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).
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