On Weighted Greedy-type Bases

TL;DR

This paper extends the theory of greedy bases by introducing a general weight framework, characterizing $oldsymbol{ ext{omega}}$-(almost) greedy bases, and establishing new equivalences and distinctions among these classes.

Contribution

It introduces a general weight scheme for greedy algorithms, characterizes $oldsymbol{ ext{omega}}$-(almost) greedy bases, and proves new equivalences and separations in the weighted greedy basis theory.

Findings

Existence of an $oldsymbol{ ext{omega}}$-greedy unconditional basis not $oldsymbol{ ext{varsigma}}$-almost greedy for any sequence.

Equivalence between $oldsymbol{ ext{omega}}$-semi-greedy and $oldsymbol{ ext{omega}}$-almost greedy bases under structured weights.

Extension of known equivalences from sequential to structured weights.

Abstract

In this paper, we study weights for the Thresholding Greedy Algorithm (TGA). While previous work focused on sequential weights $\varsigma = (s_n)_{n\in\mathbb{N}}$ on each positive integer, we study a more general weight $\omega = (w_A)_{A\subset\mathbb{N}}$ on each set $A\subset \mathbb{N}$. We define and characterize $\omega$-(almost) greedy bases. Furthermore, we leverage existing results to show that there exists an $\omega$-greedy unconditional basis that is not $\varsigma$-almost greedy for any weight sequence $\varsigma$. Last but not least, we show the equivalence between $\omega$-semi-greedy bases and $\omega$-almost greedy bases when $\omega$ is a so-called structured weight, thus considerably extending the equivalence previously known to hold for sequential weights.