Greedy-like bases for sequences with gaps

In 2018, Oikhberg introduced and studied variants of the greedy and weak greedy algorithms for sequences with gaps, with a focus on the \({{\textbf {n}}}\)-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. First, we aim to prove that for \({{\textbf {n}}}\) with bounded quotient gaps, \({{\textbf {n}}}\)-t-quasi-greedy bases are quasi-greedy bases. This generalization extends a previous result to the context of Markushevich bases and, also, completes the answer to a question by Oikhberg. 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.