1 results
Schreier families and
$\mathcal {F}$-(almost) greedy bases
- Part of
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- Journal:
- Canadian Journal of Mathematics / Volume 76 / Issue 4 / August 2024
- Published online by Cambridge University Press:
- 14 June 2023, pp. 1379-1399
- Print publication:
- August 2024
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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 Here,
$$ \begin{align*} \|x - G_m(x)\|\ \leqslant\ C \inf\left\{\left\|x-\sum_{n\in A}a_ne_n\right\|\,:\, |A|\leqslant m, A\in \mathcal{F}, (a_n)\subset \mathbb{K}\right\}. \end{align*} $$
$G_m(x)$ 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 $, 
$$ \begin{align*} \mbox{quasi-greedy}\ \Longleftarrow\ \mathcal{S}_{\alpha}\mbox{-greedy}\ \Longleftarrow\ \mathcal{S}_{\beta}\mbox{-greedy}\ \Longleftarrow\ \mbox{greedy}. \end{align*} $$
