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

TL;DR

This paper extends classical harmonic analysis theorems, such as Wiener-Tauberian and Schwartz theorems, to radial functions on symmetric spaces and complex groups, broadening their applicability.

Contribution

It proves a Wiener-Tauberian theorem for $L^1$-spherical functions on semisimple Lie groups of any real rank and establishes a Schwartz theorem for complex groups.

Findings

Wiener-Tauberian theorem for $L^1$-spherical functions on semisimple Lie groups

Schwartz theorem for complex groups

Wiener-Tauberian type theorem for compactly supported distributions

Abstract

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.