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arXiv · 2205.06839

On Weighted Greedy-type Bases

Hung Viet Chu
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Abstract

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

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Hung Viet Chu. 2023-02-09. On Weighted Greedy-type Bases. https://arxiv.org/abs/2205.06839

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