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Ukrainian Mathematical Journal

, Volume 71, Issue 4, pp 589–618 | Cite as

Approximation of Bounded Holomorphic and Harmonic Functions by Fejér Means

  • V. V. SavchukEmail author
  • S. O. Chaichenko
  • M. V. Savchuk
Article
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We compute the exact values of the least upper bounds on the classes of bounded holomorphic and harmonic functions in a unit disk for the remainders in a Voronovskaya-type formula in the case of approximation by Fejér means. We also present some consequences that are of independent interest.

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References

  1. 1.
    S. B. Stechkin, “Estimation of the remainder of Taylor series for some classes of analytic functions,” Izv. Akad. Nauk SSSR, Ser. Mat., 17, No. 5, 462–472 (1953).Google Scholar
  2. 2.
    I. Meremelia and V. Savchuk, “Approximations of holomorphic functions by generalized Zygmund sums,” Cas. J. Appl. Math., Ecol. Econ., 1, No. 1, 70–81 (2013).zbMATHGoogle Scholar
  3. 3.
    P. Butzer and J. R. Nessel, Fourier Analysis and Approximation, Birkhäuser, Basel (1971).CrossRefGoogle Scholar
  4. 4.
    J. Bustamente, Bernstein Operators and Their Properties, Birkhäuser, Basel (2017).CrossRefGoogle Scholar
  5. 5.
    A. Zygmund, “On the degree of approximation of functions by Fejér means,” Bull. Amer. Math. Soc., 51, 274–278 (1945).MathSciNetCrossRefGoogle Scholar
  6. 6.
    V. V. Savchuk, “Approximation of functions from the Dirichlet class by Fejér means,” Mat. Zametki, 81, No. 5, 744–750 (2007).MathSciNetCrossRefGoogle Scholar
  7. 7.
    V. V. Savchuk, “Approximation of holomorphic functions by Fejér sums,” in: Approximation Theory of Functions and Related Problems [in Ukrainian], Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv, 8, No. 1 (2011), pp. 162–180.Google Scholar
  8. 8.
    V. V. Savchuk, M. V. Savchuk, and S. O. Chaichenko, “Approximation of analytic functions by de la Vallée-Poussin sums,” Mat. Stud., 34, No. 2, 207–219 (2010).MathSciNetzbMATHGoogle Scholar
  9. 9.
    A. I. Stepanets, “Approximate properties of the Zygmund method,” Ukr. Mat. Zh., 51, No. 4, 493–518 (1999); English translation: Ukr. Math. J., 51, No. 4, 546–576 (1999).MathSciNetCrossRefGoogle Scholar
  10. 10.
    A. I. Stepanets and V. I. Rukasov, “Approximation properties of the de la Vallée-Poussin method,” Ukr. Mat. Zh., 54, No. 8, 1100–1125 (2002); English translation: Ukr. Math. J., 54, No. 8, 1324–1354 (2002).Google Scholar
  11. 11.
    A. I. Stepanets, Methods of Approximation Theory [in Russian], Vol. 2, Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev (2002).Google Scholar
  12. 12.
    P. Koosis, Introduction to H pSpaces. With an Appendix on Wolff’s Proof of the Corona Theorem [Russian translation], Mir, Moscow (1984).zbMATHGoogle Scholar
  13. 13.
    E. Landau and D. Gaier, Darstellung und Bergrundung Einiger Neuerer Ergebnisse der Funktionentheorie, Springer, Berlin (1986).CrossRefGoogle Scholar
  14. 14.
    V. V. Savchuk, “Best approximation by holomorphic functions. Application to the best polynomial approximation of classes of holomorphic functions,” Ukr. Mat. Zh., 59, No. 8, 1047–1067 (2007); English translation: Ukr. Math. J., 59, No. 8, 1163–1183 (2007).MathSciNetCrossRefGoogle Scholar
  15. 15.
    J. B. Garnett, Bounded Analytic Functions [Russian translation], Mir, Moscow (1984).Google Scholar
  16. 16.
    K. I. Babenko, “Best approximations of the classes of analytic functions,” Izv. Akad. Nauk SSSR, Ser. Mat., 22, No. 5, 631–640 (1958).MathSciNetGoogle Scholar
  17. 17.
    J. T. Scheick, “Polynomial approximation of functions analytic in a disk,” Proc. Amer. Math. Soc., 17, 1238–1243 (1966).MathSciNetCrossRefGoogle Scholar
  18. 18.
    V. I. Belyi, “The problem of the best linear methods for approximating functions which are analytic in the unit disk,” Ukr. Mat. Zh., 19, No. 2, 104–109 (1967); English translation: Ukr. Math. J., 19, No. 2, 216–220 (1967).Google Scholar
  19. 19.
    É. Ya. Riekstyn’sh, Estimates for the Remainders in Asymptotic Expansions [in Russian], Zinatne, Riga (1986).Google Scholar
  20. 20.
    A. I. Stepanets, Methods of Approximation Theory [in Russian], Vol. 1, Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev (2002).Google Scholar
  21. 21.
    N. K. Bari, Trigonometric Series [in Russian], Fizmatgiz, Moscow (1961).Google Scholar
  22. 22.
    S. G. Samko, A. A. Kilbas, and O. I. Marichev, Integrals and Derivatives of Fractional Order and Some Their Applications [in Russian], Nauka i Tekhnika, Minsk (1987).zbMATHGoogle Scholar
  23. 23.
    D. Khavinson, “An extremal problem for harmonic functions in the ball,” Canad. Math. Bull., 35, No. 2, 218–220 (1992).MathSciNetCrossRefGoogle Scholar
  24. 24.
    F. Colonna, “The Bloch constant of bounded harmonic mappings,” Indiana Univ. Math. J., 38, No. 4, 829–840 (1989).MathSciNetCrossRefGoogle Scholar
  25. 25.
    G. G. Lorentz, “Inequalities and the saturation classes of Bernstein polynomials,” in: On Approximation Theory, Proc. Conference Oberwolfach, 1963, Birkhäuser, Basel (1964), pp. 200–207.Google Scholar
  26. 26.
    A. F. Timan and V. N. Trofimov, “On sharp constants in some inverse theorems for the approximations by trigonometric polynomials,” Mat. Zametki, 20, No. 5, 787–792 (1976).MathSciNetzbMATHGoogle Scholar
  27. 27.
    P. Koebe, “Über das Schwarzssche Lemma und einige damit zusammenhängende Ungleichheitsbeziehungen der Potentialtheorie und Funktionentheorie,” Math. Z., 6, 52–84 (1920).MathSciNetCrossRefGoogle Scholar
  28. 28.
    V. V. Savchuk, “Best linear methods of approximation and optimal orthonormal systems of the Hardy space,” Ukr. Mat. Zh., 60, No. 5, 636–646 (2008); English translation: Ukr. Math. J., 60, No. 5, 730–743 (2008).MathSciNetCrossRefGoogle Scholar
  29. 29.
    A. S. Serdyuk, “Approximation of Poisson integrals by de la Vallée Poussin sums,” Ukr. Mat. Zh., 56, No. 1, 97–107 (2004); English translation: Ukr. Math. J., 56, No. 1, 122–134 (2004).MathSciNetCrossRefGoogle Scholar
  30. 30.
    V. I. Rukasov and O. A. Novikov, “Approximation of analytic functions by the de la Vallée-Poussin sums,” in: Fourier Series: Theory and Application [in Ukrainian], Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv (1998), pp. 228–241.Google Scholar
  31. 31.
    A. I. Stepanets, V. I. Rukasov, and S. O. Chaichenko, Approximation by the de la Vallée-Poussin Sums [in Russian], Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev (2007).zbMATHGoogle Scholar
  32. 33.
    S. M. Nikol’skii, “On the asymptotic behavior of the remainder in the case of approximation of functions satisfying Lipschitz conditions by Fejér means,” Izv. Akad. Nauk SSSR, Ser. Mat., 4, No. 6, 501–508 (1940).Google Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • V. V. Savchuk
    • 1
    Email author
  • S. O. Chaichenko
    • 2
  • M. V. Savchuk
    • 3
  1. 1.Institute of MathematicsUkrainian National Academy of SciencesKyivUkraine
  2. 2.Donbas State Pedagogic UniversitySlov’yanskUkraine
  3. 3.Institute for Training Personal of the Ukrainian State Placement ServiceKyivUkraine

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