Greedy-like bases for sequences with gaps

TL;DR

This paper investigates greedy algorithms for sequences with gaps, proving that bounded quotient gaps imply quasi-greedy bases and extending approximation properties to these sequences, thereby advancing the understanding of greedy algorithms in this context.

Contribution

It generalizes previous results to Markushevich bases and addresses open questions about greedy algorithms for sequences with gaps.

Findings

Bounded quotient gaps imply $ $-$t$-quasi-greedy bases are quasi-greedy.

Extended approximation properties of greedy algorithms to sequences with gaps.

Established relationships between new extensions and usual convergence.

Abstract

In [25], T. Oikhberg introduced and studied variants of the greedy and weak greedy algorithms for sequences with gaps, with a focus on the $\mathbf n$-$t$-quasi-greedy property that is based on them. Building upon this foundation, our current work aims to further investigate these algorithms and bases while introducing new ideas for two primary purposes. Firstly, we aim to prove that for $\mathbf n$ with bounded quotient gaps, $\mathbf n$-$t$-quasi-greedy bases are quasi-greedy bases. This generalization extends the result previously established in [7] to the context of Markushevich bases and, also, completes the answer to a question from [25]. The second objective is to extend certain approximation properties of the greedy algorithm to the context of sequences with gaps and study if there is a relationship between this new extension and the usual convergence.