# Series Analysis and Schwartz Algebras of Spherical Convolutions on Semisimple Lie Groups

## TL;DR

This paper explores the harmonic analysis of spherical convolutions on semisimple Lie groups, focusing on the Harish-Chandra transform and Schwartz algebras, and clarifies the role of the Trombi-Varadarajan Theorem in this context.

## Contribution

It provides exact descriptions of the Harish-Chandra transform of Schwartz functions and demonstrates how the Trombi-Varadarajan Theorem applies to spherical convolution transforms.

## Key findings

- Explicit characterization of the Harish-Chandra transform for Schwartz functions
- Connection established between spherical convolutions and $L^{p}$-Schwartz algebras
- Proof of the role of Trombi-Varadarajan Theorem in harmonic analysis

## Abstract

We give the exact contributions of Harish-Chandra transform, $(\mathcal{H}f)(\lambda),$ of Schwartz functions $f$ to the harmonic analysis of spherical convolutions and the corresponding $L^{p}-$ Schwartz algebras on a connected semisimple Lie group $G$ (with finite center). One of our major results gives the proof of how the Trombi-Varadarajan Theorem enters into the spherical convolution transform of $L^{p}-$ Schwartz functions.