On the Fourier transform of Schwartz functions on Riemannian Symmetric Spaces

TL;DR

This paper provides a simplified proof of the Fourier transform isomorphism for Schwartz functions on Riemannian symmetric spaces, extending previous results to K-finite functions.

Contribution

It generalizes Anker's proof from K-invariant to K-finite functions, broadening the scope of the Fourier transform isomorphism theorem.

Findings

Established L^p-Schwartz space isomorphism for K-finite functions

Extended previous proofs to a more general class of functions

Simplified the proof methodology for the Fourier transform on symmetric spaces

Abstract

Consider the (Helgason-) Fourier transform on a Riemannian symmetric space G/K. We give a simple proof of the L^p-Schwartz space isomorphism theorem (0 <p \le 2) for K-finite functions. The proof is a generalization of J.-Ph. Anker's proof for K-invariant functions.