Analogues of the Wiener-Tauberian and Schwartz theorems for radial functions on symmetric spaces

We prove a Wiener-Tauberian theorem for $L^1$-spherical functions on a semisimple Lie group of arbitrary real rank. We also establish a Schwartz theorem for complex groups. As a corollary we obtain a Wiener-Tauberian type theorem for for compactly supported distributions.