Wiener Tauberian theorem for rank one semisimple Lie groups

We prove a genuine analogue of Wiener Tauberian theorem for $L^1(G\mathrm{/}\mathrm{/}K)$, where G is a semisimple Lie group of real rank one. This generalizes the corresponding result on the automorphism group of the unit disk by Y. Ben Natan, Y. Benyamini, H. Hedenmalm and Y. Weit.