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 H A PTER 11 D ISTRIBUTIONS O F ONE VARI ABLE: FUN DAME NTAL CO N C E PTS.

2.1. Terminology and notation

We shall denote by IR the field of real numbers and by C the field of complex numbers. If a, bE IR, with a < b , then [a, b], la, b], [a, br, la, b[

denote the intervals with extreme points a, b defined respectively by the conditions a s; x s; b , a < x s; b, a s; x < b, a < x < b

They are respectively closed, closed on the right, closed on the left, and open. All of them are bounded. An interval (or more generally any set of points of IR ) is said to be compact if it is closed and bounded. By extension of language, any set that reduces to a single point a E IR is called a degenerate interval that is the interval [a, a]. However, by an interval we shall always mean a non-degenerate interval, unless the contrary is exÂ plicity stated. On the other hand, the symbols [a, + 00 [ ' la, + 00[ ' ] 00 , a] , ] 00 , a[ denote the unbounded intervals defined respectively by x ďż˝ a, x > a, x s; a , x < a. -

The first two are bounded on the left and the last two are bound ed on the right. The first is closed, the second is open, etc. Finally the unbounded interval] - 00, + 00 [ is the set IR itself; it is both open and closed (in IR). Let I be an interval in IR. We denote by C(/) - or by C if there is no danger of mistake - the set of all complex -valued functions f (x) of the real variable x which are defined and continuous on I. More generally, for any integer n > 0, we denote by cn(/) - or simply Cn the subset of C (I) formed by those functions f, which have a derivative, f(n), of order n continuous on I; in particular Co = C. The elements of cn(l) are said to be C n functions on I. The term "function" will mean "complex - valued function" wherever the range is not specified. Instead of f(n) we shall often write D nf . This notation puts in evidence the derivation operator, D, which assigns to each function f E C 1 the function Df = ff E C . So D n is the nth power of D. On the other hand, the symbol � denotes an integration operator defined by the formula:

;sf (x) =

ff(�) d�,

for all

fEe

where c is an arbitrary fixed point in I. Then � is a mapping of the set C into C 1 C C such that

D�f

for all

fEC.

This � is a right-inverse of D (but not a left-inverse) . More ge nerally:

2.1.1.

Dn�nf = f,

for all fEC,

0,1, . . .

2.2. Axiomatic introduction of distributions. (1) Let I be any interval in IR; the system of all distributions on I can be described by the following system of fundamental properties: (1) The word

"distributions" is used here with a meaning equivalent to that of "distributions of

finite order" according to L. Schwartz. We shall further introduce the concept of "global distribution" which is equivalent to that of "distribution" in the sense of Schwartz.

AXIOM 1. Every function which is defined and continuous on I is a distribution on l. AXIOM 2. To every distribution f on I there corresp onds one and only one distribution on I, which is called the "derivative of f" and denoted by Df, in such a way that, if f is a C 1 junction, then Df is the derivative of f in the ordinary sense. DEFINITION: The derivative of order n of a distribution which is denoted by Dnf, is defined as follows :

DOf=f , D nf=D(D n-lf) ,

for n= 1 , 2, . . .

AXIOM 3. To every distribution f on I there exists at least one integer n � 0 and one function F, continuous on I, such that f D nF . =

AXIOM 4. If n is an integer, n � 0, and f, g are two continuous f£!,nctions on I, then we have Dnf = Dng if and only if f g is a p oly nomial junction of degree < n. We denote by INo the set of all integers n � 0 and by ClPn' for each n E INo, the set of all polynomial functions of degree < n (restricted to I). Our immediate purpose is to prove that the preceding axioms are: 1 0 consistent; i.e. there exists at least one structure satisfying the axioms (a model) . 2 0 categorical; i.e. two such models are necessarily isomorphic. This will imply that any statement about distributions on I which is not false is a consequence of the axioms, and eventually of some supplementary definitions that have been introduced in order to simplify the language. We shall begin with the proof of categoricalness because it leads to a natural proof of consistency. -

PROOF of categoricalness Suppose that there is a model M satisfying the axioms, i.e. a set of objects f, g, . . , and an operator D such that, if we call these obj ects the distributions on I and Df, Dg, . . . , the derivatives of f, g, . . , then the axioms are satisfied. The axioms 1 and 2 along with definition 1 imply that, for any couple -

(n, F), where n E1No and FE C, there exists one and only one dis tribution f=DnF (element of M).(2) Conversely, according to axiom 3, for any f E M there exists at least one couple ( n, F) with n E1No ' FE C, such that f=DnF. However there exists more than one couple (n, F) satisfying this condition. Let ( m, G) be any couple such that:

2.2.1. and let k be any integer such that k � m, n. By axioms 1 and 2,defi nition 1 and property 2.2.1.,we have:

hence D k (�k-n F)=D k (�k -m G )

and consequently by axiom 4:

2.2.2.

�k-n F _�k- m G E CJP . k

Conversely, axiom 4 shows that 2.2.2. implies 2.2.1. These two conditions are therefore equivalent. (For example, if m � n, we can choose k=m, so that condition 2.2.1. is satisfied by all functions G of the form G = �m-n F P, where PE CJPm). Now let us denote by [n, F] the class of all couples ( m, G) � atis fying 2.2.2.,i.e., such that DmG=DnF, and let us denote by C the set of classes [n, F], with arbitrary n EINo' FE C. Then the corres pondance:

[n, F] --:. DnF

is obviously a one-to-one mapping of C onto M such that: "-

2.2.3. (2) Remember that we

write C instead of C(l) for the sake of simplicity.

In particular, the correspondence:

[0, F] � F is a one-to-one mapping of a subset

define :

....

C* of C onto C. Therefore, if we

D [n, F]· [n + 1 , F]

2.2.4.

and we identify(3) each element

F [0, F]

[ 1 , �F]

[0, F] of C* with the function F itself

. . . , then C becomes a second model,

consistent with the axioms, isomorphic to M according to

by puting

2.2.4.

2.2.3. and ....

M satisfying the axioms is isomorphic to C and

therefore, any two models M and M' are isomorphic (remember that Thus any model

M) .•

the construction of of

....

based on

2.2.2. ,

is independent of the choice

PROOF of consistency - We have just seen .... that if there is a model M of the system of axioms then the set C, described above, the existence of any previous model M, that the set

exists too and is also a model. We shall now prove, without assuming

and gives us a model of the system of axioms. Let us consider the set and

INo x

....

C actually exists

C of all couples (n, F), where nE/No

FE C. Given two such couples (n, F) and (rn, G) we shall write:

(n, F) .... (rn, G ) if and only if there exists an integer

�k-n F - �k-m GE rzp k

2.2.5. (3) By identifying

k rn, n,

such that:

•

[0, F] with F, we mean in reality that the symbol "[0, F]" and its equiva lents "[1, �F]", [2, �2F], , will denote from now the function F, i � stead of the class of couples [0, F] equivalent to (0, F). Thus the meaning of the symbol C is also changed.

...

It is easily seen that the relation " ..... " just defined is reflexive and symmetrical. We now prove that it is transitive. Observe first that if there exists an integer k � m, n satisfying 2.2.5 . , so does any other integer r, such that r?! m, n. In fact we find that : ;sr-n F- ;sr-m GE rzp

by applying to both members of 2.2.5 . the operator D k-r or ;sr-k ac cording to whether k?! r or k < r. So suppose:

(n, F) ..... (m, G ) and (m, G ) ( p, H ). .....

Then, if we choose r � m, n, p, we have:

;sr-n F ;sr-m GE rzp , ;sr-m G- ;sr-p HE rzp ' r r -

and hence, by addition:

p ;sr-n F_ ;sr- HE rzp

that is (n, F) ..... ( p, H ).

So the relation ..... is an equivalence relation and, as such, it determines a partition of the set 1No x C off all couples (n, F) into equivalence classes. For each couple (n, F), we shall denote by rn, F] the class of �l couples which are equivalent to (n, F) and we shall denote by C the set of all of these classes (the "quotient" of 1No x C by ..... ) . The correspondence [0, F] � F being a one-to-one mapping of a subset C* of C onto C, we can identify each .Element [0, F] of C* with FE C. Now, let us call the elements of C distributions on I . So Axiom 1 is satisfied by C. Moreover, we shall call [n 1, F] the derivative of rn, F] and we shall write: .....

.....

D [n, F]

[n 1 , F] .

According to this definition, there is only one derivative for each [n, F] E C. Indeed, suppose [n, F] [m, G]; this means that .....

(n, F)

.....

(m, G ), i.e. �k-nF - �k-m G E �k for any k � m, n; hence �(k+ I)-(n + I) F �(k+ I)-(m+ I) G E � , k+1 _

that is [n 1 , F] [m + 1, G] which means that D [n, F ] =D [m, G ] . Moreover if f E Cl, then D [0, f] = [1, f] [0, f'] f ', since f - � f' E � 1 • So axiom 2 is satisfied. On the other hand we have: =

[n, F] = D [n - 1, F]

= . . . = Dn[0, F ] = DnF for every

....

[n, F] EC.

So axiom 3 is also satisfied.

n Finally, if D f Dng with f, g E C, then [n, f ] [n, g], that is �k-n f - �k-n gE � k , for any k � n. Choosing k n, we see that axiom 4 is also satisfied, as we have Dnf Dng if and only if f - g E � .• n Conclusion: We have just proved that the set C gives us a model of the preceding system of axioms . We could conceive many other such models, but this would have no essential interest since we have p!oved that such models are necessarily isomorphic to C. The model C itself, after having afforded a simple proof of consistency of axio matic system, will have no further interest. From now on, all that matters will be the rules of calculus of distributions: that is, the axioms 1 - 4 and the definitions that will =

....

be convenient to add to them, as well as the propositions implied by this system of axioms and definitions. In reasoning as well as in calculation, the distributions will be denoted by the notation "D nf" or by any other that be convenient. But it will no longer be necessary to think of a distribution as a class of couples (n, f). Observe that this situation is quite similar to the one connected with the successive extensions of the number concept.

2.3. Rank of a distribution and further conventions

For each integer n � ° we shall denote by C (I) or by C , when n n there will be no danger of mistake - the set of all distributions f on I of the form: -

f = DnF, where F E C(J).

Observe that Co = C C Cl C C2 C ... . On the other hand, we shall denote by �(J) or simply !?lJ - the set of all distributions on J. Thus !?lJ is the union of all the sets C/J): -

!?lJ(J)

n=O

C n(J)

(Co

C) ,

and accordingly we may use the alternative notation Coo for !?lJ.

2.3.1. We say that a distribution f is of rank n if and only if (iff) n is the least integer such that f E C n' For example, consider the Dirac 8-distribution which can be de fined as follows : 2 8 = D J, with J

2.3.2.

{Ox,,

for x < 0

for x � 0

So, 8E C2• Suppose there exist a continuous function F, such 2 that 8 =DF. Then DF=D J, that is J = �F+P, with PE ClP'2' Hence DJ = F+P'. But this is impossible as F+P' is continuous and J has no continuous derivative (on IR). Consequently the rank of 8 is 2. It follows from this that '8(n) is of rank n+2, for n = 1 , 2, . . . .

2.3.3. An interval J is said to be the domain of a distribution f iff f is a distribution on J ; i.e. f E !?lJ(J) . We also say that f is de fined on J. 2.4. Addition of distributions The sum f +g, of two distributions f, g, on the same interval J, is to be defined so as to guarantee the following properties :

Al. If f, g E C (l) , then f + g is the sum of thefunctions in the ordinary "

sen..se.

A2. If f, g E!ilJ(l), then f+gE!!»(J) and D( f+g) =Df+Dg.

Suppose:

According to the axioms (2.2) , it is possible to represent f and g as derivatives of the same order of two continuous functions ; indeed, taking r � m, n, we have:

Now, conditions Al and A2 imply

So, 2.4.3.

f + g=DnF+ Dm G=D r(�r-nF+�r- m G ).

In this way we assign to each couple (f, g) of distributions on /, at least one distribution on /, which is denoted by f+g. We shall next prove that there is only one distribution f + g , for each couple (f, g) ; i.e. , we shall prove the sum f+g does not dep�nd on the representation 2.4. 1 . of f and g. Indeed, consider another represen tation: f=DVlP, g=DJ.ltp, Then, taking p �

Choose now k;;:: r,

J.1 we get :

with

J.1EINo'

fP, tpE C.

p. Then :

Dr(F+G)=Dk (F*+ G *), with F*=�k-n F, G*=�k-m G. DP(& +Vf)=Dk (fP*+ tp*), with lP*=�k-vlP, tp*=�k-J.ltp .

But DkF*=Dk(�k-nF)=DnF=f and DkfP*=Dk(�k-VfP)=DVfP= f. Then Dk F*= DklP* and F* - fP* E � . k

D k G* = D k tp * and G * - tp * E CfPk; hence (F* + G *) - ( l/J * + tp*) E CfPk' which means : Analogously

Thus, we have proved that the sum f + g is uniquely defined for each couple ( f, g). Besides, it is obvious that conditions Al and A2 are actually satisfied by addition defined according to 2.4.3 . . Hence addition in �(J) can be defined either implicity by the properties Al and A2 or explicity by formula 2.4.3 . . Moreover, this operation is:

I. Associative: ( f + g) + h = f + ( g + h), Vf, g, h E !!2J.

11. Commutative: f + g = g + f, Vf, g E!!2J.

for any two distributions f, g on J, there exists one and only one distribution ; on J, such that f + ; = g.

Ill. Reversible:

To prove these properties, it is sufficient to represent f, g, h as derivatives of the same order of continuous functions and to apply the corresponding properties of addition in C. The preceding properties I, 11, III along with the existence and uniqueness of f + g in!!2J , for all f, g E!!2J , can be expressed shortly by saying: 2.4.4. !!2J

is a commutative group with respect to addition.

2.5. Multiplication by complex numbers

The product, af, of a complex number a by a distribution f is to be defined so as to guarantee the two following properties:

P1.·- If f E C(J), then af is the product of a by f in �he ordinary sense. P2. - If f E !iJ(J), then afE !!2J(J) and D (af ) a(Df ) . =

Suppose f D nF, with n E 1No explicit definition: =

F E C. Then PI and P2 imply the

Thus to each couple (a, f), where aEC and fEC(/), there is assigned at least one distribution on I, which is denoted by af. It is easily seen that the product af is unique for each couple (a, f); i.e. , does not depend on the representation off. Moreover, it is quit trivial to prove that if f, g E0(/) and a, f3EC, then:

}

I.a(f+g)=af+ ag (distributive laws) f )f=af 11. (a+f3 +f3 Ill. (af3)f = a(f3f) (associative law) IV.1 路f=f 路 .

We have seen (2.4.4.) that 0(/) is a module, i.e., a commutative group with respect to addition. As usual, this fact along with proper颅 ties I-IV, can be expressed by saying : 2.5.1.

0(/) is a vector space over C (or a complex vector space).

On the other hand, the conj unction of the properties D(f +g) = =Df+ Dg and D(af) =aDf is equivalent to the property :

D ( af+f3g) =aDf + f3Dg, Va, f3EC; f, gE0(/) and it may be expressed by saying : 2.5.2.

The operator D is a linear mapping of the space !iJ (/) into itself

We shall further be concerned with the more delicate problem of defining the product of two distributions . 2.6. Translation operators

If fEC (I) , h E IR, then rh f is the function defined as follows :

2.6. 1.

rh f (x) =f (x - h)

.---

/i� I I

Z-

'thf

I+h

The graph of rh f is that of f translated by an amplitude h. In par ticular, the domain of 'rh f is the interval J = 1+ h. If h ;z! 0, we have I J if and only if I = fR. Thus rh denotes a one-to-one mapping of C(/) onto C( J), which is natural to call a translation operator. Accordingly, we shall call 'rh f the h-translate of f. Taking 2.6.1. into account it is readily seen that rh (Df) = D( rh f) if

fE C l (/)

The extension of the operator rh to distributions is defined so as to generalize this property. So we set by definition:

It is obvious that this formula actually defines a one-to-one map ping rh of §J(/) onto §J(J ) whose inverse is 'r_h• Besides, it is easily seen that for any hE fR, this operation is linear and interchangeable with D, that is:

The distribution Th 8, which we shall also denote by Dirac distribution at the point h.

8(h) is

the

Remarks about notation. If f is a function and x a point of its domain, the symbol f (x) denotes the value that f assumes at this point. When the point x is not specified, we are dealing with a variable and the expression "function f (x)" is generally used instead of "function f ". Now, it must be remembered that this is an abuse of language which is certainly convenient in many situations, but which can lead to error in other cases, especially in functional analysis. In these cases it is advisable to adopt the convention consisting on writing the accent" over the variable which is then said to be an apparent or mute variable. So the symbols f, f(x), f(t), . .. , become equivalent. For example, the expression 3x2 +x is only a variable dependent on x; meanwhile the expression 3x2+x denote.s properly the function f defined by f(x) =3x2 +x, for all x EIR . These conventions can be extended to distributions . If f is a dis足 tribution on I and x a point of I, then the symbol f(x) has generally no meaning for there is in general no value of a distribution at a point, as we shall see. But it is often convenient to use the symbol f(x) for denoting the distribution f. Accordin gly, the distribution Thf

may be sug gestively denoted by f(x - h) . In particular, we may write 8(x - a) for Ta 8 and more generally

8( n)(x - a) instead of Ta 8(n). Frequently, we shall write f(x) instead of f(x) or f. It must be remembered however that this is an abuse of writing, which we can admit for the sake of simplicity, whenever no misunderstandin g is

possible.

2.7. Restrictions operators

If fEe (I), the restriction of f to an interval 1 C I is the function f* whose domain is 1 and such that:

f*(x)=f(x), for all x E l.

We denote by PJ f the function f* , which is the restriction of f to J It is obvious that the symbol PJ denotes a linear mapping of C (I) into C(J), interchangeable with D, that is PJ(Df ) =D (PJf), for all f E C [ (I). It is then natural to put by definition: .

Thus the operator PJ becomes a linear mapping of !?lJ(/) into !?lJ(J) such that: 2.7.2.

the restriction operator PJ may reduce the rank of a distribution. For example, the distribution sin x - 3 8 + 8'( x - 3) which is of rank 3 on IR (cf. 2 . 3) becomes of rank 2 by restriction to ] - 00, 3 [ and of rank 0 by restriction to ] -00, 0 [. Observe, that

Another property of the restriction operators , which is easily shown, is the following: 2.7.3.

If I, 1, K are three intervals such that I � J � K, then PKf =PK (PJf) , Vf E� (/) .

2.8. Collecting principle. Global distributions (or distributions in the sense of Schwartz)

Let Then:

I[ and 1 2 be any two intervals in IR (distinct or coincident) .

2.8.1 . DEFINITION. Two distributions, f E �(/[) and g E� (/ 2) are said to be equal on an interval Je/[ n 12 iff PJf =PJg. Then we write f=g on J. 2.8.2. LEMMA. Let 1[ , 12 , be two open intersecting intervals in IR and f p f 2' two distributions on I[ and 12 respectively, such that f [ = f 2 on I[ n 12 . Then there exists one and only one distribution f on the interval 11 U 12 such that f = f 1 on 11 and f = f 2 on 12 .

PROOF. Suppose f, = D n F, and f2= DnF2 with F]E C (ll ) and F2 E C ( I2). Then F] -F2 equals a polynomial P of degree < n on I] n 12, Therefore if we put F = F, on 1 1 and F= F2+P on 12 , we de fine a continuous function F on I) U 12 , and the distribution f = D n F satisfies the condition of the lemma. Conversely, a distribution g = DqG satisfying this condition coincides necessarily with f, as is readily seen .•

Let I . .., I n be n open intervals whose union is again an interval I, and let fl"'" fn be n distributions on /1 "", In respectively, satisfying the conditions fj = fk, on � n Ik, whenever � n Ik is not empty ( j, k = 1, 2, . . . , n). Then there exists one and only one distribution f on I such that f = fj on � for j = 1, . . ., n. 2.8.3. Collecting principle (1st form).

PROOF. We can suppose the intervals I I , ..., In ordered in such a way that if j < k, then the left extremity of Ij precedes the left extremity of Ik or, if these extremities are coincident, the right extremity of � precedes that of Ik• Then, the successive unions are again intervals, and we can achieve the I) U 12, (I) U 12) U 13, proof by repeated application of the lemma along with 2.7.3.• •

•

2.8.4. Remark. The conclusion is no longer true, if we consider an infinite system of distributions f l' f2 , • • • on intervals 11, 12"" instead of a finite system. For example, take : I = ] n, n [ and n fn =Pl [O(x) +o'(x-1) +···+On( -I)(x -n+ 1)] for n = 1, 2, . . . . Since n the rank of f is n+ 1 for each n, there is no bound for the ranks of n f l' f2' . .• , and, therefore, there is no distribution f on IR, such that f = fn on In' for every n . But we can extend the collecting principle in the following way: -

2.8.5. Collecting principle (2nd form). Let A be any set of objects. Suppose that to each aE A is assigned an open interval I a and a

distribution fa on I a' in such a way that:

(i) the union of all these intervals is again an interval I; (ii) whenever two of these intervals, l a and l p, intersect, then fa= fp on l a nIp ;

(iii) there exists an integer r such that the rank of fa is s r for every a E A , Then, there exists one and only one distribution f on I such that f=faonla for every a EA,

PROOF, The open interval I can be expressed as the union of a sequence of compact intervals KI C K2 C ' , . , Now, according to the Heine-Borel principle, there exists for each n a finite system of open intervals 1nl, ... , i:n, belonging to the given system (la) and covering K that is such that: n

Besides , these intervals may be chosen so that in C in+l' for any n. Then I is again the union of the increasing sequence of intervals in . Let g: be the distribution of the system (fa)' assigned to i: , for every n=l , 2, . . . , and k= 1 , ..., Pn' According to 2.8 . 3 . , there is one and only one distribution g n on i n such that g n= &kon �kfor any n and k . On the other hand, by the hypothesis (iii) , there exists neces sarily for each n a function Gn E C(in), such that g n=DYGn . Since g n +l= gn on in for every n, it is easily seen that we can choose the functions Gn so that Gn +l = Gn on in for each n . But then it is obvious that there exists one and only one function G E C(l) such that G Gn on in for n = 1, . . . . Consequently, if there exists a distribution f on I such that f= fa on la for every a E A , then necessarily f= gn on in for n= 1 , 2, . and therefore f=DYG. Conversely, the distribution g= DYG satisfies the condition g= fa on I a for every a E A. In fact, if we denote by g a the restriction of g to I a and if we put ink = �k n la' whenever this intersection is not empty, then la is the union of all �k and we have g a= fa on each interval Hence, by the uniqueness property just proved, g a= fa; i.e. g = fa on la (for every a). So the proof is concluded .• Condition (iii) is obviously neces sary in theorem 2 . 8 . 5 . How ever, the preceding example (2 . 8 . 4.) suggests a generalization of the concept of distribution. First of all we shall consider more generally open sets in IR instead of open intervals. Remember that every open set Q in IR, is the union of a finite or countable system of mutually disj oint open intervals (the so-called components of Q). For the =

. .

�:.

same purpose, we could consider still more generally any set which results from any open set Q by adding one or more boundary points to Q. But this case reduces to the preceding one, as we shall see. 2.8.6. DEFINITION. Let Q be any open set in IR and let us suppose that to each compact interval le Q is assigned a distribution f/ on I in such a way that for any two compact intervals 11 and 12 contained in Q, we have fl f/2 on /1 n Iv whenever /1 n 12 is not empty nei l ther degenerate. The system (f/) of distributions defined in this way is called a global distribution on Q, and Q is called the domain of f. The distributions f/ are the components of f· We shall denote by � ( Q) the set of all global distributions on Q. Given two elements, f= (fJ and g = (g/), of � ( Q), and a com plex number a, we shall define f+g (sum of f and g), af (product of a by f) and Df (derivative of f), by the formulas : =

f+g= (f/ +g/), af= (af/), Df= (Df/) · It is immediately seen that � ( Q) is a complex vector space with respect to the first two operations and that D is a linear mapping of

� ( Q) into itself

Observe now that to each continuous function f �n Q corre sponds the global distribution (f), where f/ is the restriction of f to I and that this correspondence i� a one-to-one linear mapping of C( Q) onto a subspace C( Q) of � ( Q) such that if f E C l ( Q), then D(f/) corresponds to the derivative of f in the usual sense. Then, we can identify every function f E C ( Q) with the corresponding element

(f/) of�( Q) so that C ( Q) becomes a subspace of�( Q).

In particular, Q may be an interval. Then taking 2.8.5. into ac count, we see that the space �( Q) of all distributions on Q, can be identified, in the same way, with a subspace of� ( Q). In the general case, w e shall call any element f of� ( Q) of the n form f D F, with n E/No and F E C ( Q), a distribution on Q. It is easily seen that the set of all distributions on Q, which we shall denote by �( Q), is then a vector subspace of�( Q). Distributions may b e called global distributions of finite rank. According, a global distribution which is not distribution is said to be of infinite rank. =

The concept of restriction, as well as def. 2. 8. 1., can be extended, in a natural way, to global distributions. Then we can also extend to global distributions the Collectin g Principle 2.8.5 . , considering more generally open sets instead of open intervals and suppressing condi tion (iii) . It should be observed that there exists actually global distribu tions of infinite rank. An example is suggested in 2.8 .4.

2.9. Carrier of distribution Global distributions of infinite rank have rather a theoretical in terest mainly connected with the functional theory of L. Schwartz . So we shall hence forth confine our discussion to distributions . We say that a distribution f on a open set Q in IR is null on a open set 0 C Q iff f equals the zero function on o.

The union of all open sets 0 where a distribution is null, is a gain an open set Qo where f is null (hence the greatest open set where f is null). PROOF. Let 10 be any component of Qo . Then 10 is an open in terval which is the union of a system (la) of open intervals where f 2.9.1. LEMMA.

is null . But the zero function is also null on all intervals of the sys tem. Hence, by the collecting principle (2.8 .5.), f is equal to the zero function on 10 , and since 10 is any component of Qo ' it follows that f is null on Qo .•

2.9.2. DEFINITION. Let f be a distribution on an open set Q in IR and let Qo ' be the largest open set where f is null. Then the set Q\ Qo (complement of Qo in Q) is called the carrier of f. According to this definition, the carrier of f is always closed re latively to Q. In particular, if Q = IR, the carrier of f is a closed set.

Examples: I - If f is a continuous function on IR, the carrier of f is the closure of the set of all points x, such that f(x) � o. Thus the function f, such that f(x)=sin x when sin x > 0 and f(x) = 0 when sin x :s 0, is a continuous function on IR whose carrier is the set of all points x such that sin x � O.

1 1 The carrier of the distribution 3 8 + 8'( x + 1 ) reduces to the isolated points 0 and - 1 . -

2. 9.3. Proposition. The carrier of a distribution f on IR reduces to a s ingle point a if and only if f is a linear combination of derivatives of

8(x - a) ,

�J m

cj 8(J1(x - a), where m is an arbitrary integer > 0 and

co ' . . . , Cm are arbitrary complex constants which do not all vanish together. PROOF. This condition is obviously sufficient. Let us suppose, conversely, that f is a distribution on IR whose carrier reduces to one point a. Then f is of the form f = D n F with F E C(fR) and, since f = 0 on the set of all points x � a, F is represented by two polynomials, P I and P2 ' of degree < n for x < a and for x > a respectively. Hence, putting G = F - P I ' P = P2 - P I , we have f = D ItG, with G(x) = P(x) for x > a and G(x) = 0 for x < a . Since F is continuous on IR, so is G and hence G (a) = P (a) = O . Consequently, P must have the form:

Put now, for k = 0, 1 , . . . X

Then we have G (x) =

k n-l

{

x k,

if x > 0

0 ,

if x < O

a (x - a) ! and k! 8(x) = D k + lX! for k = 1 , 2, k

. . . . Consequently, putting n - 2 = m, (k + 1 )! a + 1 = cm _ ' we obtain

/ = D nG =

� m

ck8(k) (x - a) • .