Issuu

JOSE SEBASTIAO E SILVA

TEXTOS DIDAcTICOS

Volume III

SERVI<;O DE EDUCA<;AO E BOLSAS

FUNDA<;AO CALOUSTE GULBENKIAN

LISBOA

Reservados todos os direitos de acordo corn a lei Edic;ao da FVNDA<;AO CALOUSTE GULBENKIAN Av. de Berna - Lisboa

1999 ISBN 972-31-0971-9 Dep6sito Legal

n.O

148805100

111.1 THEORY OF DISTRIBUTIONS路

Este texto tern por base apontamentos coligidos por diversos alunos de Jose SebastHio e

Silva na sequencia de urn curso que realizou em

1958 na Universidade de Maryland, e que

posteriormente foram utilizados, e por ele revistos, na Faculdade de Ciencias de Lisboa.

CHAPTER VII

DISTRIBUTIONS OF SEVER AL VARIABLES; FUNDAMENTAL CONCEPTS

7.1. Intervals in IRn space Let n be any integer >1. Given two points, n b = (bp ..., bn) in the IR space, we shall write j=l, . . . , n, and intervals

a = (a l ' ... , an) and a < b , iff aj< bj for

a<b iff aj< bj for j=l, . . , n. Then the bounded .

]a, b[ [a, b], ] a, b], [a, b[ with the extremities a, b are to be ,

defined as in the case of one single dimension. For example, [a, b] is n the set of all points x of IR such that a s x s b (a rectangle in n = 2, n a parallelepiped if n=3, etc). In turn, the set of all points x of IR such that

a < x is the open interval ] a,

n [, unbounded on the

n right. In any case, an interval I in IR is the Cartesian product of n intervals in IR. For example, if 1=

[a, br, then 1=11

12 X

•

In' with

[ap hI [, ..., In= [an' bn[· In order to make the reciprocal of this statement also true, we

shall call every Cartesian product of intervals 11 , ..., In in IR an in n terval I in IR . Then the interval is said to be degenerate, iff at least one of the intervals 11 , ..., In is.

1 28

7.2. Distributions on an interval 1 in IRn

I be any interval in IR n , hence the Cartesian product of n in tervals Il ' . . . , In in IR , and consider the space C(I) (in short C) of all complex valued functions f (x) f (xl " " , xn ), which are defined and continuous on I. As in the case of one single variable, C(I) is a com Let

algebra), rel a tively to the usual algebraic operatio ns . For each k = 1 , , n we shall denote by Dk the partial derivation operator with respect to xk ' that is plex vector space (and even

complex c ommu tative

Dk = � ' Then, for each system r = (rl " put:

dXk

rn ) of n i ntegers rk � O ,

and denote by Cr (I) the set of all functions f such that Dkf (for k s r) exists and is continuous on I in the ordinary sense, independently of

the order in which the differentiation are performed. the other hand, considering for each arbitrarily chosen, we shall put On

a fixed point c k in Ik ,

The integration operator ,3k defined in this way, is obviously a linear mapping of the space C into itself, More generally, for every system r = (ri ' . . " rn ) of n integers, rk � O, we shall denote by ,3r the operator ,3(1 . . , ,3;11 , Obviously, ,3k is a right inverse of Dk , i.e. , Dk ,3k f f, for any f E C. More generally, for every system r (rl , ' , " rn ) of non-negative integers, we have Dr,3rf=f, VfEC. As each Dk is not defined on the whole space C (as a mapping of C into C), there arises the problem of enlarging the set C, in or-der that the operators Dl ' ' ' ' ' Dn may be extended as mappings of the enlarged set into itself according to some natural conditions, which we are going to state precisely in the form of axioms . The new =

1 29

set will be denoted by

tri butions on I. The

DJ , . . . ,

Dn ,

!W(/) and its elements will be called dis set !W(/), provided with the n basic operators

system of axioms :

is j ust defined, up to an isomorphism, by the following

AXIOM 1.

If f E e (/), then f E!W(/) .

AXIOM 2.

To each f E!W(/) and each k = 1 , . , n there corresponds an element Dk f of !W(/) (the derivative of f with respect to xk) , in such a way that: (i) if f is a function having a derivative f:k , with respect to xk ' in ordinary sense and continuous on I, then Dk f coin cides with f:k ; (ii) the operators D 1 , , Dk are mutually inte r changeable, that is: Dj Dk f = Dk � f, for all j, k = l , . , n, and all f EPfl(/). . .

•

. .

If r i s any system teg ers, then Dr = D(1 . . . D:n .

DEFINITION.

AXIOM 3.

�O

(rl ' . . . ' rn )

o f n n o n - n eg ative in

Fo r every f E§(/) there exists a system

of n integers

and a function F E C (/) such that f= DrF.

AXIOM 4.

If r is a system of n integers rk � 0 and F, G E C (I),

DrF = D r G if F - G = e1 + . . . + en ' where ek

and only if F- G is of the form is a polynomial in Xk of degree < rk whose coefficients are continuous functions on I independent of Xk (for k = l , . , n). then we have . .

More explicity, each ek considered in this axiom is of the form:

where the coefficients a k v are continuous functions on I independent of xk . We shall de n o te by �k rk the set of all functions 8k of this form ( for k = 1 , . . . , n ) and by �r the set of all functions 8 of the form

1 30

e = e[ + . . . + en with ekE CJPk rk (which we call of degree < r ) . Thus

In turn, the set of all systems

1Non .

pseudo-polynomials

r of n integers � 0 will be denoted

As for the case of one single variable, it can be proved, in a sim ilar way, that this axiomatic system is both consistent and categori cal. The only essential difference arises in the proof of consistency, about the definition of the equivalent relation. We are going to see precisely what this difference consists of. Axiom 3 says that every distribution on / is determined by a couple (r, F ) where r E 1Non and F E C (/). On the other hand, axiom 4 leads to define a relation --- in the set of all such couples in the following way : (r, F ) --- (s , G ) iff there exists a system m of integers such that m � r, s and 7.2. 1.

The difficulty arises j ust when it is necessary to prove that, if there exists at least one m > r , S satisfying 7 . 2 . 1 . , then every other system h such that h � r, s satisfies the corresponding condition. Now this can be proved with the aid of two lemmas :

LEMMA 2. If p <r.

e E CJPr and, in addition, e E C P then Dp e E CJPr_p for

In fact, suppose that these two lemmas are true and denote by IL the least system of integers such that IL > r, s , that is, IL = ( J1 I . . . , J1n ) with J1 i = sup (rp sJ, i = l , . . . , n . Then, if m is any system � r, s , satisfying 7 .2. 1 . , w e obtain, b y applying D m - p. to both members of 7 .2. 1 . and taking lemma 2 into account: '

131

Th e remaining part of the proof is analogous to the one given for the case n = 1 . So, it is easily proved that the relation ---- just de fined is an equivalence relation, and the class of all couples equiva lent to (r, F ) is denoted by [r, F ] etc. It is, however, convenient to

o bserve that the derivation operators can now be defined in general

b y putting:

Dp [r, F ] [r +p , F ] =

for every system p EINon ; in particular, D, = D( I · o

. . . .

O),

•

, D = D( o . o

. . . .

I ).

The preceding definition shows immediately that these operators are interchangeable. PROOF OF LEMMA 1 . It is almost immediate. It will be suf

fi cient to remember that ,Jp equals the product ,J p( . . . ,J P n regardless

of the order, and that, if Bk is a polynomial of degree < rk in xk ' whose

coefficients are continuous functions on I independent of x k ' then Jj ek is again a polynomial in x k ' with coefficients of the same type

and of degree < rk + 1 or < rk , according to j = k or j ;z! k • .

PROOF OF LEMMA 2. It can be reduced to the following

proposition : if B E ClPr and, in addition, B E CP, then B can be repre sented in the form e = w, + . + wn ' where wk E ClPk rk ' and, in addition, .

E C P.

In fact, this implies that D P wk E ClPk . rk - Pk ' hence D p B E ClPr_ ' by an p argument similar to the one used for lemma 1 . To prove the preceding proposition, remember that I i s the

Cartesian product of n intervals Ii ' . . . ' In in IR . Let Ck l , , Ckrk be rk points chosen arbitrary in Ik for k = 1 , . . , n. Then, to each function •

•

f E C and each k = 1 , . . . , n, corresponds one, and only one, function fk E ClPk r k ' such that:

1 32

To see this it is sufficient to apply the Lagrange interpolation

formula : 7.2.3.

where 7.2.4.

and

lPv (ck,u ) = 0 for v #-J.l , which along with 7.2.3 .

Thus 7 .2.2.

and 7 .2.4. implies

Let us denote by lik the mapping f --:. fk defined in this way for . , n. It is readily seen that lik is a projection of C onto �k rk ' that is a linear mapping of C onto <!Pk rk ' such that lik f = f, for every

k= 1,

f E <!Pk r k ·

Suppose now that EJ is a function E <!Pr having a continuous derivative DPEJ on I in ordinary sense (p s r). Then EJ is of the form

.? e, n

with e. E Qi'", . Pu t

= XI e ; since

XI el

= e" we have

EJ - w 1 = ( I - liI ) e2 + · · · + ( I - liI ) (9

( l - li1 ) EJk E <!Pk rk for k = 2, . . . , n. Put in general

133

Then it i s easily seen by repeated application of the same argument

that e = in

� n

with

E '1/'" " Finally, observe that

is a polynomial

which is obtained by repeated application of Lagrange ' s

7 .2. 3 . ; therefore, its coefficients are linear functions , which derive from e(x ) by replacing

formula of

variables

XI "

combinations

by constants . Since DP8 exists in ordinary sense

and is c onti nuou s on J, i t DPwk (k = 1 , . . , n) . •

one or more

follows that the same property holds for

7 .3. Vector operations and other fundamental concepts

Let f and g be any two distributions on an interval J in /R n, f = D r F and g = D s G, where r, s E /Non and F, G E C (J ). As in the case of one variable, we shall pu t ,

definition:

where m is any system of n integers such that m � r, s . On the other hand, we shall put, by definition:

is easily seen, as in the case of one variable, that the set §(J)

of all distributions on J becomes a complex vector space with the preceding two definitions. Moreover, it is obvious that the derivation operators Dp are linear mappings of this space into itself. Translation operators can also be defined as in the case of one variable. If f D PF, with F E C (J ), and h E /R n , then sh f = DP( Sh F ), where =

1 34

For every r E 1Non , we shall denote by Cr (/ ) - in short C - the set

of all distributions i on / of the form i = D r F, with F E C (/ ) .

7.4. Restriction operators. Global distributions

The restriction operators , for distributions on intervals in IR n , may be defined and denoted exactly as in the case of one variable and they have similar properties . In particular, if / is any interval in

IR n , we can identify every distribution i on / with its restriction to the o

interior of /, so that §"(/ ) C §J(/ ) .

B esides, the collecting . principle can be extended to distribu tions on intervals in IR n by an argument similar to the one used in the case of one variable, but it is a little more complicated ; now the proj ections llk considered in 7 . 2 . should be used for each variable x k separately in order to "collect" to each other the given distributions . 7.4. 1 . DEFINITION. If Q is a (non-empty) open set in

IR n , a global

i = ( i/ ) that may be defined by assigning to each compact interval / C Q one distribution i/ on /, in such a way that, if J is a compact subin-terval of /, then iJ A i/ .

distribution on Q is any system

We shall denote by !if (Q) the set of all global distributions on Q

and, as in the case of one variable, we shall put by definition:

( i/ ) + ( g/ ) = ( f/ + g/ ) , A ( f/ ) = ( A fl ) , Dr(fl) = (D rf/) · Then §J (Q) becomes a complex vector space and D r a linear map pin g of §J (Q) into itself. I n particular, every function f E C (Q) may be identified with the global distribution ( fl ) , where fl is the restric tion of f to each compact interval / C Q , so that C (Q) C §J (Q) .

1 35

7.4.2. DEFINITION. A global distribution f on Q is said to be of

finite rank, if and only if there exists r E /Non and f E C (Q) such that

f = D r F; otherwise, f is said to be of infi ni te rank.

In particular, if Q is an interval, it is easily seen, by the collecting p rinciple, that every distribution f on Q can be identified with a global distribution of finite rank on Q. So, in the general case, the global distributions of finite rank on Q will be called distributi ons on Q, and the set of all these obj ects will be denoted by §J(Q) . This

set, which is obviously a vector subspace of §J (Q), could also be

defined directly by a system of axioms, as in the case of intervals . (It

contrary to the case of IR , the components of an open set Q in IR n are not, in general, intervals. ) should be observed that,

In the preceding definitions, we could consider, more generally, as the domain of a distribution, any set .4 such that

Q C .4 C Q where Q is any (non-empty) open set in IR n . But as in the case of intervals, it is easily seen that every distribution on .4 can be iden tified with a distribution on Q , so that §J(.4) C i?lf(Q) . If f,

g E �(.Q) and (9 is an open set contained in Q, we write f g =

on (9, if and only if the restrictions of f and g to each interval I C (9

coincide. We say that

is null on (9, if and only if f equals the null

function on (9. From the collecting principle follows that the

union of all open sets where a global distribution f is null is again a set where f is null. That being so: 7.4.4. DEFINITION. If f is a global distribution on an open set Q in IR n and if Qo is the greatest open set where f is null, then the set

Q\ Qo is called the carrier of f.

1 36

7.5. Locally summable functions as distributions

A function f is said to be locally summable on an open set Q in IR n if and only if f is summable on each co mpact interval IC Q . The integral of f over I may be denoted by

i f.

If the extremities of

and b (b p . . . , b n ) with a s b , then we may also denote the integral by the notation

I are a

(ap . . . , an )

ff (X) dx or more explicity,

If the condition a s b is not satisfied, we shall put, by definition

where

inf(a,

b) and

p sup (a, b). =

7.5.1 . DEFINITION. If f is a locally summable function on an in颅

terval I in IR n , any function

F such that

where c is an arbitrary fixed point of I, is said to be an integral func路路 tion of f on I. It can be proved that, if F is an integral function of f on I, then

F E C (/)

and f(x) =

a a . . . - F(x) almost everywhere in ordinary aX1 aXn

sense. Moreover, if F] and F2 are two integral functions of j, then

1 37

is a function

fa of the form fa =

.? n

fa. ,

o us function on I independent of xk ' for k

where

fa.

is a continu-

1 , . . , n. As in the case of one variable, two locally summable functions f =

and g on I have the same integral function, if and only if f (x ) = g (x) almost everywhere on 1. In this case, f and g are said to be equiva lent on I, and the vector space of the corresponding equivalent classes [ f ] is defined as in the case of one variable. Besides we shall denote by r (standardized f ) the function de fined by the formula:

only at the points x for which the written derivatives exist in ordi nary sense, independently of the order, and leading to the same value. From now on, when we speak of locally summable functions, it will be in general understood that they are standard functions, and we shall replace any equivalence class [ f] by the corresponding stan dard function r. The vector space of all locally summable functions on I will be denoted by L (/) . That being so, it is easily proved, as in the case of one variable, that o

7.5.4. By assigning to each locally summable fu nction f on I the dis tribution f* DJ . . . Dn F, where F is any integral function of f, there is defined a one-to-one linear mapping of L (/) into �(I), such that: (i) if f E e(I ), then f = f * ; =

(ii) if f is absolutely continuous with respect to xk on Ik , for almost every system of values of the remaining variables, then, to the derivative f:k infunctional sense, corresponds the derivative Dk f * in distributional sense. o

That being so, it is natural to iden�ify each function f E L (I) with the corresponding distribution f * E �(/) .

1 38

An important example of a (non-standard) locally summable function is the Heaviside function on IR n (which we shall denote by H [n) ) defined as follows :

H [ n] (x ) =

if xk > 0 for all k = 1 ,

o if xk < 0 for

some k

. ., n 1, . ., n " .

�

The standardized Heaviside function H [n) is equal to H [n] at any continuity point of H [n) and is not defined at any discontinuity point of H [n). For example, if n 2 , H [n] is not defined only on the semiaxis x2 = 0, I � 0 and x I = 0, x2 � O.

�

0 - - - - - - - - - -

We shall put in general D = D 1 The Dirac distribution on

IR n,

•

c5'[n) , can c5'[n] = D H [ n).

which is denoted by

defined as the pure mixed derivative of

H [ n], that is,

In general, given a locally summable function f, even if f is not a standard function, we may denote also by D'f, where r is any .......,

system of integers, the distribution D'j For example, we may write c5' [ n) = D H [ n) . .

Remarks about notation:

I) We shall often denote simply by H the Heaviside function on IR n, whenever no mistake seems possible. In particular, no misun-

1 39

derstanding may arise, if the independent variables are written; for example, the meaning of expressions such as H(x I ' . . . ' x ) , H(x3), n etc . , becomes quite clear. It should also be observed that, in prac tice, variables appear generally without subscripts , but this gives n o trouble; for example, there will be no doubt about the meaning of expressions such as H (x, y) , H(t ), etc . , or formulas such as H (x, t) = H (x)H (t), ft (X' t) = f (x, t)H(t), etc . , when x, y, t are real variables and f a function on IR 2 . Observe that the Dirac distribution at a point a of IR n is to be defined as in the case n = 1 :

11) It must be observed that the preceding conventions about dummy variables cannot be extended, without some modifications, to distributions on IR n, Now, we shall adopt the following conventions : a) If f is a distribution on a subset of IR n and x, y, . . . are vari ables on IR n, then f(x) = f ( y ) = . . . = f. b) If f is a distribution of one single variable, then f(x l ), · · · , f (x ) n denote distinct distributions on subsets of IR n. For example, the symbols Xl' . . . , x denote n distinct functions on n IR n - the coordinate functions. In turn, H(xl ), , H(x ) denote n n distinct locally summable functions on IR n, whose product is H [n], and so forth. •

•

7.6. Measures as distributions

Let .Q be an open set in IR n. The concept of a measure J1 on .Q can be defined exactly as we did for the case n 1 . For the sake of =

simplicity we shall restrict us here to the case where interval 1. 7.6. 1 . DEFINITION.

point of 1. If we put:

Let J1 be a measure on I and

is an open

= (C l ' . . . ' C ) a n

1 40

(k = 1 ,

2 , . . . , n)

then the function F defined by

F(x) = sgn IT (xk - ck ) · f.1(Jl x J2 x . . . x Jn ) k integral function

for all

xEI

is called the

of f.1 from c .

In order to see how to derive f.1 from F, it is convenient to con sider, for every system r = (ri ' . . . , r ) of integers rk � 0 and every vec n tor h = (hl ' . . . , h ) E IR the operator n

7.6.2.

where ..4 ihi is the difference operator defined by

n=2

For example, for

2 f (X x2) ..4 1 h J f(x , x + h ) - f(x 1 , x )] 2 2 2 = f(x I + h I ' X 2 + h 2 ) - f(x 1 + h I ' X 2 ) - f(x 1 , X 2 + h 2 ) + f(x 1 , X 2 ) ·

..4 h f(x) = ..4 1 h ] ..4 =

is an integral function of the mea sure f.1 on I (in IR n), then, for every pair ofpoints a, b of I, such that a < b, we have: Now, it is easily seen that if F

7.6.3.

,u ] a, b] = ..4 h F(a ) ,

with h = b - a .

Moreover, f.1 [ a, b] = Um X """ a

f.1] a,

b[ =

f.1 ] x , b]

Um f.1 ] a, x]

......

141

and analogously for the other types of bounded intervals J, such that je l (in particular for degenerate intervals). Thus the measure J.1 can be determined entirely from its integral function F. It can also be seen that F is continuous on the right at every point a of L i. e., F (a) = F (a+ ) . 7.6.4. DEFINITION. By a primitive of a measure J.1 on

understand any function 7. 6. 3 .

F on I,

I we

shall

continuous on the right, satisfying

Obv iously, every integral function o f J.1 i s a primitive of J.1,

not conversely .

Let us put, for every interval

but

J = l a, b] with a, b E l (a < b) :

iiF(J) = iih F(a)

with h = b - a .

Then, a function F on I is said to be of bounded variation, if and

J = ] a , b ] with a , b E l, (a < b) , there corre

only if to each interval

M(J), such that, for every partition of each inter val Jk = ] ak , bk ] in a finite number of left open intervals Jki ' · · · ' Jk sponds a number

(k = l , 2, . . . , n), we have

� . . . � I iiF(JIV 1 PI

vI = l

vn = l

•

x Jnv)1 s M (J ) .

That being so, it is easily seen that: 7 .6.5. A function

F on I is a primitive of some measure J.1 on I, if and

only if F is of bounded variation on I and continuous on the right at every point a of I. Moreover, two such functions Fl and F2 are primi tives of the same measure if and only if F F if of the form (91 + . . + en where ek is a function of bounded variation on I, inde pendent of Xk ' k = 1 , 2, . . , n. I

1 42

In the set 0ll (/) of all measures on /, there is defined the structure of a complex vector space, as in the case n 1 . On the other hand, every function fE L (/) can be identified with the measure f.1f ' defined =

by lip) =

Finally, observe that every function F of bounded variation on / is locally summable on / and uniquely determined by the correspond ing standard function. Thus, applying 7 . 6 . 5 . and taking into account axiom 4 in 7.2. , we arrive at the following conclusion:

7.6.6. By assigning to each measure f.1 on / the distribution f * = DF,

where F is any primitive of f.1 , there is defined a one-to-one linear mapping of 0ll (/ ) into §(/ ) such that, if f.1 is a locally summable function on /, then f.1 f *. =

That being so, it is natural to put in the general case f.1 f * , so that 0ll (/ ) becomes a vector subspace of §(/) . This result holds, if we consider instead of an open interval /, any open (non-empty) set Q in IR n . =

7.7. Concepts of multiplication ; tensor products, concrete examples.

Let / be any interval in IR n . The product of a continuous function f on / with a measure f.1 on / can be defined as in the case of one variable . For example, we have, for every continuous function on IR n and every point a of IR n :

f (x ) 8 (x - a ) f ( a ) 8 (x - a ). =

If r is a system of n integers rk � 0, 0llr (/) - or simply 0llr - de notes the set of all distributions f D r F, where FE 0ll (/) . Obviously, �r is a vector subspace of § and C r a sub algebra of c. =

1 43

N ow we can define the product fg of a function f E c r and a distri bution g E mL , so as to satisfy the two conditions : r i) If g E mL , then fg is the product of the function f by the mea sure g in previous sense. ii ) If Dk f E C r and g E 011r , then Then if f E c r and g = D r G with G E 011 , the product fg is uniquely defined by the following formula (cf. chapter IV, 4. 1 . 1 . ) :

where

Il r - k ll = (r1 - k 1 ) + . . . + ( rn - kn ) ·

n n For example, for f E c r (IR ) and a E IR :

As in the case of one variable, it is easily proved that 0ll r becomes a module on the algebra C r . Besides, we have several different possibilities of extending this concept of product, as in the case of one variable, and even new pos sibilities. For example: Let p be, not a system of integers, but an integer � O . Then we shall denote by CP(/ ) or simply CP the set of all functions f having continuous derivatives f (k) on /, in ordinary sense, of all orders sp, that is, such that I I k ll = l k 1 1 + . . . + I kn l sp. On the other hand, we shall denote by 0R//) or simply by 0llp the set of all distributions g on /, which can be expressed as

sums

� gv of a finite (arbitrary) m

numb er of distributions gv ' belonging to mLr with r = (r1 , I r1 1 + . . . + I rn l s p.

•

rn ) and

144

Now, if we require the distributive law to be maintained, it can be shown that, if fECP and g E01tp ' the product fg is uniquely de fined by:

fg=

::? m

f g,

where fgv' is given by the previous general formula. Thus 0ltp be comes a module on the algebra Cp. Observe that, in particular, the space � of all distributions is a

module over Coo, the space of infinitely differentiable functions (on J). Another new possibility arises from the concept of "tensor product". Let J and J be two intervals respectively in IRm and IRn spaces (m, n > 0). If f and g are two continuous functions on J and J respectively, then the expression f(x)g(y)= f(x l , ···, X )g(Yl'···' Y ) n m + m defines obviously a continuous function on the interval J x JCIR n. Let now f and g be two distributions on J and J respectively,

f= DrF and g=DsG, with F EC(J ), GEC(J). Then it is readily seen that the expression 7.7.1.

DrEBs [F(x) G(y)]

denotes a distribution on J x J, uniquely determined by f and g. That being so 7.7.2. DEFINITION. The distribution 7.7.1. will be called the tensor product (or direct product) of

f by g and denoted by f®g

or by f(x)g(y). It is readily seen that this tensor product is bilinear and associa tive, but, of course, not commutative. Furthermore, it can obviously be extended to any finite system, of distributions. For example: H[m]®H[n] =H[m+n], 8[m]® 8[n] = 8[m+n],

8[3]= 8® 8® 8, etc. In a less rigorous, but more convenient notation, we may write also, for example (cf. 1.5.): 8(x, y, z) = 8(x)8(y)8(z), Dt8(x, t) = 8(x)8'(t), etc.

145

Many concrete situations lead to considering tensor products of distributions, as we have already seen in 1.5. For example, let f (x, y) be a locally summable functions on IR2, then f(x, y)8(z) will be a distribution whose carrier is contained in the x, y-plane. This may be the case of an electric charge distribution of suiface density f (x, y) on this plane. Analogously f(x, y)8'(t) may represent an electric doublet on the x, y-plane, and so forth. Similar situations may arise relating to curves, surfaces or, more generally, manifolds in IRn-spaces.

7.S. Change of variables. Concrete examples; 8-distributions of a hypersurface

Let a be a distribution on an open set Q in IRnand r a system of n

integers rk > O. Then the symbol aDr will denote the operator defined

by the formula

for all distributions f on Q such that a is multipliable by Drf. In par ticular, if aECr:tJ(Q), the domain of aDr will be �(Q). By a linear differential operator of finite order we shall under stand any operator A which can be represented as the

sum

of a finite

number of operators of the form aDr; then, the order of A is the greatest value of

Ilr 11 occurring actually

in all terms of the sum.

That being so, let us consider any two integers

rn,

n > 1 and a

mapping h of an open set Q* C IRninto an open set QC IR m. Then h is defined by a system of

reai-valued functions h l' . . , hm on Q*:

which may be written x =h(t).

146

...

Let f be now a complex-valued function on Q and suppose fEC1(Q), hiECl(Q*) for i=l, , m. Then

or else, putting hi)

7.8.1.

ah.

= _I

atj

Dt/fah)=

� hjj(Dxifah),

l =

j=l, ..., n.

a) Let us consider at first the case when m = n, and suppose that the

Jacobean ofh with respect to t (i.e. the determinant Ihijl) is different from zero on Q*:

(

)

h 1 .. . hn ;z! 0 for all . t 1 . tn

t E Q* .

Then 7.8.1. can be solved with respect to the functions Dx. fah: 1

�

(Dx,f) oh= }

a,p./f oh), i=l, ..., n

where ajjE Cl(O*) for i,j=l, . . , nand [aij ] is the inverse of the ma trix [hi ]. This result may be expressed by writing : j 7.8.2.

From now on the change of variables for distributions of ables may be defined essentially as in the case

vari

=1.

7.8.3. DEFINITION. If f=DrF, with FEC(Q) and r=(rl'0 . . ' rn) and if aijE cr(Q*) for all i, j then

1 47

foh

D;(F o h ) E Cr (Q* )

with

Uniqueness and other properties of the composition f 0 h may be proved as in the case of one variable. b) Consider now the case m < n and suppose that the characteristic of the matrix [ hij J is equal to m for all t E Q* . Then, for every t OE Q* we could solve 7 . 8 . 1 . with respect to (Dx f ) o h in some neigh

borhood of t o. B ut, in order to obtain a global solution (on Q), it is

convenient to "normalize" the system 7 . 8 . 1 . , i .e. to consider the " normal" system deduced from the first: 7.8.4.

here

� hvj D / f o h) = � hV i ( Dxi f o h) , v = l , . . . , m t ]=1 1=1 m

hVi =

� hvj hij '

]=1

v=I, . . ., m .

Then the determinant I hV i I is different from zero for all t E Q* and the system 7 . 8 .4. can be solved with respect to Dx o f 0 h , for i 1 , . , m I

(DX i f ) o h =

� aij D,/ f o h)

]=1

or in short n

DX i =

� aij Dtj ,

]=1

i=l, . ' " m ,

where the coefficients aij are C l functions on Q* uniquely deter

mined by the given functions hi '

1 48

From now on the change of variables for distributions may be defined as in the previous case_ Suppose in particular m = l _ Let us consider the change of variables defined by a function u = h (xl , , xn ) mapping an open set Q * in IR n into Q in IR (now the new variables are X I ' _ _ _ , x instead of n tl ' - , t ) - Assume h E C I (Q * ) and _

Then, every function f ( u ) of the real variable u, such that f E C I (Q), is transformed into a function f(h (x I , , xn )) = (f o h )(x) such that _

Dx . ( f o h) = h :. ( f ' o h ) , j = l ,

7.8.5.

where

f ' = Duf- From this follows : � h : . Dx . ( f o h ) = (f ' o h ) � J J � � = l = l n

(h: . Y ; J

hence putting

a. = J

( h ' ) 2 + _ _ _ + (h' ) 2 Xl

, J' = 1 , - - - , n ,

we obtain

that is 7.8.6.

Du = a D + - - - + an D I

Observe now that, for each u Eh(Q * ), the equation h (x) = u represents a hypersurface L'u in IR n + l , and that al , _ _ _ , an are the components of the vector

1 49

I grad h i

. wIth

I grad h I

g rad h

which is normal to .Eu at each point x. So 7 . 8 .6. can be written simply 7.8.7.

Du =

a I grad h i an 1

---

where a/an denotes normal derivation with respect to the hypersur face .E ' i.e. the differentiation along the unitary vector n (more pre u cisely, along the vector field n). c) Suppose finally m > n. Then h maps Q* onto a manifold V of di mension s n contained in Q and, given a distribution f on Q, the composition f 0 h exists, if and only if there exists the restriction fv of f to V, as well as fv 0 h ; then f 0 h = fv 0 h . We shall speak later about this new concept of restriction. Consider the distribution 8 on /R and a C l mapping f of an open set Q* in /R n into /R such that I grad f l ;I! 0 on Q*. Then it is easily seen that 8 0 f exists and is given by

Examples :

H ' (f(x)) =

H(f(x )) I I grad f an -

a/ an denotes the derivation along the vector field n =l grad f l - 1 grad f . Observe that H(f(x)) equals 0 or 1 according as f (x) < O or f (x) > 0; so, denoting by .E the hypersurface f (x) = 0, we could say that H(f(x )) equals 0 on the left of .E and 1 on the right of .E (.E is supposed to be oriented by means of the normal n) . On the other where

� u;y n

hand, since

0 on Q*, there exists, for every xO EI, a

bounded open interval I in /R n , containing xO E .E, such that the equa tion f (x) = 0 can be solved in I with respect to one of the variables

1 50

and Now we have (supposing, as we can, a

I grad f l -D ' f an Xl

Let l= [a l ,

b I ] x . . . x [an ' bJ

X2 x" tj/(x ) = . . . L L a

f� l > 0 on ] )

1 k(rp�y Dx , • n

be an interval such that l e ] and put

k [ rp�. <';' , . . . , S'n >F H(j(xl ' S'2 ' . . . , S'n ))dS', . . . dS'n · n

It is readily seen that, in the neighborhood ] of x o , we have a

- H (f(x)) = D tj/(x ) an

where D = Dx I Dx 2 • • • DX . Applying this argument to each x o E I, it foln lows that -

7.8.8.

- H(f(x)) is the measure on Q* assigning to each bounded

interval l, such that l e Q* , the Harea " of I n l. -

We shall denote by 8� this measure (8-distribution of the orient ed hypersurface I) and by HI the function H(f(x)) (Heavisidefunc tion of I). Hence we have 7.8.9.

It can also be shown that, if f E e 2 , then

( a )2

AH�= u� = an HI (where A = dlV grad) . 5:'

151

These considerations, except the preceding result, can be extend ed to the case where I is any oriented piecewise smooth manifold

all the classical vector and tensor analysis can be rebuilt for distribu tions with proofs which are in general more natural and more simple than the classical ones. in IR n. It should be observed that, by considerations of such type,

As an example of the 8-distributions of a hypersurface, consider the distribution 8( lx l -p), wherexEIR3, Ixl =

YXi

i +x �

and p>O.

It is easily seen that this distribution is the 8 of the sphere I x l = p. A concrete example may be a distribution of electric charge with sur face density

1/4n on the sphere,

supposed to be a conductor in elec

trostatic equilibrium. Then the charge distribution 8( l x l -p) creates

the electric field

defined by

=p2

Ixl

Hc l x l

p) which derives

from the electric potential

-- for Ix l sp 4n p

4n x II

for Ixl >p·

It is easily seen then that �V = - 4n8(l x l- p). Consider now the distribution and

8(X2_V2 t2),

3 with xEIR , t real � 0

v constant >0. We have now O(X2-V2 t2)

where

41xl

[oClxl-vt) + oclxl + vt)],

8(lxl-vt)=�H(lxl-vt)

defined only for

for each t�O. This distribution is

t �O and its carrier is just the wave cone x2- v2 t2

1 52

In turn, the carrier of 8(X 2 - v 2 t 2 + p), with p > O, is an hyperboloid of two leaves in the space IR; x IRt ' etc.

minus the origin.

7.9. Topological vector space of distributions of several variables

a compact interval in IR n , p = (p, p, . . . , p) E /N; , and consider the vector space C(/) provided with the usual norm II f ll = max lf(x) l . If we denote by Cp (/) the vector space of all dis Let

I be

tributions

of the form

f = DPF = Df ' " D: F,

where

FE C(/),

it is

natural to consider C/I ) provided with the semi-norm corresponding to the ball DP U, where U = { f : f E C(/) Il f ll = I } . Now the kernel ,

of DP is the set

� Gk. P n

of all pseudo-polynomials of degree

« p, . . . , p), and it can be proved as in the case of n = l , by means of Lagrange ' s interpolation formula, that Gp is closed in C(/). Hence C/I) is a normed space. On the other hand: 00

lW(l) =

U Cp (l) ,

p=o

and it is easily shown, as in the case n = 1 , that the injection

Cp Cp +1 is compact for all p . Hence lW(I) , considered as the inductive limit of the normed spaces C/I), is a (LN* ) space. ---+

In particular, the convergence of sequences can be defined direc tly as follows : 7.9. 1. A

sequence of distributions fk E lW(/) converges to g E lW(/) , if and only if there exists an integer p 0, a sequence of functions Fk E C(l), and a function G E C(I) such that: fk = DP� for all k, g = DpG and I I Fk- G 11 ---+ 0 . It is still readily seen that convergence in the mean on I implies convergence in distributional sense .

153

Let now

be an open set in

provided with the topology of the projective limit of spaces

§(I) ,

where

§ (D) is the (LN*)

The vector space

I is any compact interval in D,

by means of the

linear mappings PI. This means that a filter a converges to 0 in

§f(D),

if and only if PIO'converges to 0 in

interval

I CD.

§(I)

for every compact

In any of these distributional spaces, the following property is obviously true:

Examples:

where

Put

<\=H; (cf. 6.8.1.). Then it can be seen, as in the case n=l, that lim Dr 8k[n]

k-+oo

Dr 8[n]

Vr EIN;.

2-Considering primitives of the measures

k= 1,

2, .

( �),

5(x) and 5 lxl-

for

it is easily seen that

(

)

= 8(x). k-+oo 8 lxl-� k

Let

(k= 1,

U (x) k

2, ... ,

be equal to

�(

resp.

�I I )

for

Ixls

xEIR3). Then

f JI

[U/X) I xll ] __

dx�O

�(

resp.

Ixl>

�)

1 54 for every bounded interval Hence

so that

ďż˝

in distributional sense .

and, therefore (cf. 2) :

A - = - 4no(x) Ix l

IR 3 .