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Sampling Discretization of the Uniform Norm

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Abstract

Discretization of the uniform norm of functions from a given finite dimensional subspace of continuous functions is studied. We pay special attention to the case of trigonometric polynomials with frequencies from an arbitrary finite set with fixed cardinality. We give two different proofs of the fact that for any N-dimensional subspace of the space of continuous functions it is sufficient to use \(e^{CN}\) sample points for an accurate upper bound for the uniform norm. Previous known results show that one cannot improve on the exponential growth of the number of sampling points for a good discretization theorem in the uniform norm. Also, we prove a general result, which connects the upper bound on the number of sampling points in the discretization theorem for the uniform norm with the best m-term bilinear approximation of the Dirichlet kernel associated with the given subspace. We illustrate the application of our technique on the example of trigonometric polynomials.

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Acknowledgements

The authors are grateful to the referees for their useful comments and suggestions. The work was supported by the Russian Federation Government Grant No. 14.W03.31.0031.

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Authors and Affiliations

  1. Steklov Institute of Mathematics, Moscow, Russia

    B. Kashin, S. Konyagin & V. Temlyakov

  2. Lomonosov Moscow State University, Moscow, Russia

    B. Kashin, S. Konyagin & V. Temlyakov

  3. Moscow Center for Fundamental and Applied Mathematics, Moscow, Russia

    B. Kashin & V. Temlyakov

  4. University of South Carolina, Columbia, USA

    V. Temlyakov

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  1. B. Kashin
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Correspondence to V. Temlyakov.

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Communicated by Albert Cohen.

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Kashin, B., Konyagin, S. & Temlyakov, V. Sampling Discretization of the Uniform Norm. Constr Approx 57, 663–694 (2023). https://doi.org/10.1007/s00365-023-09618-4

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