Mellin Analysis and its distance concept. Applications to sampling theory

TL;DR

This paper introduces a new notion of functional distance within the Mellin transform framework, providing formulas and applications to sampling theory, especially for Mellin band-limited functions.

Contribution

It develops a general representation of the Mellin-based distance and links it to Lipschitz and Mellin-Sobolev spaces, with applications to sampling theory.

Findings

Derived a general formula for Mellin-based distance

Connected the distance to Lipschitz and Mellin-Sobolev spaces

Applied the concept to approximate sampling relations

Abstract

In this paper a notion of functional "distance" in the Mellin transform setting is introduced and a general representation formula is obtained for it. Also, a determination of the distance is given in terms of Lipschitz classes and Mellin-Sobolev spaces. Finally applications to approximate versions of certain basic relations valid for Mellin band-limited functions are studied in details.