Weight-almost greedy bases

TL;DR

This paper introduces the concepts of weight-almost greedy and weight-semi-greedy bases in Banach spaces, characterizing their properties and relationships with quasi-greedy and democratic bases, with implications depending on the space's cotype.

Contribution

It defines new classes of bases in Banach spaces and establishes their equivalences and conditions, expanding the understanding of greedy-type bases with weights.

Findings

w-almost greedy bases are characterized by being quasi-greedy and w-democratic.

w-almost greedy bases are w-semi-greedy.

The equivalence between w-semi-greedy and w-almost greedy holds in spaces with finite cotype.

Abstract

We introduce the notion of a \textit{weight-almost greedy} basis and show that a basis for a real Banach space is $w$-almost greedy if and only if it is both quasi-greedy and $w$-democratic. We also introduce the notion of \textit{weight-semi-greedy} basis and show that a $w$-almost greedy basis is $w$-semi-greedy and that the converse holds if the Banach space has finite cotype.