Variations of Property (A) Constants and Lebesgue-type Inequalities for the Weak Thresholding Greedy Algorithms

TL;DR

This paper extends the concept of property (A) constants to (A,τ)-property constants, providing new Lebesgue-type inequalities for weak thresholding greedy algorithms and analyzing their relationships with other constants.

Contribution

It introduces (A,τ)-property constants, derives new Lebesgue inequalities for weak greedy algorithms, and explores their relations with existing constants.

Findings

New estimates for Lebesgue parameters using (A,τ)-property constants

Relationships among (A,τ)-property constants and classical constants

Extended characterization of greedy bases through generalized property constants

Abstract

Albiac and Wojtaszczyk introduced property (A) to characterize $1$-greedy bases. Later, Dilworth et al. generalized the concept to $C$-property (A), where the case $C = 1$ gives property (A). They (among other results) characterized greedy bases by unconditionality and $C$-property (A). In this paper, we extend the definition of the so-called A-property constant to (A,$\tau$)-property constants and use the extension to obtain new estimates for various Lebesgue parameters. Furthermore, we study the relation among (A,$\tau$)-property constants and other well-known constants when $\tau$ varies.