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12 REGULARIZATION OF DISTRIBUTIONS 289
Deduce Sonine's formula
J,+f«<*) -
(Convolve with Yfl+i.)
3. With the notation of (17.9.4), show that Z« * Z0 = ZB+/, for all complex numbers a, /},
(Same method as in Problem 2.)
4. Consider <T(Rn) as a vector space of linear forms on «?(RB), and endow <T(RB) with the
corresponding weak topology (12.15.2). For each distribution S e J0'(R"), show that the mapping 1WS * T of <T(R") into S'QR.11) is continuous.
5. Let u be a continuous linear mapping of <i?'(Rn) into ^'(Rn).
(a) Show that the following two properties are equivalent:
(1) Y(h)«CT) = «(Y(h)T) for all T e <T(R») and all h 6 R";
(2) D; w(T) «= u(Dj T) for all T e <T(R") and 1 ^; <;«.
(Use formula (17.8.2.1) and consider, for each /e^(R"), the function
h>-> <«(v(h)T), y(h)/>; calculate its partial derivatives.)
(b) If u satisfies the equivalent conditions of (a), show that u is necessarily of the form
1W S * T, where S € ^'(Rrt). (Consider the linear mapping R»-> w(R * T) - R * u(T) of ^'(R") into ^'(R"), for a fixed distribution T e ^'(R*), and show that its kernel is the whole of &'(R"). For this purpose, observe that the Dirac measures ex (x e R") form a total set (12.13) in <j?'(Rn), by using Problem 13 of Section 12.15; then remark that the ex belong to the kernel of the linear mapping in question.)
6. Let G, G' be Lie groups and let S, T (resp. S', TO be strictly convolvable distributions
on G (resp. GO- Show that S ® S' and T ® T' are strictly convolvable distribution on G x G' and that
(S ® SO * (T ® TO = (S * T) ® (S' * TO-
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12. REGULARIZATION OF DISTRIBUTIONS
(17.12.1) Let p, m be two integers ^0 such that p^m.IfTe 0/(m)(Rrt) and
/e ^(R") (resp. ifT e r(m)(Rn) andfe ^(R")), then the distribution T */ may be identified (17.5.3) with a function in $(p~"m\Rn) such that, for each xeR",
(17.12.1.1) (T */)(*) = <T,/(x)Y> = J/(* -
The "fact that the function x\^ff(x-y)dT(y) belongs to ^""m)(Rw)
follows from the hypotheses and from (17.10.1). Next, if S = T */, then we |
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