论文标题
Schreier家庭和$ \ Mathcal {F} $ - （几乎）贪婪基地
Schreier families and $\mathcal{F}$-(almost) greedy bases
论文作者
论文摘要
令$ \ mathcal {f} $为$ \ mathbb {n} $的有限子集的遗传集合。在本文中，我们介绍并描述了$ \ Mathcal {f} $ - （几乎）贪婪基地。给定这样一个家庭$ \ nathcal {f} $，一个基础$（e_n）_n $的banach space $ x $称为$ \ nathcal {f} $ - 如果有一个常数$ c \ geqslant 1 $，则每个$ c \ geqslant 1 $，以便每个$ x \ in x $，$ m \ in \ in \ mathbbbbbbbbbbbb in \ n $ g__ g_ g_g_ -g_m（x）\ | \ \ \ \ leqslant \ c \ c \ inf \ left \ {\ left \ | x- \ sum_ { \ Mathbb {k} \ right \}。$ $ $ g_m（x）$是$ x $ $ x $ $ m $的贪婪和$ \ mathbb {k} $是标量字段。从定义来看，任何$ \ MATHCAL {F} $ - 贪婪的基础都是准绿色，因此，成为$ \ Mathcal {f} $ - 贪婪的概念在于贪婪和成为quasi-greedy之间。我们将$ \ MATHCAL {F} $ - 贪婪的基础为$ \ Mathcal {f} $ - 无条件，$ \ Mathcal {f} $ - 脱节民主和Quasi-Greedy，因此概括了Konyagag和Temlyakov的贪婪表征。我们还证明了$ \ mathcal {f} $的类似表征 - 几乎是贪婪的基础。 此外，我们提供了几个基础的示例，这些示例是非琐事$ \ Mathcal {f} $ - 贪婪的示例。对于可数的序数$α$，我们认为$ \ MATHCAL {f} = \ MATHCAL {s}_α$，其中$ \ Mathcal {s}_α$是订单$α$的Schreier家族。我们表明，对于每种$α$，都有一个基础，即$ \ MATHCAL {s}_α$ -Greedy，但不是$ \ Mathcal {s} _ {α+1} $ - 贪婪。换句话说，我们证明没有以下含义可以逆转：对于两个可数序列$α<β$，$$ \ mbox {quasi-greedy} \ \ \ \ \ \ longleftarrow \ \ m arthcal {s}_α_α_α\ mbox {-greedy} {-greedy} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \。 \ Mathcal {s}_β\ mbox {-greedy} \ \ longleftarrow \ \ \ \ \ mbox {greedy}。$$
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 $α$, we consider the case $\mathcal{F}=\mathcal{S}_α$, where $\mathcal{S}_α$ is the Schreier family of order $α$. We show that for each $α$, there is a basis that is $\mathcal{S}_α$-greedy but is not $\mathcal{S}_{α+1}$-greedy. In other words, we prove that none of the following implications can be reversed: for two countable ordinals $α< β$, $$\mbox{quasi-greedy}\ \Longleftarrow\ \mathcal{S}_α\mbox{-greedy}\ \Longleftarrow\ \mathcal{S}_β\mbox{-greedy}\ \Longleftarrow\ \mbox{greedy}.$$
