On some characterizations of greedy-type bases

TL;DR

This paper explores and extends characterizations of greedy and almost greedy bases in Banach spaces, introduces a new version of the Weak Thresholding Greedy Algorithm, and demonstrates the equivalence of convergence properties.

Contribution

It provides new characterizations of greedy-type bases and proposes a novel version of the WTGA with proven convergence equivalence.

Findings

Extended several characterizations of greedy bases.

Introduced a new version of the Weak Thresholding Greedy Algorithm.

Proved convergence equivalence of the new WTGA.

Abstract

In 1999, S. V. Konyagin and V. N. Temlyakov introduced the so-called Thresholding Greedy Algorithm. Since then, there have been many interesting and useful characterizations of greedy-type bases in Banach spaces. In this article, we study and extend several characterizations of greedy and almost greedy bases in the literature. Along the way, we give various examples to complement our main results. Furthermore, we propose a new version of the so-called Weak Thresholding Greedy Algorithm (WTGA) and show that the convergence of this new algorithm is equivalent to the convergence of the WTGA.