Lebesgue-type estimates for greedy algorithms in quasi-Banach spaces

TL;DR

This paper advances the understanding of Lebesgue-type parameters in quasi-Banach spaces by introducing new parameters, establishing bounds for semi-greedy bases, and resolving open questions about bounds in greedy algorithms.

Contribution

It introduces a new Lebesgue parameter for strong partially greedy bases and provides improved bounds for semi-greedy bases, addressing open problems in the field.

Findings

New Lebesgue parameter for strong partially greedy bases

Upper bounds for semi-greedy bases using quasi-greedy and squeeze symmetry parameters

Resolved open questions on optimal power bounds in greedy approximation

Abstract

We continue the study of Lebesgue-type parameters for various greedy algorithms in quasi-Banach spaces. First, we introduce a parameter that can be used with the quasi-greedy parameter to obtain the exact growth of the Lebesgue parameter for strong partially greedy bases. Second, we establish a new upper bound for the Lebesgue parameter for semi-greedy bases using the quasi-greedy and the squeeze symmetry parameters. Finally, we answer several open questions regarding the optimal power in various bounds proved in [F. Albiac, J. L. Ansorena, and P. M. Bern\'a, New parameters and Lebesgue-type estimates in greedy approximation, Forum Math. Sigma 10 (2022), 1-39].