Approximation spaces, greedy classes and Lorentz spaces

TL;DR

This paper characterizes the approximation spaces of various bases, including almost greedy bases, using weighted Lorentz spaces, and identifies conditions where approximation spaces and greedy classes coincide.

Contribution

It provides a new characterization of approximation spaces for a broad class of bases in terms of weighted Lorentz spaces and establishes criteria for their equivalence with greedy classes.

Findings

Approximation spaces are characterized by weighted Lorentz spaces.

Necessary and sufficient conditions for approximation spaces and greedy classes to coincide are identified.

The results apply to a broad class of bases, including almost greedy bases.

Abstract

We characterize the approximation spaces of a broad class of bases - which includes almost greedy bases - in terms of weighted Lorentz spaces. For those bases, we also find necessary and sufficient conditions under which the approximation spaces and greedy classes are the same.