To Multidimensional Mellin Analysis: Besov spaces, $K$-functor, approximations, frames

TL;DR

This paper develops a comprehensive framework for multidimensional Mellin analysis, introducing new function spaces, approximation theorems, and frame characterizations, expanding the theoretical foundation of Mellin-based harmonic analysis.

Contribution

It introduces Besov-Mellin spaces, establishes their interpolation properties, and develops approximation theorems and frame representations in the Mellin setting.

Findings

Besov-Mellin spaces are interpolation spaces between Sobolev-Mellin spaces.

Established direct and inverse approximation theorems for Mellin spaces.

Characterized Besov-Mellin spaces using Hilbert frames in the Hilbert case.

Abstract

In the setting of the multidimensional Mellin analysis we introduce moduli of continuity and use them to define Besov-Mellin spaces. We prove that Besov-Mellin spaces are the interpolation spaces (in the sense of J.Peetre) between two Sobolev-Mellin spaces. We also introduce Bernstein-Mellin spaces and prove corresponding direct and inverse approximation theorems. In the Hilbert case we discuss Laplace-Mellin operaor and define relevant Paley-Wiener-Mellin spaces. Also in the Hilbert case we describe Besov-Mellin spaces in terms of Hilbert frames.