New parameters and Lebesgue-type estimates in greedy approximation

TL;DR

This paper introduces new parameters to precisely estimate Lebesgue constants in greedy algorithms, addressing a question about their growth in Banach spaces with theoretical and computational insights.

Contribution

It develops a novel sequence of parameters that, combined with unconditionality measures, accurately characterizes the growth of Lebesgue constants in greedy approximation.

Findings

New parameters effectively modulate Lebesgue constants

Answer to Temlyakov's 2011 question on greedy parameters

Theoretical and computational validation of the approach

Abstract

The purpose of this paper is to quantify the size of the Lebesgue constants $(L_m)_{m=1}^{\infty}$ associated with the thresholding greedy algorithm in terms of a new generation of parameters that modulate accurately some features of a general basis. This fine-tuning of constants allows us to provide an answer to the question raised by Temlyakov in 2011 to find a natural sequence of greedy-type parameters for arbitrary bases in Banach (or quasi-Banach) spaces which combined linearly with the sequence of unconditionality parameters $(k_m)_{m=1}^{\infty}$ determines the growth of $(L_m)_{m=1}^{\infty}$. Multiple theoretical applications and computational examples complement our study.