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

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.

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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

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
\Bibitem{MalRyu17} \by Yu.~V.~Malykhin, K.~S.~Ryutin \paper The Product of Octahedra is Badly Approximated in the $\ell_{2,1}$-Metric \jour Mat. Zametki \yr 2017 \vol 101 \issue 1 \pages 85--90 \mathnet{http://mi.mathnet.ru/mz11281} \crossref{https://doi.org/10.4213/mzm11281} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3598753} \elib{http://elibrary.ru/item.asp?id=28172127} \transl \jour Math. Notes \yr 2017 \vol 101 \issue 1 \pages 94--99 \crossref{https://doi.org/10.1134/S0001434617010096} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000396392700009} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85015674750} 

• http://mi.mathnet.ru/eng/mz11281
• https://doi.org/10.4213/mzm11281
• http://mi.mathnet.ru/eng/mz/v101/i1/p85

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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
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
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