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.

中文翻译：

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重贪婪基数的表征

贪婪逼近理论的一个经典结果是，在有限共型的Banach空间中，几乎贪婪和半贪婪的碱基在Schauder碱基的上下文中是等效的。Dilworth等人证明了这一结果。（Studia Math 159：67–101，2003），以及最近在Berná的研究中（J Math Anal Appl 470：218–225，2019），作者证明了可以消除有限共型的条件。本文的结果之一是表明可以使用\（\ rho \）放宽Schauder的条件-可采性，在Berná等人的研究中引入的概念。（Rev Mat Complut； https：//doi.org/10.1007/s13163-019-00328-9）。另一方面，在Dilworth等人的研究中。（Tr Mat Inst Steklova 303：120–141，2018年），作者将半贪婪的概念扩展到权重的上下文，并证明了以下内容：如果w是权重，\（\ mathscr {B} \）是一个权重具有有限共型的Banach空间\（\ mathbb X \）中的Schauder基，则w-半贪婪和w-几乎贪婪是等价的。在这里，我们证明了相同的特征，但是消除了有限共型的条件。另外，我们给出了一些结果，可以改善w与w之间的关系中某些常数的行为。-贪婪类型的基地和一些w-民主性质。