(2/p)-l is a non-zero integer - the "integral case" - then t must be restricted to a sub-
space of Cg.
m n
(G) , as discussed below.)
The proof of Theorem 15.2 puts together the results of the previous three sections.
From Theorem 12.li there is a positive integer j such that the wave packet portion of
S#'*/ is approximated on A + = {g€A : gxl 1} in the desired way by the wave
packet of the j-th truncation of the Harish-Chandra expansion for the corresponding
principal series matrix coefficient. These wave packets are integrals over the line Re A =
0; shifting the integration to either Re A = 7 or Re A = - 7 will yield integral terms
which are bounded in the desired way by the growth conditions of Theorem 13.7. The
residue terms which result from this shift of integration comprise, by Theorem 14.3, com-
binations of j-th truncations of the Harish-Chandra expansions for certain discrete series
matrix coefficients. These last terms approximate the discrete portion of S ^ ' / in the
desired way by Theorem 12.1H, which completes the proof of Theorem 15.2.
A discrete analogue of Theorem 15.2, a growth estimate on fixed K-type com-
ponents of the Lp cusp forms portion of the inverse transform, is obtained in Theorem
15.5. Fortunately this result is merely an easy consequence of Theorem 8.1, the extended
Trombi & Varadarajan estimate on discrete series matrix coefficients.
In Section 16 Theorems 15.2 and 15.5 lead to an easy proof of Theorem 16.3, the
desired Fourier isomorphism theorem in the "non-integral case", i.e., when 7 = (2/p)-l is
not a positive integer. The restiction on 7 is necessary because, when 7 is a positive
integer and the K-types n and m assume certain values, the bound in Theorem 15.2 has
been established only for a subspace of Cft.mn(G) ^ e ^u^ s P a c e consists of certain
functions defined on the strip |Re A | $ 7 ; the subspace C^.m
( G )
consists of those
functions j which vanish at 7 or —7. When 7 is a positive integer, the proof of Theorem
15.2 (see (15.22)) requires that 0(A)/(A—7') be finite on Re A = 7', where 7 ' equals
either 7 or —7. This is the reason for the restriction to the subspace.
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