To multidimensional Mellin analysis: Besov spaces, K-functor, approximations, frames

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 operator and define relevant Paley–Wiener–Mellin spaces. Also in the Hilbert case we describe Besov–Mellin spaces in terms of Hilbert frames.