Wiener Tauberian theorem for rank one semisimple Lie groups

TL;DR

This paper establishes a Wiener Tauberian theorem analogue for the algebra of bi-$K$-invariant integrable functions on rank one semisimple Lie groups, extending classical results to a broader non-commutative setting.

Contribution

It introduces a genuine Wiener Tauberian theorem for $L^1(G//K)$ on rank one semisimple Lie groups, generalizing known results from the automorphism group of the unit disk.

Findings

Proves a Wiener Tauberian theorem for $L^1(G//K)$ on rank one semisimple Lie groups.

Extends classical results from the automorphism group of the unit disk to non-commutative groups.

Provides a new tool for harmonic analysis on semisimple Lie groups.

Abstract

We prove a genuine analogue of Wiener Tauberian theorem for $L^1(G//K)$, 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.