An analogue of the Wiener-Tauberian theorem for spherical transforms on semisimple Lie groups

Abstract

Let G be a semi-simple connected noncompact Lie group with finite center and K a fixed maximal compact subgroup of G. Fix a Haar measure dx on G and let I^G) denote those functions in L (Gj dx) which are biinvariant under K. The purpose of this paper is to prove that if fe /i(G) is such that its spherical Fourier transform (i.e., Gelfand transform) / is nowhere vanishing on the maximal ideal space of /^G) and / "does not vanish too fast at oo", then the ideal generated by / is dense in Ii(G). This generalizes earlier results of Ehrenpreis-Mautner for G=SL(2, R) and R. Krier for G of real rank one.