On Weighted Greedy-Type Bases - Chu, Hùng Việt (Boletim da Sociedade Brasileira de Matemática, 2023)
We study weights for the thresholding greedy algorithm, aiming to extend previous work on sequential weights ς on N to weights ω on P ( N ) . We revisit major results on weighted greedy-type bases in this new setting including characterizations of ω -(almost) greedy bases and the equivalence between ω -semi-greedy bases and ω -almost greedy bases. Some new results are encountered along the way. For example, we show that there exists an ω -greedy unconditional basis that is not ς -almost greedy for any weight sequence ς . Moreover, a basis is unconditional if and only if it is ω -greedy for some weight ω . Similarly, a basis is quasi-greedy if and only if it is ω -almost greedy for some weight ω .