Performance of the Thresholding Greedy Algorithm with Larger Greedy Sums
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 socalled $\lambda$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 greedytype bases.
 Publication:

arXiv eprints
 Pub Date:
 April 2022
 DOI:
 10.48550/arXiv.2205.00268
 arXiv:
 arXiv:2205.00268
 Bibcode:
 2022arXiv220500268V
 Keywords:

 Mathematics  Functional Analysis;
 41A65;
 46B15
 EPrint:
 23 pages. Version 02: edited based on an anonymous referee's suggestions