A Paley-Wiener theorem for the inverse spherical transform

A Paley-Wiener theorem for the inverse spherical transform is proved for no ncompact semisimple Lie groups G which are either of rank one or with a com plex structure. Let K be a fixed maximal compact subgroup of G. For each K- bi-invariant function f in the Schwartz space on G, consider the function f defined on a fixed Weyl chamber a(+) by (f) over tilde(H) := Delta(H) f(ex p H). Here Delta(H) := Pi(alpha is an element of Sigma+) (sinh alpha(H))(m alpha/2), where Sigma(+) is the set of positive restricted roots and m(alph a) is the multiplicity of the root alpha. The K-bi-invariant functions f wh ose spherical transform has compact support are identified as those for whi ch (f) over tilde extends holomorphically and with a specific growth to a c ertain subset of the complexification a(c) of a. The proof of the theorem i n the rank-one case relies on the explicit inversion formula for the Abel t ransform.