Seria I: PRACE MATEMATYCZNE X (1966) ANNALES SOCIETATIS MATHEMATICAE POLONAE Serio I: COMMENTATIONES MATHEMATICAE X (1966)
B. K
otkowski(Poznań)
Remarks on the theory of distributions
1. The subject of this note is to give a simple approach to the theory of distributions based on the definition of L. Schwartz [3]. The simpli
fication consists in extending some operations on infinitely differentiable functions to the case of distributions (x) and in avoiding the full formula
tion of the theorem on the partition of unity. So the support of a distri
bution is defined without application of the partition of unity. The proof of equivalency of this definition with the definition of L. Schwartz [1], [3] includes at the same time the proof of the fact that the complement of the support of a distribution T is the greatest open set such that if the support of <peB is contained in this set, then T(tp) = 0.
Throughout this note we shall use the following notations:
Nf — the support of the function / ;
E — the set of infinitely differentiable functions defined on Rn\ ) В — the set of infinitely differentiable functions of compact support defined on Iin]
B' — the set of distributions defined on _D;
E* — the set of distributions generated by the functions from E, i.e. the set of distributions:
f((p) = f f(oo)q)(x)dx, where feE-,
Rn
r a = { yeD: 0, J y{x)dx = 1, Ny с < -1 /д ,1 /# > }.
Rn
2. It is known that the spaces E, D, B' are complete.
2.1. T
heorem. For each positive integer q there exists a sequence of ПеГа such that
Jyi{t)cp{x+t)dt-> cp{x)
Rn
uniformly in q> for an arbitrary bounded set A a B.
(x) Products of distributions were already defined by extension by J. Miku- siński [2].
2.2.
Th e o r e m.I f the sequence ip^ of functions from В tends in В to yiz uniformly in £ eE, then for an arbitrary T e B T ( y i i t ) —> T (ip s) uniformly in £eE, where E is the set of indexes.
2.3.
Th e o r e m.I f <peB, TeB' and rj — ±1? then Tt{cp(xf-r)t))eE and [Tt(<p{x+Yit)))(a) = T i ^ f x - f y t ) ) .
We omit the proofs of the theorems 2.1-2.3.
2.4.
Th e o r e m.Let ip, <peB, T e B ' and rj = ± 1 . Then J Tt[ip{x-\-r]t))(p{x)dx — J ip(xJr 7 ]t)(p(x)dxj .
Rn Rn
P r o o f. For the sake of simplicity the proof will be given only in case n = 1 , the general case may be proved analogously.
Let ip and q> be arbitrary but fixed functions from B. There are a, b such that N v c by. Therefore
ь
^ ip{xJr rjt)cp{x)dx = j ip(x-\-rjt)(p{x)dx.
r1 a
We now form a sequence Шг) of subdivisions of the interval <a, &>;
Ili — a subdivision of the interval (a, by into 21 equal subintervals. Let xa (j — 0,1, . . . , 2г) be the jth point of a subdivision /7*. We construct a sequence of д{ еВ in the following way:
9i{t) = (b—a )2 1 ^ ipixy+rj^vixi
7 = 1
Let e > 0 and let p be an arbitrary but fixed non-negative integer.
The function ipP(x-\-rjt)(p(x) is uniformly continuous in R 1. Then there exists ip such that if \x'—x"\ < 2 ~l * * v, \t'—t"\ < 2-г*>, then
Ii p ? ] { x " + r } t ” ) q > ( x " ) — y ) ¥ ) (x'-\-r]t')(p(x')\ < e /(b — a) .
For every i > ip and for every non - negative integer 1c we have:
2 ^ 2 k
= (Ь a)2 '
I
^ ip\ ^ (ffii+Jc,(j-l)2
k+s~Srr]t)<p(Xi+ jC'0
_i)2fc+s)• f P ixu+^)<р{ха) \ < £ •
j = l S = 1
This means that the sequence {gf\ is a Cauchy sequence in B. Since В is complete there exists a function geB such that gi-> g but lim ^(l) =
г
f ip(x-\-r)t)<p(x)dx and therefore g(t) — jip(xJr r]t)(p(x)dx. Since Te B' we
R 1 R l
have
f y>(x+r]t)(p(x)docj = T(g) = Т(Иш^) = lim T(^)
Rl i i
= j Tt(xp{xĄ-r}t))(p{x)dx.
Rl
2.5. D
efinition. Let T el)' and xpeD. Then the distribution T*xp{<p) = TĄipx((p(x+t)))
is called the regularization of T by ip.
Theorem 2.4 implies
(1) T*xp(cp) = J Tt{(p(xJr t))xp(x)dx,
Rn
(
2
)T*xp(cp) — j T t[xp{x—t))(p(x)dx.
Rn
From (2) follows that the regularization of a distribution is an in
finitely differentiable function and
(3) T*xp(x) = Tt[y>{x—<)).
2.6. T
heorem:. Space E* is dense in D '.
P r o o f. Let {yfa be such a sequence from Г г that Ny. <= <—1 /(1 + i), 1 /(1 +г)> and let A be some bounded subset of D. From 2.1 and 2.2 we have:
For each e > 0 there exists a positive integer i 0 such that for each i + % and for each (peA: \T*yi(q>)— T(q>)\ < e, i.e. T * y i^ T, but from (3) follows that T*yieE*. This means that E* is dense in D'.
Let U be an operation from the Cartesian product E x E x ... x E — Ev into E .We define the operation U on E*p as follows: if /, g, ..., heE*, then by U(f, g, h) we understand the distribution generated by the function U(f,g, . .. , h).
2.7.
Th e o r e m.I f for any Cauchy sequences fa, gi, ЬгеЕ* in D' U(fa, дг, ..., hi) is also Cauchy sequence in D ’ , then U may be continuously extended in one and only one way to D '.
The easy and typical proof of the last theorem will be omitted.
Theorem 2.7 makes it possible to extend in the most natural way operations defined on E to the space D'. For example let us define in this way the convolution in D'.
2.8. D
efinition. T from D' is called a distribution of compact support if and only if there is a sequence of щ eD such that T and all supports of q>i are contained in a compact set.
Prace Matematyczne X .l 2
The distributions of compact support may be extended from D to E. First let us extend the distributions generated by the functions from 1). This can be made in the following way:
Let (peD) then for every f e E we define: <p(f) 4= f{(p). Now assume T to be of compact support. Then there is a sequence of yieD such that (Pi-> T and all supports of щ are contained in one compact set. We define:
T(f) = l i m ^ /) , i
where f e E (obviously, this limit exists and is unique).
To prove the next theorem the following lemma is useful:
2.9.
Le m m a.I f a sequence of TieD' converges in' D' and if A is a bounded subset of D, then
SUpSUp \ Ti((p)\ < oo.
i <peA
We omit the easy proof of this lemma.
2.10.
Th e o r e m.I f fa, gi from, E* are Cauchy sequences and if at least one of these sequences has all supports contained in a compact set, then the convolution fa * gi is a Cauchy sequence in D'.
P r o o f.
\(fa+k*gi+k—fa*gi)(<p)\ < |J [fi+k(t)—fa(t)][ f gi+k(a)<p(x+t)d<Ądt\ +
‘ Rn Iin
+ I /[#гЧ*(ж)-<7г(ж)][ j fa{t)(p{x+t)dĄdx |.
Rn Rn
Let all supports of {fa} be contained in a compact set K. Then by 2.9 for each a and for each bounded subset A <=. I) there exists a constant M a such that for each i, Ti,teK and for each cpeA the inequality I / 9i+k(®)<p(x Jrt)dx\ < M a holds. By assumption fa+k—fa tends to zero in D ’
Rn
uniformly in Tc. Hence we have for i > ix
I /
U i + k W — f i i l ) }[ /
g i + k ( x ) { p ( x + t ) d x ] d tI < £e.
R n Rn
Let A be a bounded subset of D. Then there exists such a compact set that all supports of ffa(t)(p(xJrt)dx for each i and each <peA are contained
» R n
in this set. Moreover, from 2.9 follows that for each a and for each bounded subset A a D there is a constant M'a such that for each i, X€Rn and <peA:
Rn
By assumption </•£+&— 0 uniformly in Jc, hence
I / [& + *И — ffi(t)(p{x+t)dt]dx\ < \s
Rn . Rn
for i ^ i 2 and the proof of the theorem is completed.
In this manner the convolution of two distribution is defined in case when at least one of these distributions is of compact support (more generally, in a convolution of n factors, n —1 distributions must be of compact supports).
The convolution of functions possesses the following properties:
(i) f*g = g*f, (ii) {f*g)*h = f*(g*h), (iii) (f+g)*h = f*hf-g*h, (iv) (f*g)(n) = f {a)*g, moreover,
(v) f*g{<p) = ft[gx(<p{x+t)))-
Hence and by theorems 2.7 and 2.10, we have for distributions:
(i') T*S = S*T, (ii') (T*S)*U == T*{S*U), (iii') ( T + S) * U = T*TJ+ 8 *U, (iv') (T* 8 )<n] = Л « ,
(v') T*S(f ) = T,(Sx(f>(s+t)j).
3. Let
oo
AqT = U and A T = П A qT
yerg q=l
where T el)'.
3.1. D
efinition. I f T is an arbitrary distribution, then the set NT = A
tis called the support of the distribution T.
3.2. T
heorem. Let TeD', 2lT = {U open: for every cpeB if N v c: U, then T(<p) — 0} and От = U Then
(a) Ж'те^т, (b) N ’
t= O
t-
P r o o f, (a) Let cp be an arbitrary but fixed infinitely differentiable
OO
function such that Nę is compact and N a N'T — U A'qT- Since the sets
— _ „ e=i ■
A'qT are open and A'
qt<=■ A'a+1>T for every q, there is q 0 such that N v c Aq0T c Ag0T = (П ^T*y or
ver4
(*) for every уеГд0, Nrp гл NT*V = 0.
Let {y*} be a sequence such that for each i, у^Гд^ and у*5* d. Then by (*): for every i T*yi{ę) = 0 and hence T((p) = limTxyfcp) — 0.
г
(b) Let £eOy. Then there exists an open set I7e2lT such that £eU.
Hence there is an r > 0 such that K{tj,r) c U{K(£,r) — a ball with the
centre in £ and radius r). Moreover, there exists an e > 0 such that К(Ё, łr)
+ < - £ , S > c z K ( t , r ) .For every (peD such that N v <= К (£, -Jr) and for every же<—e, e> we have TĄ(p(x-\-t)) = 0.
Let q0 > 1/e. Then for an arbitrary у e r qQ and for every cpeD such that Nv а К (£, \r) we have T*y(<p) = 0. Hence £eA'T and От c= A'T. From this follows that A T a 0'T and N T = A T c 0'T. By (a) we have N'T a 0 T and therefore N T = 0'T.
References
[1] L. H ó r m a n d e r , Linear partial differential operators, Berlin 1963.
[2] J. M i k u s i ń s k i, Criteria of the existence and of the associativity of the pro
duct of distributions, Studia Math. 21 (1962), pp. 235-259.
[3] L. S c h w a r t z , Theorie des distributions, Yol. I, Paris 1950.