Performance of the Thresholding Greedy Algorithm with Larger Greedy Sums

TL;DR

This paper explores how increasing the size of greedy sums affects the performance of the Thresholding Greedy Algorithm, introducing new concepts of $oldsymbol{\lambda}$-almost and $oldsymbol{\lambda}$-partially greedy bases and analyzing their properties.

Contribution

It introduces the concepts of $oldsymbol{\lambda}$-almost greedy and $oldsymbol{\lambda}$-partially greedy bases, extending classical greedy basis definitions and providing new characterizations and examples.

Findings

A basis is almost greedy iff it is $oldsymbol{\lambda}$-almost greedy for all $oldsymbol{\lambda extgreater 1}$.

Existence of unconditional bases that are $oldsymbol{\lambda}$-partially greedy but not $1$-partially greedy.

Examples of bases that are not almost greedy but are $oldsymbol{\lambda}$-almost greedy for some $oldsymbol{\lambda extgreater 1}$.

Abstract

The goal of this paper is to study the performance of the Thresholding Greedy Algorithm (TGA) when we increase the size of greedy sums by a constant factor $\lambda\geqslant 1$. We introduce the so-called $\lambda$-almost greedy and $\lambda$-partially greedy bases. The case when $\lambda = 1$ gives us the classical definitions of almost greedy and (strong) partially greedy bases. We show that a basis is almost greedy if and only if it is $\lambda$-almost greedy for all (some) $\lambda \geqslant 1$. However, for each $\lambda > 1$, there exists an unconditional basis that is $\lambda$-partially greedy but is not $1$-partially greedy. Furthermore, we investigate and give examples when a basis is 1. not almost greedy with constant $1$ but is $\lambda$-almost greedy with constant $1$ for some $\lambda > 1$, and 2. not strong partially greedy with constant $1$ but is $\lambda$-partially greedy with constant $1$ for some $\lambda > 1$. Finally, we prove various characterizations of different greedy-type bases.