On Approximation Spaces and Greedy-type Bases

TL;DR

This paper introduces new classes of greedy approximation methods in generalized approximation spaces, establishing their properties and relations, and extends several existing results in the theory of greedy bases.

Contribution

It defines $$-Chebyshev-greedy and $$-partially greedy classes, explores their connection with $$-approximation spaces, and generalizes key results on semi-greedy bases.

Findings

Established conditions for embeddings implying greedy properties.

Generalized a result by P. Wojtaszczyk.

Characterized semi-greedy Schauder bases in quasi-Banach spaces.

Abstract

The purpose of this paper is to introduce $\omega$-Chebyshev-greedy and $\omega$-partially greedy approximation classes and to study their relation with $\omega$-approximation spaces, where the latter are a generalization of the classical approximation spaces. The relation gives us sufficient conditions of when certain continuous embeddings imply different greedy-type properties. Along the way, we generalize a result by P. Wojtaszczyk as well as characterize semi-greedy Schauder bases in quasi-Banach spaces, generalizing a previous result by the first author.