k

=(f~I'r.p) + l:)fJxk(6(x -

Xk),r.p)J

k

which completes the proof of (3.1). In particular, if 0 is the Heaviside unit function (see Sec. 0.2), then

0

0' (x) = 0- (x ). (3.2) In the theory of electric circuits, the Heaviside unit function is called the unit-step junction, and the delta-function is called the unit-impulse junction. Formula (3.2) states that the unit-impulse function is a derivative of the unit-step function. 2.3.4. The following formulae hold true:

k

= OJ 1, ... , m -

1,

k> m.

(3.3)

Indeed, (xmo-(k)(x)J'P) = (-I)k(xm
= (_I)k

L

(~) (xm)(j)r.p(k-j)(x)!x=o

O~j~k J

=

{~~l)km!(~},o(k-ml(O)

= { 2.3.5.

k=O,l"m-l,

(O,lp),

k=O,I, ... ,m-I,

(-l)mm!(~) (8(k-m l ,I?),

k > m.

The trigonometric series 00

ikx 'L..J " ak e ,

(3.4)

k=-oo

converges in V'. True enough, the series aox

m+2

(m + 2)!

""'

+ L..J

k:tO

ak

ikx

{ik)rn+2 e

32

1. GENERALIZED FUNCTIONS AND THEIR PROPERTIES

fo(x)

x Figure 12

f~(x)

x -1/2 Figure 13

converges uniformly in ~ 1; hence, the series which is a derivative of it of order m + 2 converges in V' and its sum determines the sum of the series (3.4) (see Sec. 2.1.1). 2.3.6. Let us prove the formula

2~ L eikx = L 6(x k

2kll").

(3.5)

k

To do this we expand the 211"-periodic function (Fig. 12)

x2 41l" 1

x

fa (x) = 2" -

0

< X < 27l",

into a Fourier series that converges uniformly in ~ 1 :

fo(x) = 11" _ 6

ikx ~ ' " -"!"-e . 2

(3.6)

211" L- k k:#O

By virtue of 2.3.4, the series (3.5) can be differentiated termwise in V' any number of times. As a result, we get

1,o' (X ) -- -1 2

fg(x)

-

x -_

21l"

i ~ 1 ikx , ~ -e 21l" kf:O k

--

= -~ + LJ(x 21l"

k

2k1l")

o :S x < 211",

= ~ L:e ikX , 21l"

k;tO

whence follows formula (3.5). In differentiating the function f~(x) (Fig. 13), we made use of the formula (3.1). Note that the left-hand side of (3.5) is nothing other than the Fourier series of the 21l"-periodic generalized function L:k 6(x - 2k1l"), the graph of which is symbolically depicted in Fig. 14 (see Sec. 7.2 for more details).

2. DIFFERENTIATION OF GENERALIZED FUNCTIONS ~

j(X+21r)

r- r-

f~/(X)

I

I

Xl

r

4,,)

2")

)0-

:0

-21r

33

21r

41r

x

Figure 14

2.3.7. Let G be a domain in jRn with a piecewise smooth boundary Sand let n = D x be an outer normal to S at the point xES (Fig. 15). Suppose f E C 2 (G) n Cl(O) and f(x) = 0 outside G. Then for any rp E V we have the following Green's formula:

f (t~'P - 'P~f) = f (f~: - 'P ~~) dx

(3.7)

dS.

S

G

We can rewrite Green's formula (3.7) as follows in terms of the generalized functions (of a simple layer and a double layer) that were introduced in Sec. 1.7 and Sec. 2.3.2: (3.7 / )

G

D,.cl! is the classical Laplacian of f:

where

D..cl!(x)

Figure 15

={

Let us verify that the function

1

D,.TX\ True enough, the function

x E G,

not defined,

xES.

x

rt 0,

*

By f and ~ on S we mean the boundary values of f and

2.3.8.

D.f(X), 0,

l/lxl

on S from within the domain G.

I~l

in

jR3

satisfies the Poisson equation

= -41r8(x).

(3.8)

is locally integrable in IR 3 and

X:1=

o.

(3.9)

34

1. GENERALIZED FUNCTIONS AND THEIR PROPERTIES

Figure 16

Let

supp If' CUR. Then

(

I~I' 1") = C~I' 81")

8

=

J I~I~
UR

=

,~~o

!

dx

1~1~
o!:
Applying Green's formula (3.7) for f = l/lxl and G = [x: and taking into account (3.9), we obtain the formula (3.8):

= lim --1 2 ,~+o c

!

E

< Ixl < R]

(Fig. 16)

cpdS

Sf!

=

lim

€~+o

{~![tp(O) - cp(x)] dS - 41rCP(O)} E

Sr

= -41r(6,cp).

0

The equation (3.8) may be interpreted as follows: the function 1/lxl is the Newtonian (Coulomb) potential generated by the charge + 1 at the point x = o.

2 DIFFERENTIATION OF GENERALIZED FUNCTIONS

35

Similarly, L\ln Ixl 1 L\ Ixl n -

2

= 271"6(x), = -(n -

n = 2,

(3.10)

2)O'n6(x),

where Un is the surface area of a unit sphere in The function £n(x), which is equal to

m,n (see Sec. 0.6).

1 (n _ 2)O"n Ixln-2' n

=

en (x)

1

2:

3,

= 2, n = 1,

271" In Ix L

n

1

21xl,

is termed the fundamental solution of the Laplace operator. 2.4. The local structure of generalized functions. We will now prove that the space 1)' (0) is locally a (smallest) extension of the space £~c(0) such that, in it, differentiation is always possible. THEOREM. Let f E V'(O) and let an open set 0' function 9 E £00 (0') and an integer m ~ 0 such that 1

f (x)

= or ... 8~n 9 ( x),

@

O. Then there exist a

x E 0'.

(4.1 )

PROOF., According to the theorem of Sec. 1.3, there exist numbers [{ and k such that the following inequality holds:

1(!,
(4.2)

Since, for"p E V(O'), "p(x) = J~~ oj'l/Jdxj, it follows that ma~ l"p(x) I

< d ma~ IOj'ljJ(x) 1\

rEO

xEO

where d is the diameter of 0'. Therefore, applying this inequality a sufficient number of times, we obtain from (4.2) this inequality

I(!,'P)I

~ Cma~la~ xEO

... a~

(4,3)

Furthermore, for 'l/J E V(O')

tjJ(x) =

fXI ... fX'

a8".p(~

dYl ... dYn

Y1 ... Yn

-00

-00

and therefore

11/J(x) I ~

fI81" .8 1f(Y) Idy. n

0

1

From this fact and from (4.3) there follows the inequality (fOf m = k

I(f, 'P) I :S C

f lar ...

8: 'P (x) Idx

I

+ 1)

V (0').

0'

IThe derivative in (4.1) is to be understood in the sense of generalized functions.

(4.4)

1. GENERALIZED FUNCTIONS AND THEIR PROPERTIES

36

From the Hahn-Banach theorem it follows that the continuous linear functional

f*:

(4.5)

admits of an extension to a continuous linear functional on (1((')') with norm ~ C by virtue of the inequality (4.4):

I(f*, x)1

= I(f, 1fl)1 :::; Cllxllo(ol).

By a theorem of F. Riesz, there exists a function 9 E £00 (0') with norm

11911.c00 (OJ) <

C such that

= (_l)mn /

(!*,x)

g(x)X(x)dx.

0'

From this and from (4.5) we derive, for all

(f,lfl)

= (_1)mn /

g(x)81 ... a:
0'

= (ar ... a: g, rp), which is equivalent to (4.1). This completes the proof of the theorem. COROLLARY.

Under the conditions of the theorem,

in 0' ,

-,

o (4.6)

91 E C(O ).

The representation (4.6) follows from the representation (4.1) if the function g(x) is continued via zero onto the whole of IRn and if we put Xl

gl(X)

Xn

= / ... / -00

g(y) dYl ... dYn.

-00

o 2.5. Generalized functions with compact support. We introduce convergence on a set of functions C<X>(O): epk ¢:::::::>

8 cx epk(X)

~

xeo

k --+

OJ

00

in

COO (0)

J

~

0,

k --+

00

for all

Q

and

0'

~

O.

From this definition it follows that convergence in V(O) implies convergence III Ceo (0), bu t not vice versa. Suppose a generalized function fin V'(O) has compact support in 0, supp f J{ @ O. Suppose 1] E V(O), 1J(x) ::::: 1 in the neighbourhood of J{ (see Sec. 1.2). We will construct a functional f on C<X>(O) via the rule

=

(j. Ifl) =

(5.1 )

(!, TJep) ,

Clearly, j is a linear functional on Coo (0). Furthermore, since the operation ep ~ 1J1fl is continuous from Coo (0) into V( 0), it follows that is a continuous functional on C<X> (0). The functional f is an extension of the functional from V(O) onto COO(O), since for ep E "D(O)

i

(1, ep) = (f, 'flIP) = ('fIf, IP) = (f, ip) by virtue of the equality (10.2) of Sec. 1.10.

2.

DIFFERENTIATION OF GENERALIZED FUNCTIONS

37

We will show that there is a unique linear and continuous extension of f onto COO(O) (~n particular, the extension (5.1) does not depend on the auxiliary function Let j be another such extension of f. We introduce a sequence of functions {7]k} in V(O) such that 1Jk(X) 1, x E Ok (0 1 @ O 2 @ ... ,0 Uk Ok), so that 7]k --+ 1, k --+ (X) in COO(O). Therefore, for any 'P E Coo(O) we will have T]k'P --+ 'P, k --+ 00 in COO (0). Hence, 7]).

=

(1, Ifl)

=

= (j, k-+oo lim 11kV') = lim (1,1]k'P) = lim (f, 7]k'P) k-+oo k-too -

-

~

= lim (1,7]kY') = (], lim 7]k'P) = (], 'P), k-too k-tlXl

Ifl E

ClXl(O),

=

so that f I· We have thus proved the necessity of the conditions in the following theorem. THEOREM. For a generalized function f in 'V'(O) to have compact support in 0, it is necessary and sufficient that it admit of a linear and continuous extension onto Coo (0). PROOF OF S_UFFICIENCY. Suppose f E 1>'(0) admits of a linear and continuous extension I onto Coo (0). If f did not possess compact support in 1>, then it would be possible to indicate a sequence of functions {'Pk} in V( 0) such that SUPP!Pk C 0 \ Ok (0 1 @ O 2 (§ ... , Uk Ok = 0) and (I, !Pk) = 1. On the other h~nd
=

Let f be a generalized function with compact support in O. Then, by virtue of (5.1), we have

(I, 'P) = (/,7]ep),

'P E V(O).

Since TJ E V(O) and sUpPTJ (§ 0, it follows that 7] E V(O') for some 0' @ O. Therefore 7]'P E 1>(0') for all tp E 1>(0). By the theorem of Sec. 1.3 there exist numbers J( = j{(O') and m m(O') such that the following inequality holds:

=

I(f, ep)1

= 1(/, TJtp)1 < KII1J'Pllcm(o')1

tp E V(O),

whence immediately follows the inequality 'P E V(O).

(5.2)

Inequality (5.2) implies the following assertion: any generalized function with compact support in 0 has a finite order in 0 (see Sec. 1.3). We denote by £' the collection of generalized functions with compact support in ~n. It has thus been proved that. £' COO (ffi. n )'.

=

2.6. Generalized functions with point support. Generalized functions whose supports consist of isolated points admit of explicit description. This is given by the following theorem. THEOREM. If the support of a generalized function / E V' consists of a single point x = 0, then it is uniquely representable in the form

f(x) =

L

caoQo(x)

[QI:SN

where N is the order of f, and

CO'

are certain constants.

(6.1 )

38

1. GENERALIZED FUNCTIONS AND THEIR PROPERTIES

Ixi :::;

Suppose T/ E V(Ud, 1](x) ::::: 1, 1](~) f and, hence, for any cp E 1)

PROOF.

f :::::

(/,>0)

1/2. Then for any

E:

> 0 we

have

= ('7 (;) f. >0) =

(f,lJG)(>o-SNJ) + (f''7G)SN)

(6.2)

where

ao cp(O) x a

SN(X; cp) = ~ L.....J

o!

lal5 N

is the Taylor polynomial of cp at zero of degree N (see Sec. 1.8). Since 1J(~) (

(J,'7(:)(>o-SN))

'7(~)(>O-SNJ

':;C

::::: G max aa I$I<~

lal~N

L


f

max lal5 N

-

< G' -

{1](:'c) [
(j3Q:)

::::: G max Ixl<& laSN I3Sa

c"(U.)

L

fJrJ

o

- 13 [cp(X)

- SN(X)]

C

c-ll3lcN-la-l3lc

I3So.

max cN-10. 1+ 1

:::::

Gf/t:.

lol5 N

In the right-hand member of (6.2), let c ~ +0. By virtue of the resulting estimate, the first term wil~ tend to zero. _But the second term does not at all depend on E: and is equal to (/, rpN), where f is the extension of f onto Goo(O) (see Sec. 2.5). Therefore the eq uation (6.2) takes the form

(f,'P)

L

= (!,SN) =

aocp,(O) (!, xo.).

lal5 N

0:'.

Now set

( _I)la l _

(f,xO:)

I

Co:::::

0:'.

and we get the representation (6.1):

(f,
L

(-1)la1caaaep(0)=

lolSN

L

ca(aOo,cp),

cpEV.

lol5 N

We now prove the uniqueness of the representation (6.1). If there is another such representation

L

f(x):::::

c~aOo(x),

lalSN

then by subtracting we obtain

0=

L lo:lSN

(c~ - ca)aOo(x),

2. DIFFERENTIATION OF GENERALIZED FUNCTIONS

39

whence

° L (c~ =

Cf

k

c a )(8 J, X )

lal~N

L

=

(c~ -

ca

aa x Ix=o

)( -1 )Ia I

k

lal~N

= (-I)lklk!(c~ that is,

c~

=

o

and the proof of the theorem is complete.

Ck,

EXAMPLE.

Ck),

The general solution of the equation

=0

xmu(x)

(6.3)

in the class V' (~l) is given by the formula u(x)

L

=

CkJ(k)(x)

(6.4)

O
where

Ck

are arbitrary constants.

=

Indeed, if u E V'is a solution of the equation (6.3), then either u 0 or supp u coincides with the point x = O. By the theorem that has just been proved, u(x)

L

=

CkJ(le) (x)

(6.5)

O
for certain numbers Ck and integer N tuting (6.5) into (6.3), we have

~

O. Taking into account (3.3) and substi-

L

0= (-l)ffl m !

(~)ck
m
whence it follows that Ck = 0, k ~ m. Thus, in the representation (6.5) we can assume N m - I, and the formula (6.4) is proved. It remains to note that the right-hand side of (6.4) satisfies equation (6.3) for arbitrary constants Ck, k =

=

O,l, .. "m-l.

0

=

=

2.7. Generalized functions P(1l'vlxla-1). Let 1l'v(x) sgn V x, v 0, I, be a multiplicative characters of the field of real numbers ~. For ~Q > 0, the function 1l'v(x)lxl a- 1 is locally integrable in ~l and therefore it defines a regular generalized function 1l'vlxla-l by formula (6.1)

(1l'vl x la- 1 ,ip)

=

J

7l"v(x)lxla-lip(x)dx,

tp E'D.

(7.1)

The integral in the right-hand side of equality (7.1) is a holomorphic function of the complex variable Q in the half-plane ~Q' > O. We say that a generalized function fa E V' dependent on the complex parameter Q' is holomorphic (meromorphic) with respect to Q in a domain G if, for any t.p E V, the function (fa,
40

1. GENERALIZED FUNCTIONS AND THEIR PROPERTIES

This definition implies that the generalized function 1Tv lxIO'-l is holomorphic with respect to 0' in ~a > O. Let us rewrite equality (7.1) for ?Ra > 0 and N 1,2, ... in the form (1r v

=

f

lxl a - 1 ,lp) =

1rv (x)l x IO:-

l

[lp(x) - SN-:-dx;
Ixl
+L

f

1r v ( X )

Ix 10:- 1xl> dx

Ixl
k=O

+

f

(k)(O) I{) k!

1rv

(x)lxl o - l lp(x) dx,

(7.2)

Ixl>1 where SN (x; -N + 1,

f

7I'"v(x)lxl a - 1x N - 1 dx =

Ixl<1 and for ~(}

< - N + 1, k =

f

-

1rv

{

if v + N is odd

2 00' + N - l'

if v

'

+N

(7.3)

is even

0 1, ... , N - 1, I

(x)lxI

O -

1 k X

dx =

Ixl>1

2

{

OU

+ k'

1

if l/ + k is even if v + k is odd,

we deduce from (7.2) and (7.3) the following theorem.

The generalized function ?Tvlxl o- l , ~(} > 0, admits the meromorphic continuation P (1Tv I x 10 ) from V' onto the whole plane a with simple poles and residues THEOREM.

_2_ 6(2n)(x), (2n)!

(} = -2n, a

= -2n -

I,

In every half-plane

(2n ~(}

2

(p{7r IX\ct-l), I{)) = V

) 8(2n+l)(x),

+1

> - N,

n

= 0,1, ... ,

n=O,l, ... ,

!

if ?Tv (x)

= sgn x.

= 1,2, ... , it admits the representation

N

f

1l"v(x)l x IO -

l

[
Ixl
~

+2

1 (k) (u+k)k!

LJ v+k even

+

f

1T v

(x)l x la - 1

(7.4)

Ixl>l

and in any strip -N

+ 1 > ~a > -N

it admits the representation
(7.5)

3. DIRECT PRODUCT OF GENERALIZED FUNCTIONS

41

The generalized function p(7r v lxl a - 1) from V' considered outside its poles the holomorphic continuation of the function 1rv (x)lxl a - 1 from the domain ~Q' > 0 - is called the regularization of this function (compare Sec. 1.8). It follows from this definition that the regularization is unique.

3. Direct Product of Generalized Functions 3.1. The definition of a direct product. Let f(x) and g(y) be locally integrable functions in open sets 0 1 C }Rl and O 2 C IR m respectively. The function f (x)g (y) will also be locally integrable in 0 1 x O 2 • It defines a (regular) generalized g(y)f(x) in V ' (OI x ( 2 ) operating on test functions
=

(f(x)g(y), <,0) =

J J J J ! f

f(x)g(y)<,o(x, y) dx dy

01X 0 2

=

f(x)

0

g(y)iP(x, y) dy dx

O2

1

g(y)f(x)cp(x, y) dx dy

01 X0 2

==

f(x )
g(y)

O2

(')2

that is

(f(x)g(y), cp) == (f(xL (g(y),
(1.1 )

(g(y)f(x),Y') == (g(y), (f(xLcp(x,y))).

(1.1')

These equations express the Fubini theorem on the coincidence of iterated integrals and a multiple integral. We take (1.1) and (1.1') as the starting equalities for defining the direct products f(x) xg(y) and g(y) x f(x) of the generalized functions f E 'D'(Od and 9 E "D'(02):

= (f(xL (g(y), Y'(x, y))),

(1.2)

(g(y) x f(x), iP) = (g(y), (f(x),
(1. 2')

(f(x) x g(y),
where

w(x) == (g(y),
1,

( 1.3)

We now prove the following lemma.

Let 0' @ 0 1 X O 2 and 9 E 1)'(02)' Then there exist an open set == adO') <s 0 1 and numbers C C(O',g) > 0 and integer m m(O', g) ~ 0, LEMMA.

a1

=

=

I, GENERALIZED FUNCTIONS AND THEIR PROPERTIES

42

such that 1jJ E V(Ol)

if t.p E V(O');

aa.'ljJ(x) = (g(y),a~r.p(x,y)),

(1.4)

cp E 1)(0 1 x O 2 );

laa.'ljJ(x) I :S C ma~/18~aecp(x,y)1,

V(O'),

r.p E

(x ,y) EO

(1.5)

x E 0 1-

(1.6)

l.el~m

PROOF. We will prove that the function 'ljJ(x) defined by (1.3) has compact support in 0 1 . Since supp

Xk

in

V( (

(1.7)

2 ).

Indeed, supp cp(Xk' y) C O~

{} ya.( Xk,Y')

fJ,

I

0 yE 2 ==:::}

x

I

02 and

{}a ( ) yep X,Y 1

Xk--+

o

@

X .

Taking advantage of the continuity of the functional g, we obtain from (1.3) and (1.7), as Xk --+ x,

Figure 17

¢(Xk) = (g(y),cp(Xk,y)) --+ (g(y), 'P(x, y))

= ¢(x),

which is to say that the function 'I/J is continuous at an arbitrary point x. Thus 'ljJ E C(Od· Now we will prove that 'l/J E COO(OI) and that the differentiation formula (1.5) holds true. Let C1 = (1,0, ... 1 0). Then for every x E 0 1 1 ocp(x, y) Xh(Y) = h[r.p(x+he 1 ,y)-r.p(x,y)] --+ aXl

h--+O Indeed, supp XI. C O~

V(02)'

in

@

02 for sufficiently small hand

J:\a. U

XI. Y

()

YE~>2 f;la. 8
(1.8)

1

J:l

h

'

UX I

--+ O.

Since 9 E V f (02), then using (1.3) and (1.8), we get

·lj;{x+hed-t.p(x) h

1

= h[(g(y),r.p(x

+ he1,Y)) -

(g{y),t.p(x,y)

= (g(y), 'I'(x+hel'~) -'I'(x,y)) = (g, XI.) --+

( g(y),

o
Y)) '

)]

3. DIRECT PRODUCT OF GENERALIZED FUNCTIONS

whence follows the truth of formula (1.5) for first derivatives

81/J(x) l::l

UXj

= (g (y,) a
UXj

Q'

= (1,0, ... ,0) j

,

43

and, hence, for all

= 1,2, ... , n.

Again applying the same reasoning to this formula, we see that (1.5) holds true for all second derivatives, and so forth; hence for all derivatives. And since the function aa 0 that depend solely on 9 and O~, such that

laa~(x)1 = l(g(y),a~
c

ma~ 18ea;
x E 0 1,

yE0 2

1.BI::;m

D

whence follows inequality (1.6). The proof of the lemma is complete.

The operation

COROLLARY.

If'(x, y) --t ~(x) = (g(y),
2)

into 1'(Od.

Indeed, the linearity of the operation is obvious. Furthermore, if'fJ E 1'(0 1 X ( 2 ), then, by the lemma, 1/J E V(OI) so that this operation carries 1'(0 1 x ( 2 ) into V(OI). Let us prove that it is continuous. Let
C 0' <E 0 1

X

O2,

a~ae 'Pk(X, y) ~ 0,

k

--+ 00.

From this and from (1.4) and (1.6) we derive the following for the sequence (g(Y),'Pk(X,y)) , k 1,2, ... ,:

=

SUpp'fPk COl @Ol,

aa'fPk(X) ~ 0,

~dx)

=

k--+oo,

so that "pk --+ 0, k --+ 00 in V(OI). D Let us return to formula (1.2), the definition of the direct product I(x) x g(y). By the corollary to the lemma that was just proved, the operation ~ --+ t/J is linear and continuous from 1)(0 1 x (2) to 1'(Ot} and, hence, the right-hand side of (1.2), which is equal to (I, 1/J), defines a linear and continuous functional on V(OI x ( 2 ) so that f(x) x g(y) E V'(OI X ( 2 ), Similarly, using (1.2'), we can prove that g(y) x f(x) E V' ((')1 X ( 2),

3.2. The properties of a direct product. 3.2.1. Commutativity of a direct product.

I(x) x g(y)

= g(y)

Indeed, on the test functions r.p E 1'(0 1 x (

'fJ(x, y)

=

L l
(2.1)

x f(x),

ui(x)vdy)'

2)

of the form (2.2)

44

1. GENERALIZED FUNCTIONS AND THEIR PROPERTIES

the equality (2.1) follows from the definitions (1.2) and (1.2'):

(f(x) x g(y),~)

= 2:

= (g(y)

(f, Ui)(g, vd

x !(x),~).

l
In order to extend (2.1) to any test functions in V( VI X V 2 ) let us prove a lemma that states that the set of test functions of the form (2.2) is dense in V( 0 1 X ( 2 ). For any

LEMMA.

V(OI x O 2 ),


=

2:

k = 1,2, .. "

Vik E V(02),

Uik E V(Ot},

Uik(X)Vik(Y),

l~i~NIe

that converges to 'P in 1'(0 1

X ( 2 ),

Suppose supp 'P <E 8 1 X 8 2 ~ 0i x O~ ~ 0 1 X O 2 . By the Weierstrass theorem, there exists a sequence of polynomials Pk (x, y), k = 1,2, ... , such that PROOF.

-, -,

lal < k, Suppose e(x) E V)OD, e(x) the sequence of functions

= 1, x E 8 1 ; TJ(Y)

E

02'

(2.3)

= 1, Y E 8 2 .

Then

(x,y) E 0 1

V(O~), TJ(Y)

X

k=1,2, ... ,

is the required sequence. Indeed, supp
O~

<E 0

1 X

O 2 and of all k

~

10'1,

- O-

if (x,y) E 0

1 X

2,

if (x,y) E 0i x O~ \

(8 1

x

8 2 ),

a'"

for certain Ca estimated in terms of max I ~ I and max Ie"'?]!, f3 ::; a. And that means that
(J(x)

X

g(y), 'P) = lim (f(x) x g(y), 'Pk) k-+oo

= k-+oo lim (g(y) x f(X),
[f(x)

x

g(y)J

x

h(z) = f(x)

f

E V' (Od, 9 E V' (0 2 ) and h E

x [g(y) x

h(z)J.

(2.4)

3. DIRECT PRODUCT OF GENERALIZED FUNCTIONS

Indeed, if

([J(x)

X

O2

X ( 3 ),

45

then

x g(y)] x h(z),
= (/(x), (g(y) , (h(z),
= (f(x)

X

[g(y) x h(z)], cp).

Henceforth, taking into account the commutativity and associativity of the operation of a direct product, we will write (/ x g) x h = EXAMPLE.

o(x)

= J(xd

f

x 9 x h.

X J(X2) x ... x J(x n ).

3.2.3. If 9 E V' (0 2 ), the operation f --+ f x 9 is linear and continuous from V'(Od into V'(OI x ( 2 ), The linearity of the operation is obvious. Let us prove continuity. Suppose !k --+ 0, k --+ 00 in V(OI). Then for all

k --+

00.

That is, fk(X) x g(y) --+ 0, k --+ 00 in 1)/(0 1 x ( 2), Here we made use of the fact that 1/J E V(OI) by virtue of the lemma of Sec. 3.1. D 3.2.4. The following formula holds:

= supp! X suppg, If f(x) x l(y) = 0 in 0 1 x O 2 ,

(2.5)

supp(f x g) COROLLARY.

Indeed, suppose (xo, YO) E supp fx supp 9 and V (xo, Yo) is the neighbourhood of the point (xo, Yo) lying in 0 1 x O 2 . There exist neighbourhoods VI and U2 of the points Xo and Yo respectively such that U1 x V2 C U(xo, Yo). From the definition of the support of a generalized function (see Sec. 1.5(b)) it follows that there are functions
. suppfllsuPPU

,

X

suppf

I,

o

=

suppf

= 0 in 0 1 .

then I(x)

A

I 'I

Figure 18

suppg C supp(f

X

2) \

(2.6 )

g).

Let us now prove the converse inclusion. Take a test function

I

(supp f x suppg)

1

x(

2)

46

1.

GENERALIZED FUNCTIONS AND THEIR PROPERTIES

(Fig. 18). Then there is a neighbourhood U of the set supp / such that for every x E U, SUPPip(x,y) C O 2 \suppg. Therefore (see Sec. 1.5(a))

= (g(y) , ip(x, y)) = 0, and, hence, supp'l/J n supp f = 0, and so (/ X 9, ip) = (!,1f) = o. 1f(X)

Thus the zero set Ojxg contains (0 1 x ( following inclusion holds true: supp(f x g)

x E U,

(suppf x suppg) and, hence, the

2) \

c f x suppg.

This, together with the converse inclusion (2.6), proves the equality (2.5). 3.2.5. The following formulae, which are readily verifiable, hold true: if V'(Ol) and 9 E V'(02), then a~ae [I(x) x g(y)]

= a f(x) x a/3 g(y), Q

(2.7)

a(x)b(y) [J(x) x g(y)] = [a(x)f(x)] x [b(y)g(y)] ,

(/ x g)(x + xo, y + Yo) = f(x

+ xo)

x g(y

f

D E

+ Yo).

(2.8)

(2.9)

3.3. SOIne applications. We will say that a generalized function F(x, y) in V'(Ol x O 2 ) does not depend on the variables y if it can be represented in the form

= f(x)

F(x, y)

f

x l(y),

E V'(CJd

(3.1)

(and then F E V'(CJ 1 X ~m)). The generalized function f(x) x l(y) = l(y) x f(x) acts on the test functions ip in V(Ol X JRm) via the rule

(J(x)

X

l(y),l") = (f(x),

J

l"(x,y)d Y)

= (1(y) x f(x), ip) =

J

(f(x), ip(x, y)) dy.

We have thus obtained the equality

(f(x),

J

I"(x, y) dy) =

J

(J(x), I"(x, y)) dy

(3.2)

which holds for all / E V'(Ol) and ip E V(t\ x JRm). The formula (3.2) may be regarded as a peculiar kind of generalization of the Fubini theorem. Suppose F E D'(O x (a, b)). The following three statements are equivalent:

(1) F (x, y) does not depend on the variable y; (2) F(x, y) is invariant in CJ x (a, b) with respect to translations along y, that IS,

F(x, y + h) = F(x, y);

a

< y,

Y+h

< b;

(3.3)

(3) f(x, y) satisfies the following equation in 0 x (a, b):

8F(x, y) 8y

= O.

(3.4)

47

3. DIRECT PRODUCT OF GENERALIZED FUNCTIONS

:!- =

If f E V'(O) and 0, j = 1, ... n, in 0/ then f = const J in 0,' if f is invariant with respect to translations along all arguments in 0/ then f = canst in O. COROLLARY.

I

(1) ~ (2). It follows from (3.1) by virtue of (2.9), (2) -+ (3). Passing to the limit in PROOF.

=

F(x,y+ h) - F(x,y) 0 h as h ~ 0 in D'(O x (a, b)), we conclude that F satisfies the equation (3.4). (3) -+ (1). Let F satisfy the equation (3.4) in 0 x (a, b). Then, proceeding as in Sec. 2.2, for any 'P in D(0 x (a, b)) we obtain the representation 'P(x,y)= where Yo E (a, b), e

B"p(x, y) By +we(Y-Yo)

< min(yo

J

'P(x,~)d~,

(3.5)

- a, b - Yo) and

y

1/0(;1:, y) =

J

[",(x, y') - w,(y' - Yo)

J

",(x, €) d€] dy E D( 0 x (a, b)).

-00

By introducing the generalized function f(x) taken from V' (0), which function acts on the test functions X taken from V( 0) via the rule

(f, X)

= (F (x, y) , w, (y -

Yo) X(x )) ,

and by taking into account (3.4), we obtain from (3.5)

That is, F(x, y) = f(x)

X

1(y), which is what we set out to prove.

D

The proof is similar for the following assertion (compare Sec. 2.6). Suppose f E V' (0 x lR 1). The equation

yu(x, y) = F(x, y)

(3.6)

is always solvable and its general solution is of the form

(u, 'P) = (F,,,p) + (f(x) x J{y), 'P), where

f

(3.7)

is an arbitrary generalized function in 'V' (0) I

"p(x, Y) =

~ ['P(x, y) y

- ry(y)
(3.8)

=

ry(y) is an arbitrary function in V(IR 1) equal to 1 in a neighbourhood of y O. Indeed, since the operation 'P -+ "p given by (3.8) is linear and continuous from D(O x }Rl) into D(O x Il~.1), the right-hand side of (3.7) is a generalized function in V'(O X ~l) and the first term (F,,,p) satisfies equation (3.6) while the second term, the generalized function f(x) x 6(y), satisfies the homogeneous equation

yu(x, y) = 0 which corresponds to the equation (3.6).

(3.9)

48

1. GENERALIZED FUNCTIONS AND THEIR PROPERTIES

It remains to prove that f(x) x o(y), f E V' (0), is the general solution of the equation (3.9) in D'(O x JR.l). Suppose u(x, y) is a solution of the equation (3.9) in D'(O x JR.l). Then, by virtue of (3.8)

'P (x Y) = y1jJ ( x , Y) J

+ 1]( y) ep ( X , 0) ,

1/J E V (0 x JR. 1)

and therefore

(u, 'P)

= (u, y'ljJ) + (u, 1] (y) 'P (x, 0) ) = (u,1](Y)~(x,O)).

(3.10)

By introducing the generalized function f(x), taken from D'(O), that acts on the test functions X in 1'(0) via the rule

(fJx)

=

(u(x,y),1](Y)X(x)),

we obtain from (3.10)

(u,
u(x, y)

=

= (I,
X

o(y),
o

which is what we set out to prove.

3.4. Generalized functions that are smooth with respect to some of the variables. Suppose 1(x, y) is a generalized function in V' (0 1 X ( 2 ) and 'P (x) is a test function in V(OI). We introduce the generalized function Icp(Y) in D'(02) via the formula

(!cp,1jJ) =

(I, tp(x)1fJ(y))

From this definition there follows the differentiation formula (4.1 )

Indeed

= (-I)lal(!,'Poo?jJ) = (-I)lol(fcp,oo?jJ) = (aofcp,1/JL

((o~f)cp,?jJ) = (o;f.1fJ'P)

D We will say that the generalized function f(x, y) taken from D1(01 x ( 2 ) belongs to the class CP(02) with respect to Y, P 0,1, ... , if for any 'P E 1J(0I) the generalized function fcp E CP(02); but if Icp E CP(02), we then say that IE CP(02) with respect to y (compare Sec. 1.6). Suppose I E C(02) with respect to y. It then follows that for every y E O 2 there eXlsts a restriction fy (x) in V' (Od of the generalized function ! (x YLand

=

I

(4.2) Indeed, for a fixed Yo E O 2 , 'P ---+ f'P (Yo) is a linear functional on V (Ch). Let us now prove that it is continuous. To do this, note that for all sufficiently large k ~ N (Yo), the functional 'P ---+

(Jcp (y )

J

WI /

dy -

Yo)) ,

where Wl/k is the "cap" (see Sec. 1.2), belongs to D'(02). But by virtue of (7.6), Sec. 1.7, k ---+

00.

3. DIRECT PRODUCT OF GENERALIZED FUNCTIONS

49

By the theorem on the completeness of the space V'(Od (see Sec. 1.4) we conclude therefrom that the functional f
Suppose a generalized function F does not depend on the variable f(x) x l(y), f E 'D'(Od (see Sec. 3.3). Then f E coo(~m) with respect

EXAMPLE.

y, F{x, y)

=

to yand Fy(x)

= f(x).

Recall that the class Co(O), which is defined in Sec. 0.5, consists of continuous functions with compact support in O. We introduce convergence thus: !k -+ 0, k -+ 00 in Co(O) if supPik CO l LEMMA.

@

0

and

A{x) ~ 0,

k -+

00.

If f(x, y) E C(02) with respect to Y, then the operation X -+

(fy , X(z, y))

is continuous from 1>(0 1 x (2) into Co(Oz).

Let X E 'D(01 X ( 2), Then supp X C O~ x O 2, where 0i ~ 0 1 and O~ ~ O 2 . Put 'IjJ(y) (Iy , x(x, y)). We have supp'lf C O 2 ~ O 2 . We now prove that 1/J E C(02)' Let Yo be an arbitrary point in O 2 and let Yk -+ Yo, k -+ 00. Then PROOF.

=

1t/J(Yk) - t/J(Yo)

I :sl(!y" ,x(x, yo)) -

(fyo,x(x,yo))!

+ 1(!Yk,X(X,Yk) - x(x,Yo))l·

(4.4)

The first summand on the right of (4.4) tends to 0 as k -+ 00 by virtue of the continuity of the function (!y, X(x, Yo)), and the second summand, by the weak bounded ness of the set {IYk} c 'D' (0 1 ) and by virtue of the fact that

X(x, Yk) - X(x, Yo) -+ 0,

k -+

00

in

V(Od

via the lemma of Sec. 1.4. Thus 1/J E C(02) and for this reason 1/J E CO(02)' Let Xk -+ 0, k -+ 00 in 'D(01 x (2)' Then supp Xk C 0i x O 2 where 0i ~ 0 1 and O 2 @ O 2 . Putting 'l/Jk(Y) (fYI Xk(X, y)), we have SUpp"pk C O~ ~ O 2Furthermore, the set of generalized functions {!Yl Y E O~} in V'(Od is weakly bounded. For this reason, by applying the inequality (3.1) of Sec. 1.3 (see also the corollary to the lemma of Sec. 1.4), for certain K > 0, m ~ 0, and for all y E 02 we obtain

=

l1Pk(y)1

= 1(!Y,Xk(x,y))I:::; I
This completes the proof of the lemma.

~ 0,

k -+

00.

o

Let us now prove the following formula: if f(x, y) E C(02) with respect to y,

then

(I, X) i.e. !(x,y)

= fy(x).

=

J

(fy , X(x y)) dy, l

(4.5)

1. GENERALIZED FUNCTIONS AND THEIR PROPERTIES

50

Indeed, by virtue of (4.2) the equality (4.5) holds true on the test functions X of the form I:'P(x)~(y), where 'P E V(OI) and"p E V(02):

Jf LJ

(I, I:
(f 'L'P(x)1/J(y)) Y

But the set of such functions is dense in V( 0 1 by the lemma, --t

ifI:cp(x)"p(y) --7 X(x,y) in V(01 x ( formula (4.5) follows.

(y)1/J(Y) dy

(fy, ip)tjJ(y) dy

= I

(fy,L'P(X)1/J(Y))

lf

X ( 2)

(fy,x(x,y)) 2 ),

(4.6)

dy.

(see Sec. 3.2.1) and, besides,

in

0 0 (0 2 )

It is from this and from (4.6) that the

0

4. The Convolution of Generalized Functions 4.1. The definition of convolution. Let I and 9 be locally integrable functions in ~n. If the integral f f(y)g(x - y) dy exists for almost all x E ~n and defines a locally integrable function in ~n, then it is called the convolution of the functions f and 9 and is symbolized as 1* 9 so that (J

* g)(x)

=

J

f(y)g(x - y) dy

=I

g(y)f(x - y) dy = (9

* f)(x).

(1.1 )

We note two cases where the convolution f * 9 definitely exists. 4.1.1. Let f E .cl-o CI 9 E supp I c A, suppg C B, and the sets A and B are such that for any R > 0 the set

.ctoc'

y

TR = [(x,y): x E A, y E B,

Ix+yl

~ R]

is bounded in ~ 2n (Fig. 19). Then 1* 9 E 'cfoc' Indeed, using the Fubini theorem, we have, for all R> 0,

I I I * 91 dx :S Illf(y)119(X - y)1 dydx Ixl
Ixl
:S Ilf(y) Ilg(~)1 dyd~ < 00.

o

A

TR

In particular, if f of 9 is with compact support, then T R is bounded.

Figure 19

x

4. THE CONVOLUTION OF GENERALIZED FUNCTIONS

51

4.1.2. Let 1 E £P and 9 E £q if 1+1 > 1. Then I*g E [/ where 1T = 1+1_1. p qP q Indeed, choosing the numbers a ~ 0, {3 ~ 0, s ~ 1 and t > 1 such that 1 1 1 - + - + - = I, ar = p = (1 - 0:)8, f3r = q = (1 - (3)t, r s t and then pr qr p+-=r=q+-, s t and making use of the Holder inequality and the Fubini theorem, we obtain the required estimate I

Ilf * gll~c = / /

f(y)g(x

~ y) dy , dx

< / [/ If(yWlg(x - y)JPlf(yll'-Ulg(x - y)I'-P dyr dx

:s / /

If(yW' Ig(x - y)JP' dy

[j If(y)l('-u), dyr'

x [/ Ig(x - y)l(1-P)t dyr,t dx ~11/11~p

Ilgllcq·

l.e.

Ilf * gllL:r :s 11/1IL:pllgllcq· This estimate is called the Young inequality. The convolution 1 * 9 defines a regular functional on V(IR?n) via the rule

(1 * g, ip) = /(1 =

f

=/

* g)

:::: / f(y) /

g(~)
That is

(1 * g, 'P) :::: / f(x)g(y)ip(x

+ y) dx dy,

V.

(1.2)

(In deriving (1.2) we made repeatedly use of the Fubini theorem.) We will say that the sequence {17k} of functions taken from D(I~n) converges to 1 in jRn if

=

=

(a) for any compact f{ there is a number N N (I{) such that 17k (x) 1, x E f{, k ;::: N, and (b) the functions {17k} are uniformly bounded together with all their derivatives, !8 a l]k(X)1 < Ca , x E IR n , k = 1,2, .... Note that there always exist such sequences, for example:

1).(x)

= 1)(~),

where

1) E'D,

1)(x)

1,

Ixl < 1.

1. GENERALIZED FUNCTIONS AND THEIR PROPERTIES

52

Let us now prove that the equality (1.2) can be rewritten as

(f * g,
= k-+oo lim (f(x)

x g(y), 17k (x; y)
+ y)),


(1.3)

where {17k} is any sequence of functions taken from V (IR 2n) that converges to 1 in IR 2n .

Indeed, the function colf(x)g(y)
f

+ y) dx dy =

f(x)g(y)
lim

k-+oo

f

f(x)9(Y)l7k (x; y)ep(x

+ y) dx dy

which is equivalent to (1.3) by virtue of (1.2). 0 Proceeding from the equalities (1.3) and (1.2), we define a convolution of generalized functions as follows. Suppose f and 9 taken from V' (~n) are such that their direct product f(x) x g(y) admits of an extension (j(x) x g(y),
k-+oo

+ y))

= (f(x) x g(y),
+ y));

in fact that limit does not depend on the sequence {17k}. Note that for every k the function 17k (x; y)
(f

* g,
= (f(x) x g(y), ep(x

+ y))

= lim (f(x) x g(Y),l7k(x;y)ep(x k-+oo

+ y)),

(1.4)

Let us prove that the functional j * 9 belongs to V' (~n), that is, it is a generalized function. For this purpose, it suffices, by virtue of the theorem on the completeness of the space V' (see Sec. 1.4), to establish the continuity of the linear functionals

(f(x) on V

(I~ n).

X

k = 1,2, .. "

g(Y),l7k(X;Y)ep(x +y)),

Let ipv ---t 0, v ---t

00

(1.5 )

in V (lR n ). Then

l7k(X; y)epv(x + y) ---t 0,

V

---t

00

III

V(~2n)

since 17k E V(1R 2n ). From this, since the functional f{x) x g(y) on V(~2n) (see Sec. 3.1) is continuous, we obtain

(j(x)

X

g(y), TJdx; Y)
+ y))

---t 0,

v ---t

00

and this completes the proof of the continuity of the functionals (1.5) on v(~n). 0 Note that since ep(x+y) does not belong to V (IP?2n) (it is not with compact support in JR. 2n), the right-hand side of (1.4) does not exist for any pairs of generalized functions f and 9 and, thus, the convolution does not always exist. The convolution of any number of generalized functions is defined in similar fashion. For example, let I, g and h be generalized functions taken from v,(~n)

4. THE CONVOLUTION OF GENERALIZED FUNCTIONS

53

and let {11k} be the sequence of functionals from V(~3n) that converges to 1 in ~3n. The convolution 1 * 9 * h is the functional

(I * 9 * h, ip)

= (/(x)

x g(y) x h(z), ip(x

+ y + z))

= lim (/(x) xg(y) x h(Z),1]k(X;y;Z)ip(x+y+z)), k---t=

if that functional exists In applications, another definition of the convolution of generalized functions, which is equivalent to the definition above, is useful (see Kaminski [54]). Let I, 9 E V'. By their convolution we call the limit

1 *9 =

lim

k---t=

(I 11k) * 9

in

(1.7)

V' ,

if this limit exists for any sequence {1]k} converging to 1 in lR n . (In this case this limit does not depend on the sequence {T}k}.) Other definitions of the convolution can be found in Schwartz [89], Hirata, Ogata [45], Shiraishi [93]' Mikusinski [76], Dierolf, Voigt [16], Kaminski [54]. 4.2. The properties of a convolution. 4.2.1. Commutativity of convolution. If the convolution also does the convolution 9 * I, and they are equal:

1 *9

exists, then so

I*g=g*f. This statement follows from the definition of a convolution (see Sec. 4.1) and from the commutativity of a direct product (see Sec. 3.2.1):

(/*g,
= lim (g(y) x I(x), T]k(X; y)
= (g '" f,
ip

+ y))

E V.

Similarly, from the definition (1.6) we obtain

I*g*h=!*h*g=h*!*g= ...

and so forth.

4.2.2. Convolution with the delta-function. The convolution of any generalized function f in V' with the 6-function exists and is equal to 1:

(2.1 )

!*o=o*!=f.

True enough, let

and so

(I

* 6, r.p) = k---t= lim (I(x)

x 6(y), 11k(X; y)
+ y))

= klim (!(X),11k(X;O)ip(x)) = (I, r.p), ---too

which is what we set out to prove.

o

I. GENERALIZED FUNCTIONS AND THEIR PROPERTIES

54

The meaning of the formula f = f *0 is that any generalized function may be expressed in terms of o-functions, which, formally, is often written thus: REMARK.

f

f(x) =

I

f(e)o(x

-~) de·

It is precisely this formula which one has in mind when we say that every material body consists of mass points, every source consists of source points, and so on (compare Sec. 1.1). 4.2.3. The shift of convolution. If the convolution the convolution f(x + h) * g(x) for all hE lR n , and

f(x

+ h) * g(x)

= (f * g)(x

f *9 exists,

then so also does

+ h).

(2.2)

That is, the operations of shift and convolution commutej in other words, the convolution operator

is a translation invariant operator. Indeed, let {17k} be a sequence of functions in 1J(lR 2n ) that converges to 1 in ~2n. Then for any h E ~n, 17k (x - h; y) --+ 1,

k --+

00

in

~2n.

Now, using the definition of a shift (see Sec. 1.9) and of a convolution (see Sec. 4.1), we obtain, for all rp E 1J(~n),

((I * g)(x + h), rp)

= (f * g, rp(x - h)) = k-+oo lim (f(x) x g(y), 17k (x - h; y)rp(x - h + y)) = k-+oo lim (f(x + h) x g(y), 17k (x; y)
which is what we set out to prove. D Here we made use of formula (2.9) of Sec. 3.2 for the shift of a direct product. 4.2.4. The reflection of convolution. If the convolution f *9 exists, then so also does the convolution f(-x) *g(-x), and

/(-x) *g(-x) = (/*9)(-X). The proof is similar to that of 4.2.3. 4.2.5. Differentiating of convolution. If the convolution exist the convolutions 80: f * g and! * aag, and we have

80. f

* 9 = eo. (f * g) = f * 80. 9 .

(2.3)

f * 9 exists, then there (2.4)

It will suffice to prove this assertion for the first derivatives ej , j = 1, ... , n. Let c.p E 1J(IR n ) and let {17k} be a sequence of functions taken from V(lR 2n ) that converges to 1 in IR 2n. Then the sequence {17k + 8j 1]k} also converges to 1 in ~ 2n. From this fact, taking advantage of the existence of the convolution f * 9 (see

4. THE CONVOLUTION OF GENERALIZED FUNCTIONS

55

Sec. 4.1), we obtain the following chain of equalities (with 1]k _ TJk(X, y)):

(OJ(f*g),
(f(X)X9(Y).TJka'P~+Y)) Xj

- lim k-+oo

-

k~~ (1("') x g(y), a~

(00 [f(x)

= lim

k-+oo

+

['1kl"(n

lim (f(X) x g(Y)l

k-+oo

(11k + ~TJk) r.p(x + y)) VXj

- lim (j(x) x g(y), T/kf{>(X k-+oo

=

+ y))

lim (ojl(x) x g(y), 1]k'P(X + y))

k-+oo

= (OJ f

~~k) .1

+ y))

x g(y)] , TJkip(X

Xj

y)] - I"(x +y)

J

*9

l

+ f * g, 'P)

- (I * g, 'P)

'P) ,

whence follows the first equality (2.4) for OJ. The second one of (2.4) follows from the first one and from the commutativity of a convolution:

OJ (I * g) = OJ (g

* f)

= OJ 9

* 1 = f * OJ g.

From (2.1) and (2.4) there follow the equalities

f

E V'.

Note that the existence of the convolutions 0 01 f * 9 and f * oO:g for 10'1 > 1 is not yet enough for the existence of a convolution f * 9 and for the truth of (2.4). For example,

0' * 1 = 0 * 1 = 1,

but

0 * l' = B * 0 = O.

4.2.6. The operation f --+ f *g is linear on the set of those generalized functions f for which the convolution with 9 exists. This property of a convolution follows directly from the definition of a convolution (1.4) and from the linearity of the operation f --+ f x 9 (see Sec. 3.2.3). In passing we may note that the operation f --+ f *9 is not, generally speaking, continuous from V' into V', as the following example shows:

J(x - k) --+ 0l 4.2.7.

k --+

If the convolution

00

1 *9

supp(f

in

V';

however,

1 * J(x - k) = 1.

exists, then

* g)

C StiPP! + suppg.

(2.5)

Indeed, suppose {1]k} is a sequence of functions taken from V(~2n) that converges to 1 in 1R 2n and ({J E V (JR.n) is such that sUPP ({J n supp! + supp 9 = 0. Since supp(f x g)

= supp f

(2.6)

x supp 9 (see Sec. 3.2.4), we conclude that

supp[J(x) x g(y)] nSUpp[1]k(X; y)
+ y)]

C [supp f x supp g] n [(Xl y): x

+ y E supp 'P] = 0.

1. GENERALIZED FUNCTIONS AND THEIR PROPERTIES

56

And so, due to Sec. 1.5(a), we have

(J * g, 'P)

= lim (f(x) k-4OO

x g(y), 1Jk(X; y)ep{x

+ y))

=0

for all test functions I.p in 'D(IR. n ) that satisfy the condition (2.6). And this means that the inclusion (2.5) holds. D The set supp 1 + supp 9 may also be not closed. Generally speaking, there is no equality in the inclusion (2.5). For example, for the convolution 6'*8 = 6 it takes the form {O} C {x ~ O}. REMARK.

4.2.8. Associativity of convolution. Generally, the operation of convolution is not associative; for example,

(1

* 6') * () =

I'

* () = 0 *() =

0,

but

1 * (6'

*()) = 1 * 6 =

1.

However, this unpleasantness does not arise if the convolution f * 9 * h exists. To be more precise, the following assertion holds true. If the convolutions 1 * 9 and f * 9 * h exist, then so also does the convolution (I * g) * h, and we have

(2.7) Indeed, suppose {7Jk} and {~k} are sequences of functions from D(~ 2n) that converge to 1 in ~ 2n. Then the sequence

of functions taken from V(IR. 3") converges to 1 in )R.3n. From this fact and from the existence of the convolution 1 * 9 * h (see Sec. 4.1) there follows that existence of a double limit: .lim (/(x) xg(y) x h(z),TJi(x;y)6(x+y;z)
= (f*g*h,'P),

k-4OO

and, consequently, of the repeated limit

(I * 9 * h,
lim .lim (/(x) x g(y) x h(z), 1]i(X; Y)~k(X

k-+oo

1-400

+ y; z)tp(x + y + z))

= lim .lim (I(x) xg(Y),7Ji(x;y)(h(z),~k(X+Y;Z)
= lim

k~oo

((I *g)(t),

(h(Z),~k(t;Z)
= lim ((I*g)(t) x h(Z),~k(t;Z)
which proves (2.7) and the existence of the convolution (I*g)*h (see Sec. 4.1).

D

If there exist convolutions f * 9 * h, f * g, 9 * hand f * h, then there exist convolutions (f * g) * h, 1 * (g * h) and (J * h) * 9 and we have COROLLARY.

1 * g * h = (I * g) * h = f * (g * h) = (I * h) * g, which in this case means the convolution is associative.

4.

THE CONVOLUTION OF GENERALIZED FUNCTIONS

57

4.3. The existence of a convolution. Let us establish certain sufficient conditions (besides those given in Sec. 4.1), under which a convolution definitely exists in V'. Recall (see Fig. 19) that

TR

= [(x, y): x E A,

+ yl < R].

y E B, [x

For the definition of the space V'(A) see Sec. 1.5. Let f E V'(A), 9 E V'(B) and suppose that for any R > 0 the set TR is bounded in 1R 2n . Then the convolution f *g exists in V'(A + B) and may be represented as THEOREM.

(!*g,cp)

= (!(x) x g(y),e(x)7](Y)CP(x +y)),

(3.1)

where ~ and TJ are any functions in Coo that are equal to 1 in AE and BE and are equal to 0 outside A 2 E and B 2 E respectively (c is any positive number). Here the operation f --+ f * 9 is continuous from V'(A) into V'(A + B).

Let cp E V(UR) and let converges to 1 in IR 2n. Since PROOF.

{m-l be a sequence of functions in V(~2n)

that

supp(! x g) = supp! x suppg C A x B (see Sec. 3.2.4), it follows that supp{ [I(x)

x g(y)] ip(x + y)}

C

[(x, y): x E A, y E B

Ix + yl < R] = TR·

=

And since T R is a bounded set, there is a number N N(R) such that 77k(X; y) in the neighbourhood of TR for all k 2:: N. For this reason,

=1

(f * g, cp) = lim (/(x) x g(y), TJk(X; y)
= k-+oo lim ([/(x) = (f(x)

X

x g(y)ho(x

+ y), 1]k(X; y))

g(y), 1JN (x; y)cp(x + y))

and the representation

(f

* g, cp) = (/(x)

x

g(y), 1JN(X; y)cp(x + y))

(3.2)

I

is proved. Clearly, the representation (3.2) does not depend on the auxiliary function TJN(X; y). It can be replaced by the function ~(x)1J(Y). Indeed, the function ~(x)1](Y)
=

for any R

> 0 and c > 0 is bounded

in }R2n; furthermore, the function

vanishes in a neighbourhood of TR. This completes the proof of the representation (3.1). From (2.5) it follows that supp(f * g) C A

+B

I

so that the operation 1 -+ f *g carries D'(A) into V'(A + B). Its continuity follows from the continuity of the direct product f x g with respect to 1 (see Sec. 3.2.3) and from the representation (3.1): if !k --+ 0, k --+ 00 in V'(A), then

(!k * g, cp)

= (/d x ) x 9(Y),~(X)1J(Y)CP(x + y))

--+ 0

1. GENERALIZED FUNCTIONS AND THEIR PROPERTIES

58

!J

Figure 20

for all r.p ED, that is, fk is complete.

* 9 --+ 0,

k --+

00

in D'(A

+ B).

The proof of the theorem 0

Note that. t.he continuity of the convolution f * 9 relative to the collection of f and 9 may not occur, as the following simple example illustrates:

6(x

+ k)

--+ 0,

k --+

+00,

6(x - k) --+ 0,

k --+

+00.

However,

We note here an important special case of the theorem just proved. If f E V' and 9 E £', then the convolution in the form

(/ * g, ',0) = (f(x)

f *9

x g(y), 1J(Y)r.p(x

exists and can be represented

+ y)),

(3.3)

r.p E V,

where 1] is any test function taken from D that is equal to 1 in a neighbourhood of the support of g.

Indeed, in this case the boundedness condition of the set TR is fulfilled for all R> 0 (Fig. 20): if suppg C UR', then T R = [(x,y): x ERn, y E suppg,

Ix+ yl S R]

C UR+Rf

X

URI.

Similarly, if f E V' and gl, ... ,9m E £' 1 then there exists a convolution

f * 91 *

... * 9m (see Sec. 4.1) that is associative and commutative (see Sec. 4·2.1, 4.2.8) and the formula (3.3) is generalized thus:

(J

* 91 * ... * 9m, rp) = (f(x)

x gl(Y) x ... x 9m(Z),

1JI(y) .. . 1Jm(z)r.p(x + Y + ... + z)), But if f E COO and 9 E £1, then the convolution mula (3.3) takes on the form

f *9

9 is

the extension of 9 onto COO

= coo(I~n)

(3.4)

E Coo! and the for-

(f * g)(x) = (g(y), f(x - y)), where

r.p E 1).

(see Sec. 2.5).

(3.5)

4. THE CONVOLUTION OF GENERALIZED FUNCTIONS

59

True enough, as in the proof of the lemma of Sec. 3.1, it is established that the function

(g(y) , f(x - y))

= (g(y), 1J(y)f(x -

Y))

E

Cr.xJ.

Then, from the representation (3.3) we have, for all tpEV,

U * 9.
f(e)
+ y) de)

= (9(Y), / '1(y)f(x - y)
Noticing now that 1J(y)f(x-y)
D

(I * g, tp) = / (g(y), 1J(y)f(x - y))tp(x) dx, whence follows formula (3.5). In similar fashion, if f E COO (~n\ {O}) and 9 E [I, then the convolution f * 9 in IRn \ supp 9 is expressed by the formula (3.5); in particular, f * 9 E COO(IR n \ suppg).

Figure 21

4.4. Cones in IR n • A cone in IR n (with vertex at 0) is a set f with the property that if x E f, then AX too belongs to f for all >. > 0. Denote by pr f the intersection of f and the unit sphere with centre at 0 (Fig. 21). The cone f' is said to be compact in the cone r if pr f" C pc f (Fig. 21); we then write r' (§ f. The cone fiji

= [e:

(e,x)

~ 0, Vx E

r]

r

is said to be conjugate to the cone r. Clearly, fiji is a closed convex cone with vertex at 0 (Fig. 22) and (f*)* ch f; here ch f is the convex hull of r (see Sec. 0.2). A cone f is said to be acute if there exists a plane of support for ch f that has a unique common point with ch r (Fig. 22).

=

EXAMPLES OF CONVEX ACUTE CONES.

(a) an n- hed ral cone in IRn:

C

Figure 22

= [x: (e 1, x) > OJ ...• (en) x) > 0]

is acute (convex and open) if and only if the vectors e1, ... , en form a basis in ~n. Then

In particular, the positive quadrant ~+.

= [x:

Xl

> 0, ... , X n > 0],

(IR+')*

= IRt..

60

1. GENERALIZED FUNCTIONS AND THEIR PROPERTIES

(b) The future light cone in JRn+l.

v+

= [x:

(xo,x): Xo

> Ixl],

(V+)*

= V+.

(c) The origin of coordinates {O}, {O}* = }Rn. Note, however, that the cone lR~ x ~n-l [x: Xl > 0] is not acute: (d) The cone Pn C ~n2 of positive (Hermitian) n x n matrices X = (x pq ), P~ = P;t is the cone of nonnegative matrices. This follows from the assertion that, for X E Pn , it is necessary and sufficient that for all S E p;t, S #- 0,

=

(X , 3)

= Tr(X2) = L

Xpq~qp > O.

p,q

LEMMA

1. The following statements are equivalent:

(1) the cone r is acute; (2) the cone ch r does not contain an integral straight line; (3) int 0; (4) for any C J @ int r* there exists a number u u(C J ) > 0 such that

r· :/;

=

(~ , x) ~ u I~ II x I,

~ EC

J ,

x E ch r;

(4.1 )

(5) for any e E pr int r* the set

Be [x: 0 < (e, x) :::; 1, x E ch is bounded in

~n

rJ

(Fig. 22).

r contains an integral straight line x = xO+te, (lei = 1), then it also contains the straight line x = te, -00 < t < 00.

PROOF. (1) 4' (2). If the cone ch -00

< t < 00

Consequently, any plane of support for ch f must contain that straight line, but this contradicts (1). (2) --+ (3). Ifintf* = 0, then, since f* is a convex cone with vertex at 0, it lies in some (n - I)-dimensional plane (e,x) = 0 (lei = 1). For this reason, ±e E f** = ch r. But then t.he integral straight line y = te, ~OO < t < 00 too lies in chr, but this contradicts (2). (3) --+ (4). Since all points of the cone C' different from 0 are interior points relative to f"', it follows that (~, x) > 0 for all E C' and x E ch f. From this fact and also from the continuity and the homogeneity of the form (~, x) follows the existence of a number rr > 0 for which the inequality (4.1) holds true. (4) --+ (5). Let us take an arbitrary e E printf*. Then, by applying the inequality (4.1)' (e, x) ~ ulxl, x E ch r, we conclude that the set Be is bounded: Ixl :::; (e~x) ~ ~. (5) --+ (1). If for some e E pr int f'" the set Be is bounded, then the plane (e, x) = 0 cannot have any other points in common with ch f, with the exception ofO. 0

e

LEMMA

2. Let

r

be a convex cone. Then

r :::: r + r.

is obvious. Let x E r + r so that x = y + z, where y E f and z E f. Then for all A E (0,1) we have x = At + (1- A)l~>" E r and for this reason r + fer, thus completing the proof of the lemma. 0 PROOF. The inclusion

r c r +r

The indicator of the cone

r

is the function

Jlr(~)

= - xEpr inf (~,x). I'

THE CONVOLUTION OF GENERALIZED FUNCTIONS

4.

61

From the definition of the indicator it follows that /-lr(~) is a convex (see Sec. 0.2) and, hence, continuous (see, for example, Vladimirov [105, Chapter II]) and homogeneous first-degree function defined on the whole of lR n . Besides,

JLr«) < JLchr(~), j.tr«) = -Ll(~, ar*), and j.tr(~)

> 0 for
<E f*

Thus f* =

[<: /-ld~)

~

0]

so that the indicator of a cone fully defines only the closure of its convex hull, by virtue of ch f = r·· . EXAMPLE.

3. If

LEMMA

r

is a convex cone, then for any a

>0 (4.2)

The inclusion

PROOF.

r· + U a is trivial: if

f.ld<)

C

[<: ILr «)

= -

+ 6,6 E r·, 161 ~ a, then inf «, x) = - inf [(6, x) + (6, x)] x Epr r x E pr r

r·

(4.3)

-

~ -

inf (6, x) ~ a,

x Epr r

,rtt

4'D-b----t

eo

~ E

(~1

a]

<= 6

since (6, x) 2: 0, x E r. Now let us prove the inverse inclusion of (4.3). Let the point ~o be such that JLd~o) S; a. If <0 E [. or I~ol S; a, then <0 E r'" + U a' Now let ~o fI. ['" and leo I > a. Let the point 6 E [* realto r*, 8«0, r*) = I~o ize the distance from ell. Then, since is a convex cone (Fig. 23), it follows that

(b)

~

r·;

eo, 6) = O.

From the inequality (a) it follows that r*'" f' and therefore

=

a 2: Jlr(eo) = -

6 - eo

.

mf jeo, x)

xEprf'

E

Figure 23

2: - ( eo, l~61 -- eo0 I) .

e

Now the latter is equivalent, by virtue of (b), to the inequality 16 - eol ::; a. Thus, the point eo = 6 + (eo - ed is represented in the form of a sum of two terms 6 E r* and E U a that is, E r* + U a. This completes the proof of the inverse inclusion of (4.3) and also the equality (4.2). The proof of Lemma 3 is complete. 0

eo - e1

eo

62

I. GENERALIZED FUNCTIONS AND THEIR PROPERTIES

Figure 24

Suppose r is a closed convex acute cone. Set C = iut r* (via Lemma 1, C =j:. 0). The smooth (n - 1}-dimensional surface S without an edge is said to be C -like if each straight line x = Xo + te, -00 < t < 00, e E pr f, intersects it in a unique point; in other words: for any x E 5 the cone r + x intersects S in a unique point x (Fig. 24). Thus, the C-like surface S cuts ffi.n into two infinite regions S+ and S-: S+ lies above 5 and S_ lies below S; S+ U S- u S = ~n. At every point x of the surface 5, the normal OJ; is contained in the cone f* + x (Fig. 24). EXAMPLE.

The surface S in ~n+l, which surface is specified by the equation

Ivr f(x)1 < a < 1,

Xo = f(x),

xE

f E

jRn,

c1,

is V+ -like (space-like). LEMMA

4. If S is a C-like surface, than (4.4)

5+=3+f.

Suppose Xo E S+. The straight line x = Xo + te, ItI < 00, e E pr r, intersects 3 at some point Xl = Xo - i l e, il 2: 0 (Fig. 24) so that Xo = Xl + t l e, x I E 3, tIe E r, and the inelusion S + C S + r is proved. Clearly, the inelusion s+r C S+ is true; together with the inverse inclusion it leads to the equality (4.4), D thus completing the proof of the lemma. PROOF.

LEMMA

R' (R)

>0

5. Let S be a C-like surface. Then for any R

>0

there is a number

such that the set

TR =

[(x, y): x

E 5, y E f,

Ix + yl

~

R]

is contained in the ball URJ C jR2n.

Since 5 is a C-like surface. it follows that any point xES that can be represented as ~ - y, y E f, I~I ~ R, is of the form x = ~ - eT, e E pr [, where the number T T(e,~) is uniquely determined bye and { and constitutes a continuous function of the argument (e,~) on the compact e E prf, I~I ~ R. Hence the set [(y,O: y eT(e,O, e E prr, I~I::; R] is bounded and so also is the set TR. The lemma is proved. D PROOF.

=

=

We will say that a C-like surface S is a strictly C -like surface if, under the conditions of Lemma 5,

R'(R) ~ a(1

+ Rt,

v ~

1,

a

> O.

(4.5)

4. THE CONVOLUTION OF GENERALIZED FUNCTIONS

63

v o

Figure 25

= 0,

EXAMPLE. The plane (e, x) virtue of Lemma 1).

e E pre, is strictly C-like with v = 1 (by

4.5. Convolution algebras V'(f+) and V'(f). We will say that a set A is bounded on the side of the cone r if A C r + Ii, where J( is a certain compact (Fig. 25). It is clear that the sets bounded on the side of the cone {OJ are compacts in lR n . Suppose r is a closed cone in lR n . The collection of generalized functions in V' whose supports are bounded on the side of the cone r will be denoted by V' (r +). We define convergence in V'(f+) in the following manner: fk 4- 0, k 4- 00 in V' (f+), if !k --+ 0, k 4- 00 in V', and supp fk C r + j{, where the compact l{ does not depend on k. 2 Set 1J' ({O} +) £'; £' is the space of generalized functions with compact supports (compare Sec. 2.5). Let f be a closed convex acute cone, C = int f*, SaC-like surface, and S+ the region lying above S (see Sec. 4.4). If f E V'(f+) and 9 E V'(5+)1 then the convolution f * 9 exists in V' and can be represented as

=

(f *9,'P) = (f(x) x g(y),€(x)1](Y)
+ y)),

'P E V,

(5.1)

where € and 1] are any functions in Coo that are equal to 1 in (supp f)E and (supp g)€ and are equal to 0 outside (supp f)2~ and (supp g)2€ J respectively (€ is any positive number). Here, ifsuppf C f+J(, where J{ is a compact, then supp(f*g) C 5++J{ and the corresponding operations f ---+ f * 9 and 9 --+ f * 9 are continuous. This assertion follows from the theorem of Sec. 4.3 for A r +[{ and B S+ if we note that by Lemmas 2, 4 and 5 of Sec. 4.4 the set

=

TR

= [(Xl y): = [(x, y):

is bounded for all R

r + K, x E f + K, xE

=

Ix + yl ~ R] E 5 + f, Ix + yl :S R]

y E 5+ 1 y

> 0 and

r + K + S+ = r + [{ + r + S = f + S + K = S+ + K.

o 2 A similar meaning will be attached to other spaces of generalized functions as wellj for example, S'(r+}, .c~(r+) and so forth (see Sees. 5 a.nd 7 below).

64

1. GENERALIZED FUNCTIONS AND THEIR PROPERTIES

We now note an important special case of the last criterion for the existence of a convolution. THEOREM. Let r be a closed convex acute cone. If / E 1)' (r +) and 9 E V'(f+), then the convolution /*g exists in V'(f+) and can be represented as (5.1); here, the operation f ---t / * 9 is continuous from V' (f+) into V' (r +) .

Since f + J{, where J{ is a compact I is contained in S + for some Clike surface (which depends on K), it follows, by the preceding criterion, that the convolution / * 9 exists in V' and can be represented by the formula (5.1). Let us prove that / * 9 E V'(f+). Suppose suppl c r + J{l and suppg C r + J{2, where J{l and J{2 are certain compacts in jRn. Then, using the inclusion (2.5) and Lemma 2 of Sec. 4.4, we obtain PROOF.

= r +]{l + ]{2 The continuity of the operation I ---t f *9 from V' (r +) into

supp(f * g) c

r + ]{l + r

+ f{2

so that 1* 9 E V' (f +). 1)' (r +) also follows from this inclusion. The proof of the theorem is complete.

0

In similar fashion we can prove that the convolution of any number of generalized functions taken from V'(r+) (see Sec. 4.1) exists in V'(f+) and can be expressed by a formula similar to (5.1). From this and from the results of Sec. 4.2.8 it follows that the convolution of generalized functions taken from 1)' (f+) is associative. A linear set is termed an algebra if the operation of multiplication is defined on it, and the operation is linear with respect to every factor separately. An algebra is said to be associative if x(yz) (xy)z and commutative if xy YX. The results established in this subsection enables us to assert that the set of generalized functions V'(f+) forms an algebra that is associative and commutative if for the operation of multiplication we take the convolution operation *. Such algebras are called convolution algebras; the unit element here is the 6-function (see Sec. 4.2.2). Finally, note that the set of generalized functions V' (1) also forms a convolution algebra, a su balgebra of the algebra V' (r +). Indeed. jf IE V'(f) and 9 E V'(r), then

=

supp(f

* g)

C suppl

=

+ suppg C r + f

=

r

so that / * 9 E 1)1(f). (Here, we again took advantage of the inclusion (2.5) and Lemma 2 of Sec. 4.4.) 4.6. Mean functions of generalized functions. Let us extend the concept of a convolution 1* 9 when f and 9 are generalized functions taken from V' (0) and 9 is with a compact and sufficiently small support in 0: supp 9 C UF: and OF: =f:. 0 (see Sec. 0.2 and Fig. 26). In accordance with formula (3.3) we set, by definition

(/ * g, 1,0) =

(/(x) x g(y), TJ(Y)ip(x

=

+ y)),

(6.1 )

where T/ E V(Ot:), T}(Y) 1 in a neighbourhood ofsuppg. By construction, the operation 'i' ---t 1J(Y)ip(x + y) is linear and continuous from V(OF:) into V(O x Uc ). From this it follows that the right-hand side of (6.1) defines a continuous linear functional on V (0eo) so that / * g E V' (0 t: ). Furthermore ,it is easy to see (compare Sec. 4.3) that the right-hand side of (6.1) is not dependent on the auxiliary function TJ. Finally, as in Sec. 4.2, it can be established that the

4. THE CONVOLUTION OF GENERALIZED FUNCTIONS

65

convolution f *9 is commutative and continuous with respect to f and 9 separately, and / * fJ = f. In particular, if a E V (U, ) then, using the reprefJ sentation (6.1) and acting in a manner similar to that of ~~ Sec. 4.3 when deriving (3.5) we obtain a representation I for the convolution / * a; \ I \ tJt; ~I (6.2) (f * o')(x) (f(y), a(x - y», " / £ I

I

\ \!!J

=

...... _ ;

whence follows /

*a

E CO:> (0,) and

(/ * aHO) = (f(y),a(-y»)

Figure 26

= (is,f * a).

(6.3)

By virtue of (6.2), the formula (6.1) takes the form

(/ *g,cp)

= (f,g(-y) * cp),

cp E V(OE)'

(6.4)

Note that when 0 = IR n , formula (6.4) also follows from (3.3) and (6.2). Let WE (x) be the "cap" (see Sec. 1.2) and let f be a generalized function in V/(O). The convolution

IE (x)

= (f *

WE )(

x) = (f (y) , WE (X - y»)

is termed a mean function of I (compare this with the definition of a mean function for the case of I E .cloc(O)' see Sec. 1.2). By what has been proved, the mean function IE E Ceo (Oe ). Now let us prove that

IE -+

I,

c -+ +0

in

V/(O).

(6.5)

True enough, the limiting relation (6.5) follows from the relation WE (x) --+ o(x), C -t +0 in V/, (see Sec. 1.7) and from the continuity of a convolution, by virtue of

= I * w, -+ I * 0" =

IE

I,

C

-+ +0

in

V/(O).

To summarize: every generalized function taken Irom V/(O) is a weak limit of its mean function. Let us use this statement and establish a stronger result.

Every generalized function I in V/(O) is a weak limit 01 the test junctions in V(O), that is, 1)(0) is dense in 1)/(0). THEOREM.

Ie (x)

be a mean function of f. Furthermore, let 0 1 @ O 2 @ ... , Uk Ok = 0, Ck ~(Ok,80) > 0 and 1]1. E V(Ok), 1]k(X) = 1, x E Ok-I. We will prove that the sequence 1]k(X)!ete(X), k = 1,2, ... , of test functions taken from V(O) converges to f in V'(O). Indeed, Ck -+ 0, as k -+ 00 and by (6.5) for all cp E V(O) we have PROOF.

Let

=

lim

k --+- eo

(T]kl€k)

cp)

which completes the proof.

= klim (IEkl1]k
00

0

From the completeness of the space V/(O) (see Sec. 1.4) there follows a converse statement to the theorem that has just been proved: any weak limit of locally integrable functions in (:) is a generalized function in V/ (0). Therefore, it is possible to construct a theory of generalized functions by proceeding from weakly convergent sequences of ordinary, locally integrable functions. With regard to this approach, see Antosik, Mikusinski and Sikorski [2]. REMARK.

1. GENERALIZED FUNCTIONS AND THEIR PROPERTIES

66

It is appropriate at this point to mention the following analogy. The relation of generalized functions to test functions is reminiscent, in a certain sense, of the relation of irrational numbers to rational numbers: by completing the set of rational numbers by means of all possible limits of sequences of rational numbers, we obtain real numbers; by completing the set of test functions by all weak limits of sequences of test functions, we obtain generalized functions. 4.7. Multiplication of generalized functions. In order to give a formal definition of the product of generalized functions, we introduce the following definition. Let 7] E V, f 7](t) dt == 1 and Ak ---t 00, k ---t 00, Ak > O. The sequence ,sk(X) Ak'1](XAk), k ---t 00, of functions from V is called a special c5-sequence. Obviously, 6k (x) ---t 6(x), k ---t 00 in V' (compare (7.6) in Sec. 1.7). Let f E V'(O). The convolution (see (6.2) in Sec. 4.6)

=

fk(X) = / *,sf.; = (f(y).,sx(x - Y)) E C(Ok), is called the mean function for

I

Ok cO

and (see (6.5) in Sec. 4.6)

J,.(x) ---t f(x),

k ---t

00

in

1>'(0).

By the product f . 9 and 9 . I of generalized functions call the limits, respectively,

f and

(7.1) 9 from V'(O) we

(7.2) if these limits exist for any

(0) and for any special 6-sequences {dk} (and do not depend on {c5k}). By virtue of the completeness of the space V'(O) (see Sec. 1.4), I . 9 and 9 . I E V' (0). If f . 9 exists, then 9 . f also exists and they are equal,

f· 9 =

g.

f,

(7.3)

i.e., the product is commutative (see Itano [52]). Note, in particular, that if I, g and f 9 are locally integrable functions in 0, then f· 9 = /g. This fact follows from Theorem I of Sec. 1.2. If / E VI and a E Coo, then af == a . I. There exist other, more general, definitions of products of generalized functions (see Schwartz [89], Shiraishi, Hano [94], Vladimirov [105], Hano [52], Mikusinski [76], Kaminski [54]). 4.8. Convolution as a continuous linear translation-invariant operator. An operator L acting from V' to V' is said to be translation-invariant if Lf(x + h) (LI)(x + h) for all f E V' and for all translations h E ~n. Recall that the definition of convergence in the space Coo = Coo (IRon) is given in Sec. 2.5 and in the space £' in Sec. 4.5; £' is a collection of continuous linear functionals on Coo (see Sec. 2.5).

=

For an operator L to be linear, continuous and translation-invariant from £' to V', it is necessary and sufficient that it be a convolution operator, that is to say, that it be representable in the form L = 10 *, where 10 is some generalized function taken from V',. then 10, the kernel of the operator L, is unique and is expressed by the formula 10 = Ld. THEOREM.

Sufficiency follows from the results of Sec. 4.3 and Sec. 4.2, according to which the convolution operator I ---t 10 * I, 10 E V', is linear, continuous and PROOF.

4. THE CONVOLUTION OF GENERALIZED FUNCTIONS

translation-invariant from £' to V', and fa establish the following lemma.

* d = fa.

67

To prove necessity let us first

For an operator £ to be linear, continuous and translation-invariant from V to Coo, it is necessary and sufficient that it be a convolution operator £ = 10*, la E VI,. here, the kernel fa is unique. LEMMA.

PROOF.

To prove sufficiency, it remains to establish the continuity of the op-

eration
-+ fa *

(see (6.2)) from V to COO. But this follows from the inequality (see the theorem of Sec. 1.3)

*
laa(fo

(8.1)

which holds true m in (8.1) depend on Rand R 1 ). Necessity. From the assumed conditions it follows that the functional (Lip )(0) is linear and continuous on V. For this reason there exists an (obviously) unique generalized function fa E V' such that Lr.p)(O) = (fa (-x), t,O). From this, by the property of translational invariance of the operator L, for all Xa E ~n we derive

(L
+ xo))(O) = =

= (fo(-x),'P(x + xo)) (fo(x),Ip(xQ - x)) = (fo *
(L
D

thus completing the proof of the lemma.

The operator L1 L - L6* is linear, continuous and translation-invariant from £1 to V' (see proof of sufficiency). Besides, for all Xo E IR n we have PROOF OF NECESSITY OF THE HYPOTHESIS OF THE THEOREM.

=

L 1 6(x + xo) == (L6 - L6 * 6)(x + xo)

= (£8 -

L8)(x

+ xo) = 0

so that L 1 vanishes on all translations of the d-funetion. Now let r.p be an arbitrary test function in D. Then

;n

L O::;lkl::;N

~(~)J(x-~)-4~(x),

N-400

in

&',

because for any 'l/J E Coo

;n

L ~(~).p(~) -4 ! ~(x).p(x)dx,

N ----+

00.

O::;lkl::;N

Therefore, by virtue of the linearity and the continuity from £' to V' of the operator L 11 L 1 'P

= J~moo

[;n

L O~lk'~N

~(~)J (x - ~)] = 0,

Now let f be any generalized function in [/. There exists a sequence {/k} of functions in D that converges to 1 in £/ (see Sec. 4.6). From this fact and from the continuity from £' to V' of the operator L 1 we conclude that £1 f = liIllk-+oo L 1 fk = o for all f E £/ so that £1 a and, hence, L L6* 10*.

=

=

=

68

1. GENERALIZED FUNCTIONS AND THEIR PROPERTIES

The uniqueness of the kernel fo of the operator L stems from the following reasoning: if!l E V'is such that !l * f == 0 for all f E £' and, hence, for all I E V, then, by the above-proved lemma, II = O. This completes the proof of the 0 theorem. 4.9. Some applications. 4.9.1. Newtonian potential. Let

Vn ==

I E V'. The convolutions

1

1

!xl n - 1 * I, n ~ 3;

V2 = In I;j

* I, n = 2

(if they exist) are called the Newtonian (for n = 2, the logarithmic) potential with density I. The potential Vn satisfies the Poisson equation ~ ~J

= -(n -

n > 3;

2)(Tnf,

~ V2

= -21f f.

Indeed, using the formula (4.2) of Sec. 2.4 and (2.4), we obtain, for n AVn

~

3,

= A CXI~-l * f) = Alxl~-2 * f

=-(n -

2)(TnJ

* f = -(n -

2)(Tnf.

We proceed in similar fashion in the case of n = 2 as well. If f = p(x) is a function with compact support integrable on ~nJ n > 3, then the corresponding Newtonian potential Vn is called the volume potential. In this case, Vn is a locally integrable function in lR n and is given by the integral

=

Vn(x)

f Ix -

p(y) dy

yln-2

.~

(9.1)

in accordance with formula (1.1) for the convolution of a function p(x) with compact support integrable in lR n with the function Ixl- n +2 locally integrable in jRn. ve5s) be a simple layer and a double layer on Let f fla n and f == a piecewise-smooth surface S C jR n, n > 3, with surface densi ties fl and v (see Sees. 1.7 and 2.3). The corresponding Newtonian potentials

:n(

=

(0)

Vn

1

= Ixl n -

(1) _ _

Vn

-

2

* Jib's

1

I

~

Ixln-2 * on (vJ s )

are, respectively, the surface potentials of a simple layer and a double layer with densities J.L and v. If S is a bounded surface, then the surface potentials V~O) and VJ1) are locally integrable functions in lR n and can be represented by the integrals V:(O)(x) n

f _! ({)

= Ix _Ji(Y) dS, yln-2 y s

(1)

Vn (x) -

1/

y) any

1

Ix _ yln-2 dSy .

(9.2)

s For the sake of definiteness, let us prove the representation (9.2) for the potential V~l). Using the representation (3.3) and the definition of a double layer (see

4, THE CONVOLUTION OF GENERALIZED FUNCTIONS

69

Sec. 2.3), for all t.p E D we obtain a chain of equalities (the function TJ E V and 1J(x) _ 1 in a neighbourhood of S):

(V~l),
:n

(v6 8 ),
C~I~- 2 X :., ( v6s)(y) , 1/(Y)
=-

(a ! + ) =J :n J
= - an (vJs ), 7](Y) v(y)

dS.

[1/(y)

s

If'(x)

yln-2 dx dSy

v(y)

s

=

v(y)

s

dx dS y

t.p(x)

y


dS. dx,

v(y)

s

whence follows the required formula (9.2) for VJl). The change in the order of integration is ensured by the Fubini theorem, by virtue of the existence of the iterated integral

J Jl
O:y

s

Ix _ ~ln-2 dx dSy.

4.9.2. Green's formula. Let the domain G C ~n, n > 3, be bounded by a piecewise-smooth boundary S and let the function /-l E C2 (G) n C 1 (G). Then it can be represented in the form of a sum of three Newtonian potentials via Green's formula (n is an outer normal to S):

1 - (n - 2)o-n

{J Ix~u(y) G

J[Ix -

d yln-2 y - s

1 8u(y) yln-2 an

() 8

any Ix -

-

u y

=

{U(X), 0,

1

yln-2

~,

x E x ~ G.

] dS } y

(9.3)

Indeed, assuming that the function u(x) has been continued by zero for x ~ (; and taking advantage of the formula (3.7 /) and (3.10) of Sec. 2.3, we conclude that u=J*u==-

1

(n - 2)O"n 1

-,----,-----,---...,.-~

(n - 2)O"nlxln-2 1

= - (n _ 2)o-nl x ln-2

~

1

Ixl n -

2

*u

* ~u

au 65 - ana (u6 s )] . * [ .6.c1 u - an

70

1.

GENERALIZED FUNCTIONS AND THEIR PROPERTIES

Whence, using (9.1) and (9.3), we convince ourselves that the representation (9.3) holds true. 0 In particular, if the function u(x) is harmonic in the region G, then the representation (9.3) transforms into the Green's formula for harmonic functions: 1

f [Ix - vl 1

(n - 2)un

n- 2

ou(y) _ u(y) ~

5

on

1

any Ix - vl n - 2

] dS

y

= {u(x), 0,

x E G, x

rt G.

(9.4)

Formulas similar to (9.3) and (9A) occur in the case of n = 2 as well. In this case the fundamental solution - (n-2)u:lxl n 2 must be replaced by 2~ In Ixl· Green's formula (9A) expresses the values of the harmonic function in the domain in terms of its values and the values of its normal derivative on the boundary of that domain. In that sense, it is similar to Cauchy's formula for analytic functions. REMARK.

4.9.3.

A convolution equation has the form

(9.5) where a and f are specified generalized functions in V' and u is an unknown generalized function in V'. Convolution equations involve all linear partial differential equations with constant coefficients:

a(x)

I:

=

aa oa6 (x),

latS;m a

L

*u =

aaaau(x);

lal~m

linear difference equations:

a

*u =

Laau(X - x a ); a

linear integral equations of the first kind: a E

.ctOCl

a

*u =

!

u(y)a(x - y) dy;

linear integral equations of the second kind: a = 6 + IC

a *u

= u(x) +

K. E £toc,

I

!

u(y)K.(x - y) dy;

linear integra-differential equations; and so forth. The fundamental solution of the convolution operator a* is a generalized function f E V' that satisfies the equation (9,5) for f = 6, a

*£ =

6.

(9.6)

4. THE CONVOLUTION OF GENERALIZED FUNCTIONS

71

Generally speaking, the fundamental solution E is not unique; it is determined up to the summand £0, which is an arbitrary solution in V' of the homogeneous equation a * Eo = O. Indeed, a

* (E + Eo)

= a *E+ a

* Eo

=

o.

(1) The function En(x) defined in Sec. 2.3.8 is a fundamental solution of the Laplace operator: !!J.En = O. (2) The formula O(x) + C yields the general form of the fundamental solution in V' of the operator d~ = 0'* (see Sec. 2.2 and Sec. 2.3.3). EXAMPLES.

Let the fundamental solution E of the operator a* in V' exist. We denote by A( a, E) the collection of those generalized functions f taken from V' for which the convolutions E * f and a * £ * f exist in V'. The following theorem holds. Suppose I E A(a, E). Then the solution u of the equation (9.5) exists and can be expressed by the formula THEOREM.

(9.7)

u=E*f. The solution of (9.5) is unique in the class A(a, E).

=

The generalized function u £ * f satisfies (9.5) since, by virtue of the commutativity and the associativity of a convolution (see Sec. 4.2.8) (the convolutions £ * j, a * E * f and a * £ = 0 exist): PROOF.

a *u = a Uniqueness: if a u

* (£ * 1)

*u =

= a

*£ *I

= (a

* £) * f

= 0 * I = f.

0 and u E A(a, f), then

= u * 0 = u * (a * E) = u * a * £ = (u * a) * £ = 0 * E = 0,

which is what we set out to prove. The proof is complete.

o

We can give the solution u = £ * I, (9.7), the following physical interpretation. Let us represent the source I(x) in the form of a "sum" of point sources f(~)c5(x -~) (see Sec. 4.2.2), REMARK.

The fundamental solution E(x) is the disturbance due to the point source o(x). Whence, by virtue of the linearity and translational invariance of the convolution operator a* (see Sec. 4.8) it follows that each point source I(~)o(x -~) generates a disturbance f(~)£(x - ~). It is therefore natural to expect that the "sum" (superposition) of these disturbances

will yield a total disturbance due to the source I, that is, the solution u of the equation (9.5). This nonrigorous reasoning is brought into shape by the theorem proved above.

1. GENERALIZED FUNCTIONS AND THEIR PROPERTIES

72

4.9.4. Equations in convolution algebras. Let A be a convolution algebra, for example V'(r+L 'D/(r) (see Sec. 4.5). Let us consider the equation (9.5) in the algebra A, that is, we will assume that a E A and f E A; the solution u will also be sought in A. In the algebra A, the above theorem takes the following form: if the fundamental soLution £ of the operator a* exists in A, then the soLution u of equation (9.5) is unique in A, exists for any 1 taken from A, and can be expressed by the formula u = [ * f. The fundamental solution [; of the operator a* in the algebra A is conveniently denoted as a-I so that, by (9.6),

(9.8) In other words, a-I is the inverse element of a in the algebra A. The following proposition is very useful when constructing fundamental solutions in the A algebra: if all and a2"l exist in A, then

(9.9) Indeed, (a1 * a2)

* (all * a 21) =

(a2

* ad * (all * a 21)

= a2 * ((a 1 * all) * a2" 1) = a2 * (0' * a2"l) = a2 * a2"l = O'. Formula (9.9) forms the basis of operational calculus. 4.9.5. Fractional differentiation and integration. Denote by V~ the algebra V'(~~).

We introduce the generalized function fa., taken from real parameter a, -00 < a < 00, via the formula

a> a

V~

I

that depends on a

0,

< O.

Let us verify that fa. Indeed, if Q'

* ff3 = fo:+f3'

(9.10)

> 0 and fJ > 0, then (see Sec. 4.1)

!

x

fa. * 1(3

_

-

O(x) r(a)r(fJ)

o a f3 = O(x)x + -

1

r(a)r(f3)

=

Y

a.-I

!

(x - y)

dy

1

r:x-l(1-t)f3- 1 dt

o

O(X)Xa.+ f3 -1

r(a)f(f3) B(a,fJ) O(X)Xa.+ f3 -1 ---:....-.:---- = 101+13'

r(a + f3)

j3-1

4. THE CONVOLUTION OF GENERALIZED FUNCTIONS

Now if Q'

::;

0 or {3 ::; 0, then, by choosing integers m fa

* ffJ

= f~~~

* f~~n

(m+n)

= (!Ot+m

> -(}

and n

> -{3,

73

we obtain

* ffJ+n)(m+ n )

F

= f a+fJ+m+n = J a+f3 which is what we set out to prove.

J

0

Let us consider the convolution operator 1a* in the algebra V~. Since 10 = 0' 6, it follows from (9.10) that the fundamental solution 1;;1 of the operator fa* exists and is equal to I-a: 1;1 = I-a. Furthermore, for integer n < 0, In = 6(-n), and for this reason fn * U = 6(n) * u u(n), which means the operator fn* is the operator of n-fold differentiation. Finally, for integer n > 0,

=

=

* u)(n) = f-n * (In * u) = (1-n * In) * U = 6 * u = u, which is to say that In *u is an antiderivative of order n of the generalized function (In

u (see Sec. 2.2). By virtue of what has been said, the operator ja* is termed the operator of fractional differentiation of order -(}' for (}' < 0 and the operator of fractional integration of order (}' for (} > 0 (it is also called the Riemann-Liouville operator). EXAMPLE.

Let

f

E V~. Then

J x

81/ 2 I

= 8(11 2 * f) = _1 .!!/ ft dx

f(y) dy .

.Jx - y

o

4.9.6. Heaviside's operational calculus is nothing but analysis in the convolution algebra V~. To illustrate, let us calculate in the algebra V~ the fundamental solution £(t) of the differential operator d) dm dm- 1 P ( dt = dtm + al dt m - 1 + ... + am, where form

aj

are constants. In the V~ algebra the corresponding equation takes the

*[: = 0, P(6)(t) = o(m)(t) + Ul0(m-l)(t) + ... + umo(t).

P(o)

If, in the V~ algebra3 , we factor P(6), P(6)

= *II (6' -

Aj 6)k j

,

j

and take advantage of (9.9), we obtain

p-l (o)(t)

=Eft) = [*J} (0' -

Ajo)k;]

-1

= *J} (0' -

AjW k;,

(9.11)

eAt

(9.12)

But it is easy to verify that

*(6' - A6)-k = *[(6' _ A6)-lt =

*8(t)t

k

1 -

(k-l)!

'

74

1. GENERALIZED FUNCTIONS AND THEIR PROPERTIES

whence, by continuing the equalities (9.11), we derive

£(t)

=*

n O(t)t . (k j

-

J

k;-l

I)!

e

Ajt

.

(9.13)

The convolution (9.13) admits of explicit calculation. By decomposing the righthand side of (9.11) into partial fractions in the V'-t algebra, we obtain £(t) =

* II(JI -

AjJ)-k,

j

AjJ)-k j + ... + Ci,t

= L[Cj,kj * (JI -

* (6 ' -

Aj 6)-1],

j

whence, using (9.12), we finally derive tkj-1 _ j

£(t) = O(t) ~ [ Cj,kj (k J

]

1)1 + ... + Cj ,1 e

Ajt

(9.14)

.

We thus have the following rule for finding the fundamental solution of the operator p{ft): substitute p for set up the polynomial P(pL decompose the expression ~ into partial fractions:

it,

P;P)

= U(p - Aj)-k = ~[Cj,kj(P j

)..j)-k j

+ ... + Cj,l'(p _

)..j)-l] ,

J

J

and with each partial fraction (p - A)-k associate the right-hand side of formula (9.12). As a result, we obtain the formula (9.14). EXAMPLE.

Find E if E" + w2 £

We have p2

1

+w2

=~ ( 2Wl

P-

= J.

1. _ +1.) lW

P

lW

foot

O(t~

(eiwt _ e-iwt)

= O(t) sin wi = £(t). W

2Wl

5. Tempered Generalized Functions 5.1. The space 8 of test (rapidly decreasing) functions. We refer to the space of test functions S = S(lR n ) all functions infinitely differentiable in lR n that decrease together with all their derivatives, as Ixl ---1 00, faster than any power of Ixl-t. We introduce in S a countable number of norms via the formula
p=O,l, ....

Clearly, If! E

S.

(1.1 )

We define convergence in S as follows: the sequence of functions 1f!1 J 'P2, ... in S converges to 0, 'Pk --+ 0, k -+ 00 in 8 , if for all p = 0, 1, ... II'Pkllp --+ 0, k -+ 00. In other words, 'Pk --+ 0, k ---1 00 in S, if for all (} and j3 xa{)fJ'Pdx) x~n 0,

k --+

00.

It is clear that V C S, and if
5. TEMPERED GENERALIZED FUNCTIONS

75

Yet V is dense in 8, that is, for any r.p E 8 there is a sequence {If'k} of functions in V such that I.pk -t cp, k -t 00 in 8. Indeed, the sequence of functions, in 'D, k = 1,2, ... ,

=

where TJ E V, TJ(x) 1, Ixl < 1, converges to r.p in S. 0 Let us denote by Sp the completion of S in the pth norm; Sp is a Banach space. The following imbeddings hold:

So => 8 1 :J ... .

(1.2)

Each imbedding

8 p +1 C 8 p ,

p

= 0,1, ... ,

is continuous, by (1.1). We will now prove that this imbedding is totally cantin uous (compact), that is, it is possible, from each infinite bounded set in Sp+l, to choose a sequence that converges in Sp. Indeed, let M be an infinite set bounded in Sp+l, Ilcpllp+l < C, 'P EM. From this, for all If' E M and lal < p, we obtain

a~. aOlcp(x) < C,

j = 1, ...• n;

J

(1 Suppose Rk, k

+ IxI2y/2aal.p(x) -t OJ

Ixl-+ 00.

= 1.2, ... , is an increasing sequence of positive numbers such that (1.3)

By the Ascoli-Arzehi lemma there is a sequence {If'jl)} offundions in M that converges in CP(U R I ); furthermore, by the same lemma there is a subsequence {tp)2)} of the functions {cpJl)} that converges in CP(UR-J, and so on. It remains to k

remark that by virtue of (1.3) the diagonal sequence {
IxI 8 cp(x) P

01

So that r.p E Sp

-+ 0 for Ixl-t 00

it is necessary and sufficient that r.p E CP and and 10:1 S P, so that

Necessity is obvious. Let us prove sufficiency. Suppose

0; there exists a number R = R(c:) such that PROOF.

=

(1 + IxI2r/2IaOlr.p(x)1 < c,

Ixl> R, [0:1 < p. [xl < R + 1, k ~ N 1.

(1.4)

Let N l be a number such that TJk(X) = 1, Finally, from Theorem II of Sec. 1.2 it follows the existence of a number N ~ N l such that for all k ~ N, Ixl ~ R + 1, and 10:1 S p, the following inequality holds true:

(1 + I x I 2 )p/2I aa cp(x) - 8 a CPl/k(X)1 < E.

(1.5)

1. GENERALIZED FUNCTIONS AND THEIR PROPERTIES

76

Now, using (1.4) and (1.5) for k

Ilcp -

~

N, we obtain

'Pl/k7]kllp == suPx(l + Ix]2y/2I aa [cp(x) - Ipl/k"1k(X)] !al:Sp

<£+

sup (1 Ixl>R+l lal:Sp

I

[18 alp (x)1 + /3
+ Ixl2y/2

-

< 2e + C; sup (I + IxI2)P!2

{}fJ /Wl/k(Y)CP(X - y) dy

Ixl>R+l IfJlsp

~ 2£ + C;

I

a

sup / Wl/k (y)( 1 + IxI2)p/21 13 cp(x - y) dy I x l>R+l 1~I:Sp

~ 2e + C;

sup / wl/dy) Ixl>R+l IlJlsp

< 2£ + Cpe + Cpe / ~

[(1 + jx -

Wl!k (y) (1

y12y/2

+ IvI P ]

la lJ
+ lyl2) dy

(2 + 3Cp )c,

which is what we set out to prove. The proof of the lemma is complete.

0

It follows from the lemma that S is a complete space and

S = np~osp.

(1.6)

The operations of differentiation cp -+ aa cp and of the nonsingular linear change of variables cp(x) -+ If'( Ax + b) are linear and continuous from S to S. This follows directly from the definition of convergence in the space S. On the other hand, multiplication by an infinitely differentiable function may take one outside the domain of S, for example, e- 1x12 e1xl2 1 f/:. S. Suppose the function a E Coo grows at infinity together with all its derivatives not faster than the polynomial

=

laaa(x)j < CO'(1

+ Ixl)mc..

(1.7)

We denote by OM the set of all such functions. This is called the set of multipliers in S. The operation cp -+ acp, where a E eM, is continuous (and, obviously, linear) from S to S. Indeed, if cp -+ 8, then acp E Coo and, by virtue of (1.7),

Ilacplip = sUPx(1 + IxI2)P/2Iaa(a
< suPx(1 + I x I 2)p/2 1001:Sp

~ K p suPx (1

L (;)

1

8fJ cp(x)aa- fJ a(x) I

l3:sa

+ IxI2)P/2+N,,/2Iaacp(x)1

lal:Sp = Kpll
P

= 0, 1, ... ,

where N p is the smallest integer not less than maxlal

5. TEMPERED GENERALIZED FUNCTIONS

77

5.2. The space S' of tempered generalized functions. A tempered generalized Junction is any continuous linear functional on the space S of test functions. s,(~n) the set of all tempered generalized functions. Clearly, We denote by S' S' is a linear set and S' C V'. We define convergence in S' as weak convergence of a sequence of funetionals: a sequence of generalized functions It I h, ,.. taken from S' converges to the generalized function! E S', !k -t I, k -t 00 in S', if for any 'P E SI (fk, If') -t (I, ep) k -t 00. The linear set S' equipped with convergence is termed the space S' of tempered generalized functions. From the definition it follows that convergence in S' implies convergence in V',

=

THEOREM (L. Schwartz). Let M' be a weakly bounded set of functionals from S', that is, I(I, ep) I < Crp for all I E M' and

0 such that

f

EM',

(2.1 )

PROOF. If the inequality (2.1) does not hold, then there will be sequence {/k} of functionals from M' and sequences {If'k} of functions taken from S such that

I(A,i;?k)1 > kll'Pkllkl

k=1,2,....

(2.2)

The sequence of functions

epk(X)

1/Jdx)

k=1,2, ... ,

= YkII'Pkllk'

tends to 0 in S because for k

~

P IIlf'k

1 lip <_.

lIl/Jkllp = v'k11'Pkllk

-yfk

The sequence offunctionals {!k} is bounded on every test function If' taken from S. For this reason, we have an analogue of the lemma of Sec. 1.4 according to which (A, l/Jk) -t 0, k -t 00. On the other hand, the inequality (2.2) yields

f(fk,1/Jk)l

1

= Ykll Vk. o

The resulting contradiction proves the theorem.

From the Schwartz theorem we have just proved there follow a number of corollaries. COROLLARY 1. Every tempered generalized function has a finite order (compare Sec. 1.8), that is to say, it admits of an extension as a continuous linear functional from some (least) conjugate space S:n; then, for I, the inequality (2.1) takes the form

1(/, 'P)I where

IIfll-m

:s Ilfll-mll'Pllm,

is the norm of the functional

f

(2.3)

in S:n and m is the order of f·

Thus, the following relations hold true:

Sb

c

S~

c S~

C ... ,

S' =

US;. p~O

They are duals of (1.2) and (1.6).

(2.4)

1. GENERALIZED FUNCTIONS AND THEIR PROPERTIES

78

Also note that every imbedding

s; C s; +1 ,

P

= 0, 1, ... ,

is totally continuous (see Sec. 5.1); in particular, every (weakly) convergent sequence of functionals taken from S; converges in norm in S;+1' COROLLARY 2. Every (weakly) convergent sequence of tempered generalized

functions converges weakly in some space S; and, hence, converges in norm in

S;+1'

This follows from the Schwartz theorem since every (weakly) convergent sequence of functionals taken from S' is a weakly bounded set in S'; it also follows 0 from the remark referring to Corollary 1. COROLLARY

3. The space of tempered generalized functions is complete.

This follows from the completeness of the conjugate spaces

S; and from Corol-

lary 2.

[]

5.3. Examples of tempered generalized functions and elementary operations in S'. A function f (x) is called a tempered function in ~n, if, for some m> 0

f

A tempered function Sec. 1.6,

If(x)l(1

+ lxj)-m dx < 00.

f defines a regular functional f in S' via the formula (6.1) of (I, rp) =

!

I(x)rp(x) dx,

'P E S.

Not every locally integrable function defines a tempered generalized function, for example, eX rt. S'. On the other hand, not every locally integrable function taken from S' is tempered. For example, the function (cos eX)' = _eX sin eX is not a tempered function, yet it defines a generalized function from 5' via the formula

((coseX)','P) = -

!

rp E S.

cosexrp'(x)dx,

However, there can be no such unpleasantness as regards nonnegative functions (and even measures), as we shall now see. A measure Jl specified on ~n (see Sec. 1.7) is said to be a tempered measure if for some m > 0

f (1 + f

Ixl)-mJl(dx) <

00.

It defines a generalized function in 8' via formula (7.2) of Sec. 1.7,

(Jl, rp) =

.,,(x)Jl(dx) ,

." E 5.

If a nonnegative measure Jl defines a generalized function in S' then p is tempered. Indeed, since Jl E S', it follows from the Schwartz theorem that it is of finite order m so that rp E

S.

(3.1 )

5. TEMPERED GENERALIZED FUNCTIONS

Let {17k} be a sequence of nonnegative functions in V that tend to 1 in Sec. 4.1). Substituting into (3.1)


79

~n

(see

+ IxI2)-Tn/2

and making use of the nonnegativity of the measure

~,

we obtain

where C does not depend on k. From this, by virtue of the Fatou lemma, it follows that the measure J.1- is tempered. 0 If f E f', then f E S', and

(f,CP) = (f,1]cp)'

(3.2)

'P E S,

where TJ E 'D and 1] = 1 in the neighbourhood of the support of 1 (compare (10.2) of Sec. 1.10). Indeed, since the operation

(f(Ay+b),ep) = ( f,

cp(A-l(x - b)]) IdetAI '

cP E S,

is a continuous linear functional on S (compare Sec. 1.9). 0 If f E Sf and a E OM, then af E 8', and the operation f -+ af is continuous (and linear) from S' to S'. Indeed, since the operation cP -+ atp is linear and continuous from 8 to S (see Sec. 1.5) I it follows that the right-hand side of the equality

(af, If!) = f(, acp),

tp E 8,

is a continuous linear functional on 8 (compare Sec. 1.10). 0 Thus, the set eM contains all multipliers in Sf (actually, it consists of them; prove it). EXAMPLE.

If lak I ~ C(1

+ Ikl)N,

L k

then

ak,s(x - k) E 8'.

1_ GENERALIZED FUNCTIONS AND THEIR PROPERTIES

80

5.4. The structure of tempered generalized functions. We will now prove that the space 5' is a (smallest) extension of the collection of tempered functions in JRn such that in it differentiation is always possible (compare Sec. 2.4). Hence this explains the name of S' as the space of tempered generalized functions (tempered distributions, according to L. Schwartz [89]).

If f E S', then there exist a tempered continuous function 9 in IR n and an integer m 2': 0 such that THEOREM.

= 8-r ... 8:g(x).

/(x)

(4.1 )

Let f E 5'. By the theorem of L. Schwartz (see Sec. 5.2) there exist numbers K and p such that for all r.p E S PROOF.

I(I, rp) I ~ Kllrpllp
max!

lerl~p

la

1 •••

an [(1

+

I

I

x I 2)P/2 aer
that is

1(/, r.p)1 ~

/181" .On [(1 + !xj2y/2a er rp(x)] II· 1 1:9

(4.2)

K max 0

£1

With every function cp of S we associate a vector function {"po} with components

lal ~

(4.3)

p.

In this way we define a one-to-one mapping t.p --+ {"pa} of the space S into the direct sum E:Blerl~p £1 with norm

II{/alll = lal~p max II/all£l. On the linear subset [{"pa}, 'P E S] of the space E9 l al

r:

(/. , {1/Ja})

= (f, r.p).

(4.4)

By virtue of the estimate (4.2),

!(f*, {7/Jer}) I = 1(/,
the functional f* is continuous. By the Hahn-Banach and F. Riesz theorems there exists a vector function {X a} E Eel erl ~p .1:'::10 such that

(f*, {tPcr})

=L

J

Xa(x)"pa(x) dx_

IQI~p

That is to say, by virtue of (4.3) and (4.4), we have

(/,
!

L

Xer(x)ih ...

an

[(1 + Ix I2)P/2 aa
dx,

lal~p

Integrating the right-hand side of this equation by parts, we are convinced of the existence of continuous tempered functions ga, lal p + 2, such that

:s

(/,g)

= (_l)pn!

L lerl$(p+2)n

ga(X)8f+2 ... 8~+2
5. TEMPERED GENERALIZED FUNCTIONS

81

whence follows the representation (4.1) for m = p + 2. The proof of the theorem is complete. D If f E 8' then there exists an integer p ~ a such that for any c > 0 there are functions ga,F: lal ::; P, which are continuous tempered in jRn and vanish outside the c -neighbourhood of the support of I so that COROLLARY.

I

I

I

f(x)

=L

(4.5)

aaga,E(X),

lal :Sp

=

Indeed, suppose c > a and 1] E OM, 1](X) = I, x E (sUppJ)~/3, and 1](x) 0, x f/:. (supp J)~. (By the lemma of Sec. 1.2, such functions exist.) Then, taking into account the representation (4.1) and using the Leibniz formula (see Sec. 2.1), we have f(x)

= TJ(x)f(x) =1](x)8r ... 8:9(x) =ar ... a: [TJ(x)g(x)] +

L

TJo(x)aOg(x),

[al:Smn-l

=

where 1]a E OM and 1]a (x) 0, x ~ (supp J)t. Each term in the last sum is again transformed in that fashion, and so on. Then, in a finite number of steps, we arrive at the representation (4.5) with p mn and 90,t Xo:9, where Xa are certain functions taken from 8M with support in (supp J)t. D

=

=

5.5. The direct product of tempered generalized functions. Let f(x) E S' (~n) and g(y) E 8' (ffi m ). Since S' C V', the direct product f(x) x g(y) is a generalized function in D'(ffin+m) (see Sec. 3.1). We will prove that f(x) x g(y) E

S'(lR n +m ). By the definition of the functional f(x) x g(y) (see Sec. 3.1),

(f(x) x g(y), ',0) = (/(x), (g(y), cp(x, y))).

(5.1)

We will now prove that the right-hand side of (5.1) is a continuous linear functional on S(Ir:t n +m ). To do this, we set up the following lemma that is similar to the lemma of Sec. 3.1. LEMMA. If E

S', then for all a

a01/J(x) = (g(y), a~
(5.2)

and there is an integer q ~ 0 such that

111/Jllp

:s Ilgll-qllcpllp+q,

p = 0, 1, ... ,

(5.3)

so that the operation r.p --t 1/J = (g(y), r.p(x, y)) is continuous (and linear) from S(Ir:t n +m ) to S(Ir:t n ).

As in the proof of the lemma of Sec. 3.1, we establish the truth of the equality (5.2) for all Q and the continuity of the right-hand side. Consequently, 1/J E C=. Let q be the order of g. Applying (2.3) to the right-hand side of (5.2), we obtain for all x E ~n the estimate PROOF.

lao1/J(x) [

:s IIgll-qsupy(1 + !yj V/ 2

l.ol~q

2

j8:a:
82

1. GENERALIZED FUNCTIONS AND THEIR PROPERTIES

whence follows (5.3):

111/Jllp = sUPr(1 + IxI2)P/2I aa 1/J(x)1 lal:Sp

::; Ilgll-q

sUP(r,y)

(1 + IxI2)PI2(1 + IYI2)q/218~ae
lal::;p,ll3l:Sq

< Ilgll-qll
p = 0, 1, ...

o

The proof of the lemma is complete.

From this lemma it follows that the right-hand side of (5.1), which is equal to (I,,,n is a continuous and linear functional on S(IR n +m ) so that f(x) x g(y) E S'(IR n +m ) (compare Sec. 3.1). All the properties of a direct product that are listed in Sec. 3.2 for the space V' hold true also for the space S'. This assertion follows from the density of V in S (see Sec. 5.1). In particular, the operation f(x) ---+ f(x) x g(y) is continuous from s,(~n) to S'(IR n+m ). Finally, the formula (3.2) of Sec. 3.3 holds true for f E s'(~n) and t.p E s(~n+m):

(J(xl,j I"(x, y) dY)

=

! (J(x), I"(x,

V)) dy

(5.4)

5.6. The convolution of tempered generalized functions. Let f E S', 9 E S' and let their convolution f * 9 exist in V' (see Sec. 4.1). Now: When does f * 9 E S' and when is the operation f ---+ f * 9 continuous from S' to S'? In accordance with (1.7) of Sec. 4.1, we can assume the following definition of the convolution of tempered generalized functions, which is equivalent to (1.4) of Sec. 4.1 and is convenient for computations. Let f, 9 E S. By their convolution' f * 9 E SI we call the limit

f * 9 = k-+oo lim U'f/k) * 9

in

S'

if this limit exists for any sequence {'f/k} converging to 1 in W. n (in this case it does not depend on {'f/k}). Then f * 9 E S and there exists the convolution 9 * f and they are equal each other, f*g=g*f.

We state three sufficient criteria for the existence of a convolution in S'. 5.6.1. Let f E S' and 9 E ['. Then the convolution f * 9 belongs to SI and can be represented as

(I*g,
(6.1)

where TJ is any function from V eqJjal to 1 in a neighbourhood of the support of g,. here. the operation f ---+ f * 9 is continuous from S' to S', and the operation 9 ---+ f * 9 is continuous from [' to S' . Indeed, the convolution f >I< 9 E VI and the representation (3.3) of Sec. 4.3 holds true on the test functions in V. Since f(x) x g(y) E S'(I1t 2n ) (see Sec. 5.5), and the

5. TEMPERED GENERALIZED FUNCTIONS

operation
---1-

'1(y)
83

s(~n)

to S(~2n):

+ y) lip::; sup(x,y) (1 + IX[2 + IYI 2)p/2I ao [7](Y)
11'1(Y)
lal:Sp

< Cp sup(x,y)(1 + Ix + yI2y/2 Iaa
Cpll
10:1 :Sp

it follows that the right-hand side of (6.1) defines a continuous linear functional on S so that f * 9 E S'. 0 5.6.2. Let f be a closed convex acute cone in ~n with vertex at 0, C int f"', S a strictly C-like surface, and S+ the domain lying above S (see Sec. 4.4). If f E S' (f+) and 9 E S' (S +), then the convolution f * 9 exists in Sf and can be represented as

=

(J *g,'P) = (f(x) x g(y),~(x)1J(Y)

(6.2)

where ~ and 1] are any COO-functions, laa~(x)1 ~ ca , laO:1J(Y) I < C a , equal to 1 in (supp J)~ and (supp gy and equal to 0 outside (supp J)2~ and (supp 9 )2~ respectively (E is any positive number4. Here, if supp f C f + J{, where f{ is a compact, then the operation f ---1- f * 9 is continuous from S' (f + J{) to S' (S + + K). To prove this assertion, it remains - by using the representation (5.1) of Sec. 4.5 and by reasoning as in Sec. 5.6.1 - to establish the continuity of the operation r.p ---1- X = ~(x)1](Y)r.p(x+y) from S(I~n) to S(n:t 2n ). For all 'P E S we have

Ilxllp

= sup(x,y) (1 + Ixl 2 + lyI2y/2I arx,y) [~(x)1J(Y)
:S

C;

sup _ xEr+K+U2~

(1 + IxI 2 + IYI2)P/2Ia~,y)
YES+, lol:Sp

~ 2PC~ supz (1 + Ixl 2 + IYI2y/2Iaor.p(~)1 , xET(z)

lal:Sp

where T ( z)

= [x:

x E

r + J{ + U 2~,

X = Z - Y Y E 5+]. 1

Since S is assumed to be a strictly C-like surface, it follows that the set T( z) is contained in a ball of radius a(l + 1~lr, lJ ~ 1 (see Sec. 4.4). Therefore, continuing our estimates, we obtain Ilxllp

~ C; supz [1 + Izl 2 + a2(1 + Izl) 2L1y/2 Iaa r.p(z) I 10:1 :Sp

< Cp ll
P

= 0,1, ... ,

0 which is what we set out to do. From the criterion obtained it follows, in particular, that the set of generalized functions S' (f +) forms a convolution algebra, a subalgebra of the algebra V f (f +); in the same way, Sf (f) also forms a convolution algebra, a subalgebra of the algebras S'(f+) and V'(f). 4 According

to the lemma of Sec. 1.2, such functions exist.

84

1. GENERALIZED FUNCTIONS AND THEIR PROPERTIES

5.6.3. Let f E S' and 17 E S. Then the convolution be represented in the form [compare (6.2) of Sec. 4.6]

f * 1] exists in OM and can (6.3)