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.

C HA PT E R I I I S P ECIAL TY P ES O F D I STRI B UTI O N S

3.1. Locally summable functions A function f is said to be locally surnmable on an open set Q in IR iff f is summable on every compact interval contained in Q. For example, the function x - 1 (x -I )- 1 I3 is locally surnmable on the interval ]0, + 00 [ or even on the set Q of all points x ďż˝ 0 ; but it is not 'locally summable on IR, for it is not surnmable on any interval containing O. On the contrary, log I x I is locally summable (thought not summable)

on lR. Instead of an open set, we may consider more generally any set which results from an open set by adding to it one or more of its boundary points. If f is a function locally surnmable on an interval I, we shall call a primitive of f any function F of the form :

VxE I where c is an arbitrary point of I and K an arbitrary complex number. From the properties of the Lebesgue integral, the following theoÂ rem is deduced:

If F is a primitive of a locally summable function f on I, then F is continuous on I and has a derivative a. e. (almost everywhere) in the ordinary sense such that: F ' (x) f(x), almost everywhere on I. 3.1.1.

It should be observed that the converse of theorem 3 . 1 . 1 . is not true. There are examples of continuous functions which have a deri vative a.e. in the ordinary sense on an interval I and which are primi tives of no locally summable functions on I. The functions which are primitives of locally summable functions are said to be absolutely continuous. A direct characterization of such functions was given by Vitali . Theorem 3 . 1 . 1 . suggests calling the function f a derivative of its primitive F. B ut then F would have, of course, infinitely many deri vatives (of 1 st order) .

Two locally summable functions f l and f2 on I have the same primitive F if and only if fl (x) = f/x) almost everywhere on I. 3.1.2.

In such a case the functions are said to be equivalent and it is written f l "-' f2 (on I). It is readily seen that this is actually an equivalence relation. The class of all functions which are equivalent, in this sense, to a given function f, locally summable on I, will be denoted by [ f] . That being so, if F is any primitive of f, it will be natural to call the class [ f ] the derivative of F and to write :

DF = [ f] . So the derivative of F is uniquely defined as one class of func tions instead of a single function. On the other hand it is natural to define the sum of two such classes [ f ] and [ g ] and the product of [ f ] by a complex number a according to the formulas :

[ f ] + [ g ] = [ f +g ] , a [ f ] = [ af ]

It is readily seen that with these definitions the set of all such classes [ f ] becomes a complex vector space. Finally, if f is a con tinuous function on I, it is natural to identify [ f ] with f, so that C(/ ) becomes a subspace of the preceding vector space.

However, it will be troublesome to have to speak throughout of cl asses of functions. To avoid this we shall use a simple device. Let f be any locally summable function on I and let us place :

d dx

f (x) = �

Xf(�) d� L

(with

cE / ).

Then f is defined only at the points x of I for which the preceding derivative exists in the ordinary sense. On the other hand, we have, of course : '"'-'

f ,..., f and f = f . We shall call the operation f � f , standardization, and the functions f such that f = f, standard functions . For example, the Heaviside function : '"'-'

�

H(x)

{ I,

for x � °

0, for

x<O

is not a standard function . By standardization of standardized Heaviside function:

'"'-' { 1 ,

H(x) =

H we obtain the

for x > °

0, for

x<O

which is not defined at x = O. In particular, all continuous functions are standard functions . It is natural to replace any equivalence class [ f ] by the standard function f belonging to this class. From now on, when we speak of locally summable functions, it will be understood that they are standard functions . We shall denote by L (/) , or simply L , the set of all (standard) locally summable functions on I. According to the preceding remarks, L is a complex vector space and C is a linear subspace of L . '"'-'

3.2. Locally summable functions as distributions Let f be any (standard) locally summable function on I and let us denote by F one primitive of f:

F(x) = K +

ff(�) d� , (a E l,

KE C) .

Since F is a continuous function on J, there exists a distribution which is the derivative of F. We shall denote by F ' ( = f ) the deriva tive of F in the functional sense and by DF the derivative of F in the distributional sense. Subsequently, we shall identify DF with F ' , this identification being based on the following theorem: . o

By assigning to each function f EL (J) the distribution f* = DF where F is a primitive of f, there is defined a oneto-one linear mapping of L (J) into !iJ(J) such that: (i) if f is continous on J, then f = f *; (ii) if f if absolutely continuous on J, then Df * corresponds to f ' .

3.2. 1. THEOREM.

PROOF. First of all, it must be observed that the distribution DF assigned to each function f E L does not depend on the choice of the primitive F of f. In fact, if G is another primitive of f, then F - G is a constant function, so that DF = DG . Now consider two functions f, g E L ; we have to prove that if, to j and g corresponds the same distribution, then f = g . Let F, G be primitives of f, g respectively and suppose DF = DG . Then F - G E CJP1 is a constant on J, and therefore, F and G have the same derivatives in the functional sense (as a standard function) , that is o

f = g·

For the remaining parts of the theorem, the proof is quite trivial . • This theorem shows that we can identify every distribution DF, where F is an absolutely continuous function, with the locally sum mable function j, which is the derivative of F in the functional sense 3 . 1 . 1 . We then write:

DF = F ' = f · Since every locally summable function f is a distribution, f will have derivatives of all orders (in distributions sense) . Conversely,

ev e ry distribu tion may be expressed in the form D nf where n E fNo and fEL. For simplicity of notation, even if a locally summable func tio n f is !!ot a standardfunction, we shall denote by D nf the dis trib ution D n f . For example, the 8 distribution may be defined as the derivative of o

the (standardized) Heaviside function, and we may write in general :

3.3. Functions which are not distributions and pseudofunctions Consider for example the function

�

Since this function is

continuous on the set of all points x � 0, it is a distribution on this open set. But it is not a locally summable function on fR, and as we shall next see, it may not be interpreted as a distribution on fR. This function is the derivative in the ordinary sense (not defined at 0) of all functions

f of the form:

{ log l x l f(x) =

C l ' for x > O log l x l + c2 ' for x < O +

or shortly :

f(x) = log l x l + aH(x) + b with a = C I - C2 '

b = c2 ' where C l and c2 are arbitrary complex numbers .

� , any function f of this form is locally summable x on fR, hence it is a distribution on fR, whose derivative is :

Now, contrary to

Df = D log l x l + a8 where the symbol

D log I x I denotes the derivative of the locally sum mable function log I x I on fR, in distributions sense.

Thus, there exist infinitely many distinct distributions on IR which are the derivatives of the functions f . But for any f , we have:

-l- , on the set 0/ all x ;z! O. x

Therefore the function

�

is a distribution on this set, but may

not be interpreted as a distribution on The distribution and denoted by

IR.

D log I x I on IR is called the finite part of

-l x

Pi -l- . But Pi � is not a /unction, as � is not a x x x

distribution. (4) More generally the

finite part of -l-n is defined to be the disx

tribution:

(_ l ) n - l n "D log l x l , n 1 , 2 . . . Pf �n = (n - l )! x 1

This belongs to an important class of distributions which are called pseudofunctions by L. Schwartz. We shall further see other examples of pseudofunctions.

3.4. Measures and functions of bounded variation We have already discussed the concept of measure in chapter I. It is not dificult to see that an equivalent definition is the following : A measure ji is defined on IR iff to every bounded interval J in IR is assigned a complex number, called the ji-measure of J and denoted by ji (J) or jiJ, in such a way that: (4) The expression "finite part" is connected with the concept of finite part of certain diver gent integral which L. Schwartz used for defining this distribution . Note that th ere is no special reason to identify the function

Pf l lx + a o,

with

a � O.

l /x

with

Pf llx

rather than with a distribution

M l . If J is expressed as the union of two disj oint intervals th en: M2. If J is the union of the intervals

J.l (J)

J1 and J2 ,

J1 C J2 C . . , then : .

J.l (In) nHm -+ oo

M3. For each bounded interval J, there exists a positive number m (J) su ch that for every partition of J into a finite number of intervals In , we have : J1 , J2 ' • • • ,

� 1 J.l (Jk ) 1 n

s m (J) .

Observe that the variable interval J which we are now consider ing may be a degenerate interval [a, a] .

We can define analogously the concept of measure on any open set A C IR or even on a more extensive class. But then we must con sider bounded intervals 1, such that J C A. 3.4.1. DEFINITION. If J.l is a measure on the interval I on IR, a pri mitive of f.1 will be any function F defined on I by putting :

F(x) where

+ f.1

[c, x] , if x � c k - f.1 ] x, c[ , if x < c

c is an arbitrary point of I and k an arbitrary complex number.

From this definition and M 1 follows :

3.4.2.

F(b) - F(a) = J.l ] a, b] , whenever a < b.

On the other hand, from Ml and

3 .4.2.

[c,

M2 we have :

F (a) - F (a ) = J.l [a, a] , for all a E I. -

To see this we consider the case c < a and it suffices to express a [ as the union of a sequence of intervals [c, xn ] such that

c < X 1 < . . . < X n < . . . < a and x n � a . Then by M2 : ,u [c, a [ = Hm ,u [c, x n ]

lim

Xn -+ a

F(x n ) - k = F(a -) -k . Formula 3 .4 . 3 . is

analogously proved in the case a :s c. Finally, from 3 .4.2. and 3 .4. 3 . follows :

F (b)

�

F (a ) ,u [a, b] -

3.4.4.

F(b -) - F (a) = ,u] a, b [ for every pair of points

a, b in I such that a < b. Consequently :

If,u is a measure on I and if F is any primitive of,u, then ,u is uniquely determined by according to formulas 3 .4.2. , 3 .4 . 3 . and

3.4.5. 3 .4.4.

Now we need a characterization of the functions which are the primitives of the measures on I. Let F be such a function. From M l and M2 it can be easily deduced (as in 3 .4 . 3 . ) that F(a) = F(a +) for any a E l ; i.e. F is continuous on the right at every point of I. In addition, M3 implies that to each compact interval J = [a, b] , there exists a number m (J) such that, for every partition of J by means of points a = x < x1 < . . . < xn = b, we have :

�lI F (Xk)- F (Xk-l ) 1 sm(J). F n

But this means that is a function of locally bounded variation on I. Conversely, it is easily seen that these two properties are suffi cient to characterize primitives of measures . Thus :

F on I to be a primitive of a measure ,u on I, is that F be of locally bounded vari 3.4.6. A necessary and sufficient condition for a function

ation on I and continuous on the right at every point of I. Moreover two such functions F] and F2 are primitives of the same measure ,u if and only if F] - is constant on I.

We shall denote by 011 (/ ) , or simply 011 , the set of all measures on I . Th e sum f.l + v of two measures and the product a f.l of a co mp lex number a by f.l are defined by the formulas : (f.l + v)(J) = f.l (J) + v (J), (af.l)(J) = a(f.lJ). -

for each bounded interval J such that J C l. Then 011 (/) becomes a complex vector space.

3.5. Measures as distributions. Order of a distribution Let 1 be any (non-degenerate) interval on fR . Observe that to each -

f E L (/ ) and each bounded interval J such that J C I, there corresponds the number JJ f and this correspondence J JJ f is a function

ďż˝

measure, f.lf ' whose primitives are just the primitives of the function o

f. Thus, every function f E L (/) determines one measure f.lf E 0R (/), and if f.lf = f.l g , then f = g, since f and g have the same primitives . Besides it obvious that f.l ( f+ g ) = f.l f + f.l g and f.l a f = af.l f for any a E C. Thus it is natural to identify each f E L (/) with f.lf so ihat L (/) beo

comes a vector subspace of 011 (/) .

Now, remember that every function of locally bounded variation

on 1 is Riemann integrable on each compact subinterval J C I; hence

locally summable on I. Then taking 3 .4 . 6 . into account, it is easily shown that:

3.5.1. By assigning

to each measure f.l on 1 the distribution DF, where F is any primitive of f.l, there is defined a one-to-one linear mapping of 011 (/) into !2J(/) such that if f.l is a locally summable function f on I, then f.l corresponds to DF = f. The proof is quite similar to the one of 3 . 2 . 1 . It should however

be observed that the primitive F of a measure is not in general a stanÂ dard function; but since

ji is

defined

only at the continuity points of

F, with the same values, F -+ F , is one-to-one. ,....,..,

it is readily seen that the correspondence

Recording 3 . 5 . 1 . , it is natural to identify every measure J1 on I with the distribution DF, where F is a primitive of /1 and to write /1 = DF. S o mL (/) becomes a vector subspace of !!lJ(/) and more pre cisely of C2 (I) :

C C L C mL C C2 c !!lJ. o

For every integer n > 0, we shall denote by mL /I) the set of all distributions f such that f =D n/1 with /1 E mL(/). It is obvious that 00

!!lJ = U mL n · 1

3.5.2. DEFINITION. Given a distribution f on I the least n such that f E mL n is called the order of f. For example 0, which is a distribution of rank 2 (see order 0 (i.e. is a measure) . In general o(n) is of order n .

2 . 3 . ) is of

3.6. Product of a continuous function by a measure and the Stieltjes integral Consider f E C(l) and /1 E mL(l). Let J be any bounded interval such that iC I and let P be any partition of J into a finite number of (mutually disj oint) intervals Jp , , , , In • Denote by N(P ) the greatest lenght of the intervals J1 • • • , In • Let x k be an arbitrary point in Jk and put: '

S/J ) =

k f(xk ) /1 (Jk ) n

•

Then it is a classical result that Sp(J) tends to a finite limit, S(J), as N(P) -+ 0; that is, to every 8 > 0, corresponds an £ > 0, such that:

N(P ) < e implies I S(J) - S/J) I < o.

Moreover, it can be shown that the correspondence

--?

S (J) is a

measure on I. This measure is called the product of f by J1 and denoted by Thus, by definition:

fJ1 .

( fJ1 ) (J) = S (J) . Previously (1. 3 .1 . ) we have adopted the convention that the J1-measure of an interval J should be called the integral of J1 on J and

denoted by f, J1 . Thus:

S (J) = ( fIl ) (J) =

L fll ·

Remember that fJ1 is usually called the Stieltjes integral of f with respect to J1 . The notation fdJ1 is commonly used instead of fJ1 , but that notation in the theory of distributions may induce in error. If F is a primitive of J1 , it is quite natural to denote the integral

of f with respect to J1 by :

and since

L f(x) dF(x) ,

J1 = F ' (in the distribution sense) we could also write:

L f(x) dF(x) =L f(x) F ' (x) dx =L fll ·

But then we should have:

L f(x) dll (x) =L fll ' ,

and this is the integral of f with respect to the distribution J1 ' that we shall define later on. As an example, let us calculate f 8, where f E C(/R). If we con sider a bounded interval J such that ° E J and a partition P of J, into intervals J1 , In , then one and only one of these will contain 0, say � . Thus for every choice of xk E Jk : '

•

S/J) = Therefore: lim

N(P)

that is (f8)(J) O $. J. Hence :

......

� f(xk ) 8(Jk ) = f(x) .

S/J) = lim f(x) = f(O) ; Xj --+

f (O) , if 0 E J. It is easily seen that (f8)(J) = 0, if

f8 = f(0) 8 .

3.6. 1 .

More generally : 3.7.

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

Derivatives of peace-wise smooth functions We begin with the following proposition :

be afunction on an interval 1 = ]a, b[. Sup pose that f is absolutely continuous on two intervals ]a, c[ and ]c, b[ and tends to finite limits as x � c- and as x � c+. Then, df!noting by f' the derivative of f in the ordinary sense (not necessarily defined at c) and putting s = f(c +) - f(c-), we have: 3.7. 1 .

THEOREM. Let f

Df = [ f ' ] + s8(x- c) . PROOF : Since f is absolutely continuous on ]a, c[ and ]c, b[ and has finite limits f(c-) and f(c+) i t i s easily seen that [ f ' ] i s locally summable on I. Hence if we put:

g (x) =f(c -) + the function

f [ f ' l @ dq ,

g will be absolutely continuous on I and:

g = f , g (x) = Hence

\;/x E I

{f(X)

for

a <x < c . f (x) s, for c < x < b ,

f(x) = g(x) + sH(x- c) and Df = [ f ' ] + s 8 (x- c) . •

Consider now a similar situation concerning a finite number of p oints C l ' . . , c in I and put sk = f (c; ) - f (c; ). Then : .

Supp ose more generally that: i)

f has a derivative of order n > O . in the ordinary sense except for a set of isolated points, ck (k 0, ± 1 , ± 2 , . . ) ; ii) f (n- I ) is absolutely continuous on each subinterval of I not con taining any point ck ; iii) for every k 0, ± 1 , . . and every j 0, . . . , n - 1 there exists finite limits f(j ) (c; ) and f ( j ) (c; ). =

Then it i s easily deduced from 3 . 7 . 1 . :

D n! [ j<n) ] + =

3.7.2.

� � sln- l ) -j § ( j)(x- c, ) , + 00 n- l

where sf f(j ) (c;) - f (j ) (c;). The last term in 3 . 7 . 2 . (involving even tually a sum of infinitely many distributions) denotes the distribution whose restriction to each of the compact intervals J C I is the sum of =

the distributions

�J st1 )-i § (j )(x- c, ) n- l

where

c, E l (in finite number) .

For example, it is easily seen that:

D2 1 x l = 20 D 3 1 x 2 -1 1 = 4 ( 0 1 - 0 _1 +0�_1 + 0� 1 ) ) () ( ) 4 D 2 (x 4/3 + I x l ) = _ X-2/3 + 2 0 . 9

Remarks about notation: In the preceding considerations when

it has been necessary to distinguish the derivatives of a function f in the ordinary sense from its derivatives in the distributional sense, we have used the notation fT in the first case and Df in the second. B ut n whenever no confusion is possible we shall consider f (n) and D f as perfectly equivalent.