Best approximation with wavelets in weighted Orlicz spaces

TL;DR

This paper investigates the approximation properties of wavelet bases in weighted Orlicz spaces, establishing conditions for greediness and characterizing approximation spaces with sharp embeddings, especially in Lebesgue and Besov spaces.

Contribution

It provides a characterization of democracy functions for wavelet bases in weighted Orlicz spaces and identifies when these bases are greedy, along with embedding results for approximation spaces.

Findings

Wavelet bases are greedy if and only if the space is a Lebesgue space.

Sharp embeddings for approximation spaces are established in terms of weighted Lorentz spaces.

In Lebesgue spaces, approximation spaces coincide with weighted Besov spaces.

Abstract

Democracy functions of wavelet admissible bases are computed for weighted Orlicz Spaces in terms of its fundamental function. In particular, we prove that these bases are greedy if and only if the Orlicz space is a Lebesgue space. Also, sharp embeddings for the approximation spaces are given in terms of weighted discrete Lorentz spaces. For Lebesgue spaces the approximation spaces are identified with weighted Besov spaces.