Trudy Matematicheskogo Instituta imeni V.A. Steklova
 RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Forthcoming papers Archive Impact factor Guidelines for authors License agreement Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Trudy Mat. Inst. Steklova: Year: Volume: Issue: Page: Find

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

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

Full text: PDF file (185 kB)
References: PDF file   HTML file

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

Bibliographic databases:

UDC: 517.518

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, Trudy Mat. Inst. Steklova, 293, MAIK Nauka/Interperiodica, Moscow, 2016, 217–223; Proc. Steklov Inst. Math., 293 (2016), 209–215

Citation in format AMSBIB
\Bibitem{Mal16} \by Yu.~V.~Malykhin \paper Relative widths of Sobolev classes in the uniform and integral metrics \inbook Function spaces, approximation theory, and related problems of mathematical analysis \bookinfo Collected papers. In commemoration of the 110th anniversary of Academician Sergei Mikhailovich Nikol'skii \serial Trudy Mat. Inst. Steklova \yr 2016 \vol 293 \pages 217--223 \publ MAIK Nauka/Interperiodica \publaddr Moscow \mathnet{http://mi.mathnet.ru/tm3715} \crossref{https://doi.org/10.1134/S0371968516020151} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3628481} \elib{https://elibrary.ru/item.asp?id=26344480} \transl \jour Proc. Steklov Inst. Math. \yr 2016 \vol 293 \pages 209--215 \crossref{https://doi.org/10.1134/S0081543816040155} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000380722200015} \elib{https://elibrary.ru/item.asp?id=27120103} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84980002417} 

• http://mi.mathnet.ru/eng/tm3715
• https://doi.org/10.1134/S0371968516020151
• http://mi.mathnet.ru/eng/tm/v293/p217

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
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
3. A. R. Alimov, “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
•  Number of views: This page: 214 Full text: 16 References: 25 First page: 3