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Mat. Zametki, 2017, Volume 101, Issue 1, Pages 85–90 (Mi mz11281)  

This article is cited in 3 scientific papers (total in 3 papers)

The Product of Octahedra is Badly Approximated in the $\ell_{2,1}$-Metric

Yu. V. Malykhina, K. S. Ryutinb

a Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
b Lomonosov Moscow State University

Abstract: We prove that the Cartesian product of octahedra $B_{1,\infty}^{n,m}=B_1^n\times …\times B_1^n$ ($m$ factors) is poorly approximated by spaces of half dimension in the mixed norm: $d_{N/2}(B_{1,\infty}^{n,m},\ell_{2,1}^{n,m})\ge cm$, $N=mn$. As a corollary, we find the order of linear widths of the Hölder–Nikolskii classes $H^r_p(\mathbb T^d)$ in the metric of $L_q$ in certain domains of variation of the parameters $(p,q)$.

Keywords: Kolmogorov width, vector balancing.

Funding Agency Grant Number
Russian Foundation for Basic Research 14-01-00332
This work was supported by the Russian Foundation for Basic Research under grant 14-01-00332.


DOI: https://doi.org/10.4213/mzm11281

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English version:
Mathematical Notes, 2017, 101:1, 94–99

Bibliographic databases:

ArXiv: 1606.00738
UDC: 517.5
Received: 09.06.2016

Citation: Yu. V. Malykhin, K. S. Ryutin, “The Product of Octahedra is Badly Approximated in the $\ell_{2,1}$-Metric”, Mat. Zametki, 101:1 (2017), 85–90; Math. Notes, 101:1 (2017), 94–99

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. A. S. Romanyuk, “Trigonometric and linear widths for the classes of periodic multivariate functions”, Ukr. Math. J., 69:5 (2017), 782–795  crossref  isi  scopus
    2. G. Byrenheid, T. Ullrich, “Optimal sampling recovery of mixed order Sobolev embeddings via discrete Littlewood-Paley type characterizations”, Anal. Math., 43:2 (2017), 133–191  crossref  mathscinet  zmath  isi  scopus
    3. S. Dirksen, T. Ullrich, “Gelfand numbers related to structured sparsity and Besov space embeddings with small mixed smoothness”, J. Complexity, 48 (2018), 69–102  crossref  mathscinet  zmath  isi  scopus
  • Математические заметки Mathematical Notes
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