Characterization of Weight-Semi-greedy Bases

Abstract

One classical result in greedy approximation theory is that almost-greedy and semi-greedy bases are equivalent in the context of Schauder bases in Banach spaces with finite cotype. This result was proved by Dilworth et al. (Studia Math 159:67–101, 2003) and, recently, in the study of Berná (J Math Anal Appl 470:218–225, 2019), the author proved that the condition of finite cotype can be removed. One of the results in this paper is to show that the condition of Schauder can be relaxed using the $\rho$-admissibility, notion introduced in the study of Berná et al. (Rev Mat Complut; https://doi.org/10.1007/s13163-019-00328-9). On the other hand, in the study of Dilworth et al. (Tr Mat Inst Steklova 303: 120–141, 2018), the authors extend the notion of semi-greediness to the context of weights and proved the following: if w is a weight and $\mathscr {B}$ is a Schauder basis in a Banach space $\mathbb X$ with finite cotype, then w-semi-greediness and w-almost-greediness are equivalent notions. Here, we prove the same characterization but removing the condition of finite cotype. Also, we give some results improving the behavior of some constants in the relation between w-greedy-type bases and some w-democracy properties.