On Weighted Greedy-type Bases

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.