Lebesgue inequalities for the greedy algorithm in general bases

TL;DR

This paper investigates Lebesgue constants associated with the thresholding greedy algorithm in Banach spaces with general bases, providing improved estimates and demonstrating their optimality in certain cases.

Contribution

It offers new, sharper bounds for Lebesgue constants of the greedy algorithm in general bases, extending and refining previous results.

Findings

Establishes optimality of certain Lebesgue constant estimates.

Provides improved bounds compared to earlier literature.

Recovers and slightly enhances existing estimates.

Abstract

We present various estimates for the Lebesgue constants of the thresholding greedy algorithm, in the case of general bases in Banach spaces. We show the optimality of these estimates in some situations. Our results recover and slightly improve various estimates appearing earlier in the literature.