Democracy functions and optimal embeddings for approximation spaces

TL;DR

This paper establishes optimal embeddings for nonlinear approximation spaces using weighted Lorentz sequence spaces, linking basis democracy functions to approximation quality, and applies these results to wavelet approximation and greedy classes.

Contribution

It introduces a new framework connecting democracy functions of bases to optimal embeddings in approximation spaces, extending known results and analyzing greedy classes.

Findings

Optimal embeddings for nonlinear approximation spaces are proven.

Applications include recovering known embeddings for wavelet approximation.

Analysis of greedy classes reveals new insights into their structure.

Abstract

We prove optimal embeddings for nonlinear approximation spaces in terms of weighted Lorentz sequence spaces, with the weights depending on the democracy functions of the basis. As applications we recover known embeddings for $N$-term wavelet approximation in Lebesgue, Orlicz, and Lorentz norms. We also study the "greedy classes" introduced by Gribonval and Nielsen.