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± • • •

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.