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 I₁(G) denote those functions in L¹(G,dx) which are biinvariant under K. The purpose of this paper is to prove that if f ∈ I₁(G) is such that its spherical Fourier transform (i.e., Gelfand transform) f is nowhere vanishing on the maximal ideal space of I₁(G) and f “does not vanish too fast at ∞”, then the ideal generated by f is dense in I₁(G). This generalizes earlier results of Ehrenpreis-Mautner for G = SL(2,R) and R. Krier for G of real rank one.