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Tr. Mat. Inst. Steklova, 2016, Volume 293, Pages 217–223 (Mi tm3715)  

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

Relative widths of Sobolev classes in the uniform and integral metrics

Yu. V. Malykhin

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

Abstract: Let $W^r_p$ be the Sobolev class consisting of $2\pi$-periodic functions $f$ such that $\|f^{(r)}\|_p\le1$. We consider the relative widths $d_n(W^r_p,MW^r_p,L_p)$, which characterize the best approximation of the class $W^r_p$ in the space $L_p$ by linear subspaces for which (in contrast to Kolmogorov widths) it is additionally required that the approximating functions $g$ should lie in $MW^r_p$, i.e., $\|g^{(r)}\|_p\le M$. We establish estimates for the relative widths in the cases of $p=1$ and $p=\infty$; it follows from these estimates that for almost optimal (with error at most $Cn^{-r}$, where $C$ is an absolute constant) approximations of the class $W^r_p$ by linear $2n$-dimensional spaces, the norms of the $r$th derivatives of some approximating functions are not less than $c\ln\min(n,r)$ for large $n$ and $r$.

Funding Agency Grant Number
Russian Science Foundation 14-50-00005
This work is supported by the Russian Science Foundation under grant 14-50-00005.


DOI: https://doi.org/10.1134/S0371968516020151

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English version:
Proceedings of the Steklov Institute of Mathematics, 2016, 293, 209–215

Bibliographic databases:

UDC: 517.518
Received: October 7, 2015

Citation: Yu. V. Malykhin, “Relative widths of Sobolev classes in the uniform and integral metrics”, Function spaces, approximation theory, and related problems of mathematical analysis, Collected papers. In commemoration of the 110th anniversary of Academician Sergei Mikhailovich Nikol'skii, Tr. Mat. Inst. Steklova, 293, MAIK Nauka/Interperiodica, Moscow, 2016, 217–223; Proc. Steklov Inst. Math., 293 (2016), 209–215

Citation in format AMSBIB
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\paper Relative widths of Sobolev classes in the uniform and integral metrics
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\bookinfo Collected papers. In commemoration of the 110th anniversary of Academician Sergei Mikhailovich Nikol'skii
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\vol 293
\pages 217--223
\publ MAIK Nauka/Interperiodica
\publaddr Moscow
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    This publication is cited in the following articles:
    1. A. R. Alimov, “Selections of the metric projection operator and strict solarity of sets with continuous metric projection”, Sb. Math., 208:7 (2017), 915–928  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    2. A. A. Vasil'eva, “Widths of weighted {S}obolev classes with constraints $f(a)=\cdots= f^{(k-1)}(a)=f^{(k)}(b)=\cdots=f^{(r-1)}(b)=0$ and the spectra of nonlinear differential equations”, Russ. J. Math. Phys., 24:3 (2017), 376–398  crossref  mathscinet  zmath  isi  scopus
    3. Alimov A.R., “Continuity of the Metric Projection and Local Solar Properties of Sets: Continuity of the Metric Projection and Solar Properties”, Set-Valued Var. Anal., 27:1 (2019), 213–222  crossref  mathscinet  zmath  isi  scopus
  • Труды Математического института им. В. А. Стеклова Proceedings of the Steklov Institute of Mathematics
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