Bernstein昏睡定理的子空间条件
Subspace Condition for Bernstein's Lethargy Theorem
摘要：
本文考虑子空间上的一个条件，以改进Banach空间中Bernstein的Lethargy定理（BLT）中给出的界。设$d_1 \geq d_2 \geq \dots d_n \geq \dots > 0$是收敛于$0$的无穷数列，$Y_1 \subset Y_2 \subset \dots\subset Y_n \subset \dots \subset X$是Banach空间$X$中的闭套子空间序列，且对所有的$n\ge1$都有$\overline{Y}_{n}\subset Y_{n+1}$.我们证明，对于任何$c \in (0,1]$，存在一个元素$x_c \in X$，使$$ c d_n \leq ρ(x_c, Y_n) \leq \min (4, \tilde{a}) c\, d_n. $$在这里，$ρ(x, Y_n)= \inf \{ ||x-y||: \,\,y\in Y_n\}$，$$\tilde{a} =\sup_{i\ge1}\sup_{\left \{ q_{i} \right \}}\left \{ a_{n_{i+1}-1}^{-3}\right \}$$，其中序列$\{a_n\}$定义为：对于所有的$ n \geq 1 $，$$ a_n = \inf_{l \geq n} \, \inf_{q \in \langle q_l, q_{l+1},\dots \rangle} \frac{ρ(q,Y_l)}{||q||} $$，其中每个点$q_n$取自$Y_{n+1} \setminus Y_{n}$，并满足$\inf\limits_{n\ge1} a_n > 0$。序列$\{n_i\}_{i\ge1}$由定理\ref{100}，$\{n_i\}$满足（\ref{ni}）和$n_{i}\leq n<n_{i+1}$给出。$$ n_1=1;~n_{i+1}= \min \left \{ n\ge1 : \frac{d_n}{{a_{n}^{2}}} \leq d_{n_{i}}\right \},~i\geq 1. $$
英文摘要：
In this paper, we consider a condition on subspaces in order to improve bounds given in the Bernstein's Lethargy Theorem (BLT) for Banach spaces. Let $d_1 \geq d_2 \geq \dots d_n \geq \dots > 0$ be an infinite sequence of numbers converging to $0$, and let $Y_1 \subset Y_2 \subset \dots\subset Y_n \subset \dots \subset X$ be a sequence of closed nested subspaces in a Banach space $X$ with the property that $\overline{Y}_{n}\subset Y_{n+1}$ for all $n\ge1$. We prove that for any $c \in (0,1]$, there exists an element $x_c \in X$ such that $$ c d_n \leq ρ(x_c, Y_n) \leq \min (4, \tilde{a}) c\, d_n. $$ Here, $ρ(x, Y_n)= \inf \{ ||x-y||: \,\,y\in Y_n\}$, $$\tilde{a} =\sup_{i\ge1}\sup_{\left \{ q_{i} \right \}}\left \{ a_{n_{i+1}-1}^{-3}\right \}$$ where the sequence $\{a_n\}$ is defined as: for all $ n \geq 1 $, $$ a_n = \inf_{l \geq n} \, \inf_{q \in \langle q_l, q_{l+1},\dots \rangle} \frac{ρ(q,Y_l)}{||q||} $$ in which each point $q_n$ is taken from $Y_{n+1} \setminus Y_{n}$, and satisfies $\inf\limits_{n\ge1} a_n > 0$. The sequence $\{n_i\}_{i\ge1}$ is given by %Theorem \ref{100}, $\{n_i\}$ satisfying (\ref{ni}) and $n_{i}\leq n<n_{i+1}$. $$ n_1=1;~n_{i+1}= \min \left \{ n\ge1 : \frac{d_n}{{a_{n}^{2}}} \leq d_{n_{i}}\right \},~i\geq 1. $$
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