F. Dai, A. Prymak, V. N. Temlyakov, S. Yu. Tikhonov, “Integral norm discretization and related problems”, Russian Math. Surveys, 74:4 (2019), 579

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Russian Mathematical Surveys, 2019, Volume 74, Issue 4, Pages 579–630
DOI: https://doi.org/10.1070/RM9892 (Mi rm9892)

This article is cited in 39 scientific papers (total in 39 papers)

Integral norm discretization and related problems

F. Dai^a, A. Prymak^b, V. N. Temlyakov^cde, S. Yu. Tikhonov^fgh

^a University of Alberta, Edmonton, Canada
^b University of Manitoba, Winnipeg, Canada
^c University of South Carolina, Columbia, USA
^d Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
^e Lomonosov Moscow State University
^f Centre de Recerca Matemàtica, Barcelona, Spain
^g ICREA, Barcelona, Spain
^h Universitat Autònoma de Barcelona, Barcelona, Spain

English version PDF (887 kB) Citations (39) Russian version article

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DOI: https://doi.org/10.1070/RM9892

Abstract: The problem is discussed of replacing an integral norm with respect to a given probability measure by the corresponding integral norm with respect to a discrete measure. This problem is investigated for elements of finite-dimensional spaces. Also, discretization of the uniform norm of functions in a given finite-dimensional subspace of continuous functions is studied. Special attention is given to the case of multivariate trigonometric polynomials with frequencies (harmonics) in a finite set with fixed cardinality. Both new results and a survey of known results are presented.
Bibliography: 47 titles.

Keywords: trigonometric polynomials, discretization, Marcinkiewicz-type theorems.

Funding agency	Grant number
Natural Sciences and Engineering Research Council of Canada (NSERC)	RGPIN 04702-15 RGPIN 04863-15
Ministry of Education and Science of the Russian Federation	14.W03.31.0031
Ministerio de Ciencia e Innovación de España	MTM 2017-87409-P 2017 SGR 358
Generalitat de Catalunya
The first author's research was partially supported by NSERC of Canada Discovery Grant RGPIN 04702-15. The second author's research was partially supported by NSERC of Canada Discovery Grant RGPIN 04863-15. The third author's research was supported by the Russian Federation Government Grant No. 14.W03.31.0031. The fourth author's research was partially supported by MTM 2017-87409-P, 2017 SGR 358, and the CERCA Programme of the Generalitat de Catalunya.

Received: 20.12.2018

Bibliographic databases:

Document Type: Article

UDC: 517.5

MSC: 65J05, 65D32, 41A55, 41A17, 41A63, 42A10, 42B99

Language: English

Original paper language: Russian

Citation: F. Dai, A. Prymak, V. N. Temlyakov, S. Yu. Tikhonov, “Integral norm discretization and related problems”, Russian Math. Surveys, 74:4 (2019), 579–630

Citation in format AMSBIB

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\jour Russian Math. Surveys

\yr 2019

\vol 74

\issue 4

\pages 579--630

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Linking options:

https://www.mathnet.ru/eng/rm9892

https://doi.org/10.1070/RM9892

https://www.mathnet.ru/eng/rm/v74/i4/p3

This publication is cited in the following 39 articles:

Michael I. Ganzburg, “Discretization Theorems for Entire Functions of Exponential Type”, Journal of Mathematical Analysis and Applications, 2025, 129510
A. P. Solodov, V. N. Temlyakov, “Vosstanovlenie po znacheniyam v tochkakh v funktsionalnykh klassakh so strukturnym usloviem”, Matem. zametki, 117:4 (2025), 543–560
Egor Kosov, Sergey Tikhonov, “Sampling discretization in Orlicz spaces”, Journal of Functional Analysis, 2025, 110971
E. Kosov, V. Temlyakov, “Sampling discretization of the uniform norm and applications”, Journal of Mathematical Analysis and Applications, 2024, 128431
F. Dai, V. Temlyakov, “Random points are good for universal discretization”, J. Math. Anal. Appl., 529:1 (2024), 127570–28
I. V. Limonova, Yu. V. Malykhin, V. N. Temlyakov, “One-sided discretization inequalities and sampling recovery”, Russian Math. Surveys, 79:3 (2024), 515–545
András Kroó, “On Bernstein-Markov property for multivariate polynomials”, Journal of Mathematical Analysis and Applications, 2024, 129179
Lars Becker, Ohad Klein, Joseph Slote, Alexander Volberg, Haonan Zhang, “Dimension-free discretizations of the uniform norm by small product sets”, Invent. math., 2024
V. N. Temlyakov, “Sparse sampling recovery in integral norms on some function classes”, Sb. Math., 215:10 (2024), 1406–1425
B. Kashin, S. Konyagin, V. Temlyakov, “Sampling discretization of the uniform norm”, Constr. Approx., 57:2 (2023), 663
Y. Xu, A. Narayan, “Randomized weakly admissible meshes”, Journal of Approximation Theory, 285 (2023), 105835
F. Dai, V. Temlyakov, “Universal sampling discretization”, Constr. Approx., 58 (2023), 589–-613
D. Freeman, D. Ghoreishi, “Discretizing $L_p$ norms and frame theory”, Journal of Mathematical Analysis and Applications, 519:2 (2023), 126846
F. Dai, E. Kosov, V. Temlyakov, “Some improved bounds in sampling discretization of integral norms”, Journal of Functional Analysis, 285:4 (2023), 109951
F. Dai, A. Prymak, “Optimal polynomial meshes exist on any multivariate convex domain”, Found Comput. Math., 2023
V. N. Temlyakov, “Sampling discretization error of integral norms for function classes with small smoothness”, Journal of Approximation Theory, 293 (2023), 105913
V. N. Temlyakov, “On Universal Sampling Recovery in the Uniform Norm”, Proc. Steklov Inst. Math., 323 (2023), 206–216
F. Dai, A. Prymak, “Polynomial approximation on $C^2$ -domains”, Constr. Approx., 2023
A. Kroó, “On discretizing integral norms of exponential sums”, J. Math. Anal. Appl., 507:2 (2022), 125770, 18 pp.
A. Kroó, “On discretizing uniform norms of exponential sums”, Constr. Approx., 56 (2022), 45–73

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

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Abstract page:	846
Russian version PDF:	145
English version PDF:	92
References:	103
First page:	53

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