Skip to main content
Log in

Exact upper bounds of the deviations of Bernstein sums from functions of Hölder classes

  • Published:
Ukrainian Mathematical Journal Aims and scope

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Literature cited

  1. S. N. Bernstein, “Sur un procéde de sommation des séries trigonométriques,” C. R. Acad. Sc.,191, 976–979 (1930).

    Google Scholar 

  2. A. N. Kolmogorov, “Zur Grössenordnung des Restgliedes Fourierscher Reihen differenzierbarer Functionen,” Ann. of Math.,36, No. 2, 521 (1935).

    Google Scholar 

  3. V. T. Pinkevich, “The order of the remainder term of the Fourier series of functions differentiable in the sense of Weyl,” Izv. Akad. Nauk SSSR, Ser. Matem.,4, No. 5 (1940).

  4. S. M. Nikol'skii, “Asymptotic behavior of the remainder under the approximation by Fejér sums of functions satisfying a Lipschitz condition,” Izv. Akad. Nauk SSSR, Ser. Matem.,4, No. 6 (1940).

  5. S. M. Nikol'skii, “Certain methods of approximation by trigonometric sums,” Izv. Akad, Nauk SSSR, Ser. Matem.,4, No. 6 (1940).

  6. S. M. Nikol'skii, “An asymptotic estimate of the remainder under approximation by Fourier sums,” Dokl. Akad. Nauk SSSR,22, No. 6 (1941).

  7. S. M. Nikol'skii, “Approximation of periodic functions by trigonometric polynomials,” Trudy Matem. In-ta im. V. A. Steklova AN SSSR,15 (1945).

  8. A. V. Efimov, “Approximation of certain classes of continuous functions by Fourier sums and Fejér sums,” Izv. Akad. Nauk SSSR, Ser. Matem.,22, No. 1 (1958).

  9. A. V. Efimov, “Approximation of continuous functions by Fourier sums,” Uspekhi Matem. Nauk,14, No. 2 (1959).

  10. A. V. Efimov, “Approximation of functions with given modulus of continuity by Fourier sums,” Izv. Akad. Nauk SSSR, Ser. Matem.,23, No. 1 (1959).

  11. A. V. Efimov, “Approximation of periodic functions by de la Vallée-Poussin sums. I,” Izv. Akad. Nauk SSSR, Ser. Matem.,23, No. 5 (1959).

  12. A. V. Efimov, “Linear methods of summation of Fourier series of periodic functions,” Dokl. Akad. Nauk SSSR,131, No. 2 (1960).

  13. A. V. Efimov, “Approximation of continuous periodic functions by Fourier sums,” Izv. Akad. Nauk SSSR, Ser. Matem.,24, No. 2 (1960).

  14. A. V. Efimov, “Approximation of periodic functions by de la Vallée-Poussin sums. II,” Izv. Akad. Nauk SSSR, Ser. Matem.,24, No. 3 (1960).

  15. A. V. Efimov, “Linear methods of approximation of certain classes of continuous periodic functions,” Trudy Matem. In-ta im. V. A. Steklova AN SSSR,62 (1961).

  16. B. Nagy, “Sur une classe génerale de procédés de sommation pour les séries de Fourier,” Hungarica Acta Math.,1, No. 3 (1948).

  17. A. F. Timan, “Approximation properties of linear methods of summation of Fourier series,” Izv. Akad. Nauk SSSR, Ser. Matem.,17, No. 2 (1953).

  18. S. A. Telyakovskii, “Approximation of differentiable functions by de la Vallée-Poussin sums,” Dokl. Akad. Nauk SSSR,121, No. 3 (1958).

  19. S. A. Telyakovskii, “Approximation of functions differentiable in the sense of Weyl by de la Vallée-Poussin sums,” Dokl. Akad. Nauk SSSR,131, No. 2 (1960).

  20. S. A. Telyakovskii, “Approximation of differentiable functions by linear means of their Fourier series,” Izv. Akad. Nauk SSSR, Ser. Matem.,24, No. 2 (1960).

  21. S. A. Telyakovskii, “Norms of trigonometric polynomials and approximation of differentiable functions by linear means of their Fourier series. I,” Trudy Matem. In-ta im. V. A. Steklova AN SSSR,62 (1961).

  22. S. A. Telyakovskii, “Norms of trigonometric polynomials and approximation of differentiable functions by linear means of their Fourier series. II,” Izv. Akad. Nauk SSSR, Ser. Matem.,27, No. 2 (1963).

  23. S. A. Telyakovskii, “Integrability conditions for trigonometric series and their application to the study of linear summation methods of Fourier series,” Izv. Akad. Nauk SSSR, Ser. Matem.,28, No. 6 (1964).

  24. W. Rogosinski, “Über die abschnitte trigonometrischer Reihen,” Mathematische Annalen,95 (1926).

  25. N. P. Korneichuk, “Some questions in the approximation of periodic functions by trigonometric polynomials,” Author's Abstract of Candidate's Dissertation, Dnepropetrovsk (1959).

  26. N. P. Korneichuk, “Approximation of periodic functions satisfying a Lipschitz condition by Bernstein-Bogosinski sums,” Dokl. Akad. Nauk SSSR,125, No. 2 (1959).

  27. N. P. Korneichuk, “An estimate of approximations of the class Hα by trigonometric polynomials,” in: Investigations in Modern Problems of the Constructive Theory of Functions [in Russian], Fizmatgiz, Moscow (1961).

    Google Scholar 

  28. V. K. Dzyadyk, V. T. Gavrilyuk, and A. I. Stepanets, “Approximation of functions of Hölder classes by Rogosinski polynomials,” DAN URSR, Ser. A, No. 3 (1969).

  29. V. K. Dzyadyk, V. T. Gavrilyuk, and A. I. Stepanets, “The exact upper bound of approximations of differentiable periodic functions by means of Rogosinski polynomials,” Ukrainsk, Matem. Zh.,22, No. 4 (1970).

  30. V. K. Dzyadyk and A. I. Stepanets, “Asymptotic equalities for the exact upper bounds of approximations of functions of Hölder classes by means of Rogosinski polynomials,” Ukrainsk. Matem. Zh.,24, No. 4 (1972).

  31. V. T. Gavrilyuk and A. I. Stepanets, “Approximation of differentiable functions by Rogosinski polynomials,” Ukrainsk. Matem. Zh.,25, No. 1 (1973).

  32. N. P. Korneichuk, “Extremal properties of periodic functions,” DAN URSR, No.8 (1962).

Download references

Author information

Authors and Affiliations

Authors

Additional information

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 25, No. 2, pp. 158–169, March–April, 1973.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gavrilyuk, V.T., Stepanets, A.I. Exact upper bounds of the deviations of Bernstein sums from functions of Hölder classes. Ukr Math J 25, 129–138 (1973). https://doi.org/10.1007/BF01096971

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01096971

Navigation