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Sampling Discretization of Integral Norms

Abstract

The paper is devoted to discretization of integral norms of functions from a given finite dimensional subspace. Even though this problem is extremely important in applications, its systematic study has begun only recently. In this paper we obtain a conditional theorem for all integral norms \(L_q\), \(1\le q<\infty \), which is an extension of known results for \(q=1\). To discretize the integral norms successfully, we introduce a new technique, which is a combination of a probabilistic technique with results on the entropy numbers in the uniform norm. As an application of the general conditional theorem, we derive a new Marcinkiewicz-type discretization for the multivariate trigonometric polynomials with frequencies from the hyperbolic crosses.

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Acknowledgements

The work was supported by the Russian Federation Government Grant No14.W03.31.0031. The paper contains results obtained in frames of the program “Center for the storage and analysis of big data”, supported by the Ministry of Science and High Education of Russian Federation (contract 11.12.2018No13/1251/2018 between the Lomonosov Moscow State University and the Fund of support of the National technological initiative projects). The first named author’s research was partially supported by NSERC of Canada Discovery Grant RGPIN-2020-03909. The second named author’s research was partially supported by NSERC of Canada Discovery Grant RGPIN-2020-05357. The fourth named author’s research was supported by the Russian Federation Government Grant No. 14.W03.31.0031. The fifth named author’s research was partially supported by MTM 2017-87409-P, 2017 SGR 358, MON RK AP08856479, and the CERCA Programme of the Generalitat de Catalunya. The last two authors were supported by the Simons Foundation and Pembroke College visiting scholar grants.

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Correspondence to S. Tikhonov.

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Communicated by Wolfgang Dahmen.

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Dai, F., Prymak, A., Shadrin, A. et al. Sampling Discretization of Integral Norms. Constr Approx 54, 455–471 (2021). https://doi.org/10.1007/s00365-021-09539-0

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  • DOI: https://doi.org/10.1007/s00365-021-09539-0

Keywords

  • Marcinkiewicz-type discretization
  • Entropy numbers
  • Hyperbolic cross polynomials

Mathematics Subject Classification

  • Primary 65J05
  • Secondary 42A05
  • 65D30
  • 41A63