On the Paley-Wiener theorem in the Mellin transform setting

TL;DR

This paper extends the Paley-Wiener theorem to the Mellin transform context, offering two proofs—one using complex analysis and the other employing a Bernstein inequality for Mellin derivatives.

Contribution

It introduces a Mellin transform version of the Paley-Wiener theorem with two distinct proofs, enriching the theoretical framework of Mellin analysis.

Findings

Established a Mellin transform Paley-Wiener theorem

Provided complex analysis and Bernstein inequality-based proofs

Enhanced understanding of Mellin transform properties

Abstract

In this paper we establish a version of the Paley-Wiener theorem of Fourier analysis in the frame of the Mellin transform. We provide two different proofs, one involving complex analysis arguments, namely the Riemann surface of the logarithm and Cauchy theorems, and the other one employing a Bernstein inequality here derived for Mellin derivatives.