Elementary proofs of Paley-Wiener theorems for the Dunkl transform on the real line

TL;DR

This paper provides elementary proofs of Paley-Wiener theorems for the Dunkl transform on the real line, extending results to L^2-functions and establishing identities for L^p-functions, with potential novelty even for Fourier transforms.

Contribution

It introduces new elementary proofs of Paley-Wiener theorems for the Dunkl transform and related identities, including for L^2 and L^p functions, with possible novelty in the Fourier case.

Findings

Elementary proof of Paley-Wiener theorem for Dunkl transform

Extension of the theorem to L^2-functions

New identities in the spirit of Bang for L^p-functions

Abstract

We give an elementary proof of the Paley-Wiener theorem for smooth functions for the Dunkl transforms on the real line, establish a similar theorem for L^2-functions and prove identities in the spirit of Bang for L^p-functions. The proofs seem to be new also in the special case of the Fourier transform.