On Weighted Greedy-Type Bases

We study weights for the thresholding greedy algorithm, aiming to extend previous work on sequential weights sigma on N to weights omega on P(N). We revisit major results on weighted greedy-type bases in this new setting including characterizations of omega-(almost) greedy bases and the equivalence between omega-semi-greedy bases and omega-almost greedy bases. Some new results are encountered along the way. For example, we show that there exists an omega-greedy unconditional basis that is not sigma-almost greedy for any weight sequence sigma. Moreover, a basis is unconditional if and only if it is omega-greedy for some weight omega. Similarly, a basis is quasi-greedy if and only if it is omega-almost greedy for some weight omega.