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Kolmogorov Widths of the Besov Classes \(B^1_{1,\theta}\) and Products of Octahedra

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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 \(\ell_q\).

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Acknowledgments

The author thanks the anonymous referee for useful comments on Theorem 2 and A. I. Tyulenev for pointing out the paper [2].

Funding

This work was supported by a grant of the Government of the Russian Federation (project no. 14.W03.31.0031).

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Authors and Affiliations

  1. Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991, Russia

    Yuri V. Malykhin

  2. Laboratory “High-Dimensional Approximation and Applications,” Lomonosov Moscow State University, Leninskie Gory 1, Moscow, 119991, Russia

    Yuri V. Malykhin

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  1. Yuri V. Malykhin
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Correspondence to Yuri V. Malykhin.

Additional information

Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2021, Vol. 312, pp. 224–235 https://doi.org/10.4213/tm4136.

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Malykhin, Y.V. Kolmogorov Widths of the Besov Classes \(B^1_{1,\theta}\) and Products of Octahedra. Proc. Steklov Inst. Math. 312, 215–225 (2021). https://doi.org/10.1134/S0081543821010132

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  • DOI: https://doi.org/10.1134/S0081543821010132

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