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Doklady Mathematics

, Volume 98, Issue 2, pp 464–467 | Cite as

Fejér Sums and Fourier Coefficients of Periodic Measures

  • A. G. Kachurovskii
  • I. V. Podvigin
Mathematics
  • 2 Downloads

Abstract

The Fejér sums of periodic measures and the norms of the deviations from the limit in the von Neumann ergodic theorem are calculating in terms of corresponding Fourier coefficients, in fact, using the same formulas. As a result, well-known estimates for the rates of convergence in the von Neumann ergodic theorem can be restated as estimates for the Fejér sums at a point for periodic measures. In this way, natural sufficient conditions for the polynomial growth and polynomial decay of these sums can be obtained in terms of Fourier coefficients. Besides, for example, it is shown that every continuous 2π-periodic function is uniquely determined by its sequence of Fejér sums at any two points whose difference is incommensurable with π.

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References

  1. 1.
    R. Edwards, Fourier Series: A Modern Introduction, 2nd ed. (Springer-Verlag, New York, 1979), Vol. 1.Google Scholar
  2. 2.
    I. P. Kornfel’d, Ya. G. Sinai, and S. V. Fomin, Ergodic Theory (Nauka, Moscow, 1980) [in Russian].Google Scholar
  3. 3.
    I. A. Ibragimov and Yu. V. Linnik, Independent and Stationary Sequences of Random Variables (Nauka, Moscow, 1965; Wolters-Noordhoff, Groningen, 1971).zbMATHGoogle Scholar
  4. 4.
    A. G. Kachurovskii and K. I. Knizhov, Dokl. Math. 97 (3), 211–214 (2018).CrossRefGoogle Scholar
  5. 5.
    A. G. Kachurovskii, Russ. Math. Surv. 51 (4), 653–703 (1996).MathSciNetCrossRefGoogle Scholar
  6. 6.
    A. G. Kachurovskii and V. V. Sedalishchev, Sb. Math. 202 (8), 1105–1125 (2011).MathSciNetCrossRefGoogle Scholar
  7. 7.
    A. Zygmund, Trigonometric Series, 2nd ed. (Cambridge Univ. Press, New York, 1959), Vol. 1.Google Scholar
  8. 8.
    N. K. Bari, A Treatise on Trigonometric Series (Fizmatgiz, Moscow, 1961; Pergamon, Oxford, 1964).Google Scholar
  9. 9.
    R. Edwards, Fourier Series: A Modern Introduction, 2nd ed. (Springer-Verlag, New York, 1982), Vol. 2.Google Scholar
  10. 10.
    A. G. Kachurovskii and I. V. Podvigin, Trans. Moscow Math. Soc. 77, 1–53 (2016).CrossRefGoogle Scholar
  11. 11.
    A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis (Nauka, Moscow, 1976; Dover, New York, 1999).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Sobolev Institute of Mathematics, Siberian BranchRussian Academy of SciencesNovosibirskRussia
  2. 2.Novosibirsk State UniversityNovosibirskRussia

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