Abstract

We find the decay orders of the Kolmogorov widths of some Besov classes related to \(W^1_1\) (the behavior of the widths for the class \(W^1_1\) remains unknown): \(d_n(B^1_{1,\theta}[0,1],L_q[0,1])\asymp n^{-1/2}\log^{\max\{1/2,1-1/\theta\}}n\) for \(2<q<\infty\) and \(1\le\theta\le\infty\). The proof relies on the lower bound for the width of a product of octahedra in a special norm (maximum of two weighted \(\ell_{q_i}\) norms). This bound generalizes B. S. Kashin’s theorem on the widths of octahedra in \(\ell_q\).

中文翻译：

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Besov类$$ B ^ 1_ {1，\ theta} $$的Kolmogorov宽度和八面体的乘积

摘要

我们发现相关的一些Besov类的柯尔莫哥洛夫宽度的衰减订单\（W ^ 1_1 \） （宽度为类的行为\（W ^ 1_1 \）依然是未知）：\（D_N（B ^ 1_ { 1，\ theta} [0,1]，L_q [0,1]）\渐近n ^ {-1/2} \ log ^ {\ max \ {1 / 2,1-1 / \ theta \}} n \）表示\（2 <q <\ infty \）和\（1 \ le \ theta \ le \ infty \）。该证明依赖于八面体乘积的宽度的下界，该下界是一个特殊的规范（两个加权\（\ ell_ {q_i} \）的最大值）。此边界将BS Kashin定理推广到\（\ ell_q \）中八面体的宽度。