1. ABSTRAC T HARMONIC ANALYSIS: A SURVEY 11
T H E O R E M 8. If T is a distribution of order k with support g, then T has the
form
\a\k
For a proof of this theorem see Hormander ([80], vol. I, p. 46). The argument is
basically an application of Taylor's theorem in several variables. If X is again a
finite dimensional vector space over R, then the Schwartz space S(X) can be given
a more concrete description. For the discussion below we may assume X = R n .
DEFINITIO N 14. The Schwartz space S(R n ) is the set of all $ e C£°(R) such
that
i/aj/3($) = sup \xf3da^(x)\ oo
x
for all multi-indices a ( a i , . . . , an) and f3 = (/?i,..., (3n), where x@ =
x%n
and da = (d/dxi)ai {d/dxn)OCn. The topology in S(Rn) defined by the semi-
norms va,(3($) makes S(Rn) into a Frechet space.
The primary importance of the Schwartz space is that the Fourier transform
F : $ l is an isomorphism of S(Rn). We repeat here the definition of tempered
distributions in this context.
DEFINITIO N 15. A tempered distribution is a continuous linear form on
S(Rn). The set of all tempered distributions is denoted by S;(Rn).
It is not difficult to prove by the method of regularization that the space
C£°(Rn) is dense in the Schwartz space 5(R n ).
DEFINITIO N 16. If T e
S'(Rn)
is a tempered distribution, the Fourier
transform T is defined by
f ( * ) = T ( $ ) , $GS(R n ).
An important result of the theory is the following.
T H E O R E M 9. The Fourier transform is an isomorphism of S'(Rn) (with the
weak topology), and Fourier's inversion formula T =
{2ir)nT
holds for all T G
^ ( R " ) with f($(x)) = T($(-x)).
The proof of this theorem can be found in Hormander ([80], vol. I, p. 164).
The importance of the space S;(Rn) of all tempered distributions results from the
fact that it is the smallest subspace of £'(Rn) containing Li(R n ) and which is
invariant under differentiation and multiplication by polynomials.
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