1. ABSTRAC T HARMONIC ANALYSIS: A SURVEY 17

/ $(a)xO)

1^/i =

n /

®v(av)Xv(av) 1d/jJV

JG

v

JGV

is a finite product, depending on x, each of whose factors is bounded by

I /

§v(av)Xv{av)~ldyiv\

/ \$v(av)\dfj,v +00,

JG„ JG„

and

fJGv$v{avYv)xv(av)~ldnv

G

$v{a = 1 for almost all v. Hence the Fourier transform

is a product function, and $(x) £ Li(G). With these preliminaries, and using the

inversion formula applied to Gv and Gv, we obtain the following result.

T H E O R E M 11. With notations as above, let $ : G — C be a continuous

function with l G Li(G) and $ G L\{G). Then the inversion formula

JG

holds with the self dual Haar measure given by the product measure

dx = Y[dxv.

V

The Schwartz-Bruhat space on a restricted direct product of locally compact

groups {Gv} is the span of linear combinations of product functions

V

where each $v G S(GV) and $v — charH± for almost all v.

T H E O R E M 12. The Fourier transform

$ •- Hx) = / $(a)x(a)d//(a)

induces an isomorphism from the Schwartz-Bruhat space S(G) onto the space S(G).

In the main applications we deal with groups G that are self dual, i.e. G

is isomorphic to G. In those situations the Fourier transformation induces an

isomorphism on the Schwartz-Bruhat space S{G) which extends to the space of

tempered distributions S'(G).