Schreier families and $\mathcal{F}$-(almost) greedy bases

TL;DR

This paper introduces and characterizes $$-(almost) greedy bases in Banach spaces, generalizing classical greedy basis theory by incorporating hereditary collections like Schreier families, and provides examples illustrating the hierarchy of these bases.

Contribution

It generalizes the concept of greedy bases using hereditary collections, characterizes $$-greedy bases, and explores the hierarchy of $$-greedy bases related to Schreier families.

Findings

Characterization of $$-greedy bases via $$-unconditionality and $$-disjoint democracy.

Existence of bases that are $$-greedy for certain Schreier families but not for others.

Hierarchy of $$-greedy bases showing strict inclusions between different Schreier families.

Abstract

Let $\mathcal{F}$ be a hereditary collection of finite subsets of $\mathbb{N}$. In this paper, we introduce and characterize $\mathcal{F}$-(almost) greedy bases. Given such a family $\mathcal{F}$, a basis $(e_n)_n$ for a Banach space $X$ is called $\mathcal{F}$-greedy if there is a constant $C\geqslant 1$ such that for each $x\in X$, $m \in \mathbb{N}$, and $G_m(x)$, we have $$\|x - G_m(x)\|\ \leqslant\ C \inf\left\{\left\|x-\sum_{n\in A}a_ne_n\right\|\,:\, |A|\leqslant m, A\in \mathcal{F}, (a_n)\subset \mathbb{K}\right\}.$$ Here $G_m(x)$ is a greedy sum of $x$ of order $m$, and $\mathbb{K}$ is the scalar field. From the definition, any $\mathcal{F}$-greedy basis is quasi-greedy and so, the notion of being $\mathcal{F}$-greedy lies between being greedy and being quasi-greedy. We characterize $\mathcal{F}$-greedy bases as being $\mathcal{F}$-unconditional, $\mathcal{F}$-disjoint democratic, and quasi-greedy, thus generalizing the well-known characterization of greedy bases by Konyagin and Temlyakov. We also prove a similar characterization for $\mathcal{F}$-almost greedy bases. Furthermore, we provide several examples of bases that are nontrivially $\mathcal{F}$-greedy. For a countable ordinal $\alpha$, we consider the case $\mathcal{F}=\mathcal{S}_\alpha$, where $\mathcal{S}_\alpha$ is the Schreier family of order $\alpha$. We show that for each $\alpha$, there is a basis that is $\mathcal{S}_{\alpha}$-greedy but is not $\mathcal{S}_{\alpha+1}$-greedy. In other words, we prove that none of the following implications can be reversed: for two countable ordinals $\alpha < \beta$, $$\mbox{quasi-greedy}\ \Longleftarrow\ \mathcal{S}_\alpha\mbox{-greedy}\ \Longleftarrow\ \mathcal{S}_\beta\mbox{-greedy}\ \Longleftarrow\ \mbox{greedy}.$$