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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2022, Volume 319, Pages 106–119
DOI: https://doi.org/10.4213/tm4258 (Mi tm4258) 		 
This article is cited in 4 scientific papers (total in 4 papers)

Sampling Discretization of Integral Norms and Its Application

F. Daia, V. N. Temlyakovbcde

a Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, T6G 2G1, Canada
b University of South Carolina, Columbia, SC 29208, USA
c Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia
d Lomonosov Moscow State University, Moscow, 119991 Russia
e Moscow Center for Fundamental and Applied Mathematics, Moscow, 119991 Russia
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DOI: https://doi.org/10.4213/tm4258
Abstract: The paper addresses a problem of sampling discretization of integral norms of elements of finite-dimensional subspaces satisfying some conditions. We prove sampling discretization results under two standard kinds of assumptions: conditions on the entropy numbers and conditions in terms of Nikol'skii-type inequalities. We prove some upper bounds on the number of sample points sufficient for good discretization and show that these upper bounds are sharp in a certain sense. Then we apply our general conditional results to subspaces with special structure, namely, subspaces with tensor product structure. We demonstrate that the application of theorems based on Nikol'skii-type inequalities provides somewhat better results than the application of theorems based on entropy numbers conditions. Finally, we apply discretization results to the problem of sampling recovery.
Keywords: sampling discretization, entropy numbers, Nikol'skii inequality, recovery.
Funding agency Grant number
Natural Sciences and Engineering Research Council of Canada (NSERC) RGPIN-2020-03909
Ministry of Education and Science of the Russian Federation 14.W03.31.0031
The first named author's research was partially supported by the NSERC (Canada) Discovery grant RGPIN-2020-03909. The second named author's research was supported by the Government of the Russian Federation, grant no. 14.W03.31.0031.
Received: September 18, 2021
Revised: January 21, 2022
Accepted: February 15, 2022
English version:
Proceedings of the Steklov Institute of Mathematics, 2022, Volume 319, Pages 97–109
DOI: https://doi.org/10.1134/S0081543822050091
Bibliographic databases:
Document Type: Article
UDC: 517.5
MSC: Primary 65J05; Secondary 42A05, 65D30, 41A63
Language: Russian
Citation: F. Dai, V. N. Temlyakov, “Sampling Discretization of Integral Norms and Its Application”, Approximation Theory, Functional Analysis, and Applications, Collected papers. On the occasion of the 70th birthday of Academician Boris Sergeevich Kashin, Trudy Mat. Inst. Steklova, 319, Steklov Math. Inst., Moscow, 2022, 106–119; Proc. Steklov Inst. Math., 319 (2022), 97–109
Citation in format AMSBIB
\Bibitem{DaiTem22}
\by F.~Dai, V.~N.~Temlyakov
\paper Sampling Discretization of Integral Norms and Its Application
\inbook Approximation Theory, Functional Analysis, and Applications
\bookinfo Collected papers. On the occasion of the 70th birthday of Academician Boris Sergeevich Kashin
\serial Trudy Mat. Inst. Steklova
\yr 2022
\vol 319
\pages 106--119
\publ Steklov Math. Inst.
\publaddr Moscow
\mathnet{http://mi.mathnet.ru/tm4258}
\crossref{https://doi.org/10.4213/tm4258}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=4563388}
\transl
\jour Proc. Steklov Inst. Math.
\yr 2022
\vol 319
\pages 97--109
\crossref{https://doi.org/10.1134/S0081543822050091}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85148470301}
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    • This publication is cited in the following 4 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    		Труды Математического института имени В. А. Стеклова Proceedings of the Steklov Institute of Mathematics
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