4 WILLIAM H. BARKER

(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

n

( G )

0

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.