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Convergence to Zero of Exponential Sums with Positive Integer Coefficients and Approximation by Sums of Shifts of a Single Function on the Line

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Abstract

We prove that there is a sequence of trigonometric polynomials with positive integer coefficients, which converges to zero almost everywhere. We also prove that there is a function f: ℝ → ℝ such that the sums of its shifts are dense in all real spaces L p (ℝ) for 2 ≤ p < ∞ and also in the real space C0(R).

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Authors and Affiliations

  1. Laboratory “Multivariate approximation and applications”, Department of Mechanics and Mathematics, Moscow State University, Moscow, 119991, Russia

    P. A. Borodin & S. V. Konyagin

  2. Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkin Street, Moscow, 119991, Russia

    S. V. Konyagin

Authors
  1. P. A. Borodin
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  2. S. V. Konyagin
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Correspondence to S. V. Konyagin.

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Dedicated to the memory of Jean-Pierre Kahane

The research of both authors was supported by the grant of the Government of the Russian Federation (project 14.W03.31.0031).

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Borodin, P.A., Konyagin, S.V. Convergence to Zero of Exponential Sums with Positive Integer Coefficients and Approximation by Sums of Shifts of a Single Function on the Line. Anal Math 44, 163–183 (2018). https://doi.org/10.1007/s10476-018-0204-2

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  • DOI: https://doi.org/10.1007/s10476-018-0204-2

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