28 Approximation of Distributions by Cutting and Regularizing

28 Approximation of Distributions by Cutting and Regularizing In this chapter, we are going to show that every distribution T in an open set D of Rn is the limit of a sequence of functions belonging to %‘r(Q)and, furthermore, that there is a standard procedure for constructing this sequence from T . Let {Q,} (k = 0, 1,...) be a sequence of open subsets of D whose union is equal to Q and such that QkPl C Dk(K = 1,2, ...). For each k, select a function gk E ‘3?“(D) which is equal to one in D, . Now, given any distribution T in 52, it is clear that the distributions g,T converge to T in 9’(D), say for the strong dual topology, although this is not important, as strong and weak convergences in 9’(D) are one and the same thing for sequences, as we shall soon see. Anyway, if 37 is a bounded subset of %?:(D), there is a compact subset K of D such that 37 C %‘;(K) (Proposition 14.6); there is an integer k ( K ) such that K C D , for all K 2 K(K), therefore such that gkp = p for all K 3 K(K) and all p E 9. Then, for all p E 37, < g 3 ,v> = < T ,gkv)

= < T ,v>3

which proves that gkT -+ T in 9’(D). The fact that the functions g , are identically equal to one in open sets which form an expanding sequence is not at all necessary to reach the conclusion that g,T -+ T . In connection with this, we propose to the student the following exercise: Exercise 28.1. Let {g*} be a sequence in ym(Q) which converges to the function identically one in R. Prove that, given any distribution T i n Q, the sequence of distributions g,T converges to T i n W ( Q )and , that, if T has compact support, g,T converges to T i n &‘(Q).

Going back to the considerations above, we can take the Qk to be compact and the ‘%“ functions g k with compact support. We may then state: 298

CUTTING AND REGULARIZING

299

THEOREM 28.1. Let T be a distribution in Q. There is a sequence of distributions with compact support, {T,} ( k = 0, 1,...), such that, given any relatively compact open subset Q’ of Q, theie is an integer k(Q‘) > 0 such that, for all k 2 k(Q’), the restriction of Tk to Q’, T, I Q’, is equal to the restriction of T to Q’, T I Q’. Let T and the T, be as in Theorem 28.1. If K is any compact subset of 52, there is an integer k ( K ) such that, for k 3 k ( K ) , T and Tk are equal in some neighborhood of K and, in particular, Knsupp T z K n s u p p Tk. Remark 28.1. The operation just described, of multiplying a distribution by V“ functions with compact support, g,, equal to one in relatively compact open subsets of Q which expand as k - + co and ultimately fill Q, is the extension to distributions of the “cutting operation” on functions. I n the latter case, if we deal with some function f i n Q, we regard f as the 11-nit of the functions f , equal to f in Q, and to zero outside 52k; then obviously the fk converge to f uniformly on every compact subset of 9.Note that fk is the product o f f by the characteristic function tpnk of 5 2 k . Of course, we cannot multiply a distribution by tpRk since this function is not smooth; thus we must multiply the distribution by a V“ function (with compact support) which coincides with pR, in Qk, that is to say, which is equal to one in Q, . We have approximated an arbitrary distribution in 52 by a sequence of distributions in Q which have compact support. The next step is to approximate any distribution with compact support by a sequence of test functions. This is done by convoluting the given distributions with : W functions which converge to the Dirac measure 6: We shall therefore use the properties of the convolution of distributions, established in the preceding chapter (Chapter 27). We begin by considering the convolution T*F of a distribution T with a V“ function p, one of the two having compact support. We may regard p as a distribution, in which case T * tp is the distribution % : 3 *

- ( T * v , * ) =(T,+**),

where +(x) = p( -x). But T q is, in fact, a V” function, precisely the function (Theorem 27.5). x ( T , ,~ ( x y))

-

The approximation result which we are seeking will be a consequence of the following lemma:

300 LEMMA 28.1. Let

p

be the function defined by

=

For

[Part I1

DUALITY. SPACES OF DISTRIBUTIONS

E

> 0,

1 llzl
call pz the function x

-

x

1)'

E-"P(X/E).

Then the sequence

CpIIj} ( j = 1,2,...) converges to the Dirac measure 6 in the space &'of

distributions with compact support in Rn. About the function pI , see Chapter 15, p. 155.

Prmf of Lemma 28.1. We advise the student to take a look at the proof of Lemma 15.2; the arguments here and there are closely related. Let f be a V" function in Rn. We have

We may assume that

E

< 1. Then, if I x I < E ,

therefore,

This shows that, iff remains in a bounded set of V"(Rn), in fact, in a Q.E.D. bounded set of V1(Rn), ( p , ,f) converges to f(0). Observe that we do not need the precise information which we have about the functions pI . In connection with this, we propose the following exercise to the student: Exercise 28.2. Let {pk} be a sequence of Radon measures in R",having the follm.ng properties: (1) supp pk i s contained in a ball of radius rk centered at x

k-,+co;

=

0 such that

rk

-P

0 as

Chap. 28-41

CUTTING A N D REGULARIZING

30 1

0 such that, for all k = 1, 2, ..., and all functionsf E q(R"),

(2) the numbers (3)

Under these hypotheses, prove that pk converge to 6 in B'(R"). What rnodijications could be made on the hypotheses ;f we were only to require that the pLr converge to 6 in 9'(R") ?

We may now easily prove the following result:

THEOREM 28.2. Let T be a distribution in the open set SZ. There is a sequence of functions v k E gT(SZ) (k = 1, 2,...) With the following properties: (i)

vk converges to

T in Z(Q) and, i f the support of T is compact, converges to T in &'(a); (ii) for every compact subset K of SZ, K n supp p)k converges to K n supp T and, i f supp T is compact, suppq~,, converges to vk

a P P T.

-

A sequence {&} of subsets of a metric space E (with metric (x, y ) d(x, y ) ) is said to converge to A C E if, to every e > 0, there is k(e) such that, for every k 2 k(&), A , c {X E E ; d(x, A ) < &},

A C {x E E ; d(x, A,)

If B C E, we have set d(x, B) = inf,,,

< e}.

d(x, y).

Proof of Theorem 28.2. We begin by selecting a sequence of distributions Tk (k = 0,1, ...) with compact support, as in Theorem 28.1; for every relatively compact open subset SZ' of SZ there is k(SZ') such that, if k 3 k(G'), Tkl SZ' = T (Q'. Next we consider a sequence of functions plli

*

'

observing that supp(pIIj* Tk)C supp plli

+ supp Tk

(Proposition 27.5).

For each k we select j according to two requirements: j 3 k (in order to j sufficiently large so that the neighborhood of ensure that j + +a); order lfi of supp Tk is a compact subset of SZ. If we then call jk the integer thus selected, we contend that the test functions T k = p1ljk* Tk converge to T in 9'(Q).

302

DUALITY. SPACES OF DISTRIBUTIONS

[Part I1

Indeed, let L3 be B bounded subset of %‘2(9);there is a compact subset K of 9 such that supp v C K for all functions rp E L3. Let 9‘ be a relatively compact open subset of 9 containing K ; for k 2 k ( 9 ’ ) , we have Tkl 9‘ = T 19‘.On the other hand, for 1/k < d ( K , C Q’),

*

for all p E a.

s u p p ( ~p) ~C ~ 52’ ~ ~

We recall that j k 2 k and, also, that p’ (p)k -

T , p)

= (Tk

9

PMk

= p.

Then, for all

E g,

* p> - < T ,p>

* P) - ( T , V> = , = < T ,p l i j k

where g is an arbitrary function which belongs to %:(9) and is equal to one in 9’.This shows ttat, for all v E L3, and k sufficiently large, (Vk

- T , P) = (P115,

* ( g T ) - ( g v ?v>.

But g T is a fixed distribution with compact support, and p1ljk+ 6 in €“(Rn); by applying Theorem 27.6, we conclude that p l l j k * ( g T ) + g T uniformly on g, which is what we wanted to prove. When the support of T is compact, it is easy to see that the vk converge to T in df”(9). First of all, we may take Tk = T for all k, and v k = P l / k * T for Ilk < d(supp T,C 9). As the P l / k converge to 6 in B’(9), a fortiori in B’(9),it follows immediately from Theorem 27.6 that vk converges to T in €’(sZ). Property (ii) in Theorem 28.2 is obvious, by inspection of the definition of the functions vk .

COROLLARY. Let 9 be an open subset of in &(Q) and in g‘(9).

Rn;Vp(9) is sequentially dense

There is no need to underline the close relationship between Theorem 15.3 and Theorem 28.2; the properties stated in these theorems are often referred to as “approximation by cutting and regularizing.” Convolution of a distribution with a Vrn function is often called regularization (or smoothing). The word cutting refers to the multiplication of a distribution T by V“ functions which are equal to 1 in some relatively compact open set 9’and equal to zero outside a neighborhood of the closure of 9‘. Definition 28.1. A space of distributions in 9, d,is said to be normal if V;(9) is contained and dense in d,and if the injection of %‘2(9)into d is continuous.

Chap. 28-63

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303

We recall (Definition 23.1) that a space of distributions in 52 is a linear subspace of 9'(52)carrying a locally convex topology finer than the one induced by 9'(52).

PROPOSITION 28.1. If d is a normal space of distributions in SZ, the strong dual of d is canonically isomorphic to a space of distributions in 52. We have already essentially proved this statement (in Chapter 23) and often used it: let j : %';(Q) -+ d be the natural injection; since the of j is a continuous one-to-one image of j is dense, the transpose linear map of the strong dual d'of d into W(SZ). Examples of normal spaces of distributions in SZ: VF(52) (0 m +a)(Corollaries 1 and 2 of Theorem (i) %?(a), 15.3); (ii) L p ( s 2 ) (1 p < +a)(Corollary 3 of Theorem 15.3); (iii) B'(SZ),S'(L2)(Corollary of Theorem 28.2.).

< <

<

Example of spaces of distributions which are not normal:

L"(SZ) (we suppose SZ nonempty!); H(Rn), space of functions in Rn which can be extended to C n as entire functions (with the topology carried over from the space of entire functions in C", H(Cn)).Note that L"(S2) contains V:(Q) whereas the intersection of H(R") with c&,"(S2) is reduced to the zero function. Remark 28.2. T h e dual of a normal space of distributions is not necessarily a normal space of distributions (although it is a space of distributions, by Proposition 28.1), as shown by the example of d = L'(Sz), d' = L"(SZ). Exercise 28.3. Let {gx} be a sequence of W: functions in R" such that g&) = 1 for I s 1 < k (k = 1 , 2 ,...), and such that, to every n-tuple p , there is a constant C , > 0 such that, for all k = 1 , 2 ,...,

Prove that, given any tempered distribution S in R",g,S converges to S in Y'(R") as k 4 co. (The student may simply prove that gkSconverges weakly to S in 9';we shall strong and weak convergences coincide.) see that, for sequences in Y', Construct a sequence of functions g, with the above properties.

Remark 28.3. A corollary of Exercise 28.3 is that 9" is a normal space of distributions in Rn. We already knew that 9' is a normal space of distributions in Rn (Theorem 15.4).

304

DUALITY. SPACES OF DISTRIBUTIONS

Exercises 28.4. Prove the following lemma: LEMMA28.2. The sequence of functions ( k / ~ ' /exp( ~ ) -K*IxI*), ~

k

=

1, 2,...,

converge to the Dirac measure 6 in 9(Rn). (Cf. Lemma 15.1 and Lemma 28.1). 28.5. Prove the following result: THEOREM 28.3. Let Q be an open subset of R". Any distribution in Q is the limit of a sequence of polynomial functions. (Hint: Make use of Theorem 28.1 and Lemma 28.2; cf. Corollary 2 of Lemma 15.1 and Exercise 27.2).