A fresh approach to the Paley-Wiener theorem for Mellin transforms and the Mellin-Hardy spaces

TL;DR

This paper introduces a novel Mellin analysis approach to the Paley-Wiener theorem, utilizing polar-analytic functions and Mellin-Bernstein spaces, with applications to optical physics sampling formulas.

Contribution

It presents a new method for the Paley-Wiener theorem in Mellin analysis that avoids complex Riemann surface techniques, introducing polar-analytic functions and Mellin-Hardy spaces.

Findings

Developed a new Mellin analysis framework for Paley-Wiener theorem

Defined Mellin-Hardy spaces with applications to sampling formulas

Provided insights relevant to optical physics sampling techniques

Abstract

Here we give a new approach to the Paley--Wiener theorem in a Mellin analysis setting which avoids the use of the Riemann surface of the logarithm and analytical branches and is based on new concepts of "polar-analytic function" in the Mellin setting and Mellin--Bernstein spaces. A notion of Hardy spaces in the Mellin setting is also given along with applications to exponential sampling formulas of optical physics.