Skip to main content
SpringerLink
Log in

Approximation Properties of Repeated de la Vallée-Poussin Means for Piecewise Smooth Functions

  • Published:
Siberian Mathematical Journal Aims and scope Submit manuscript

Abstract

Basing on Fourier’s trigonometric sums and the classical de la Vallée-Poussin means, we introduce the repeated de la Vallée-Poussin means. Under study are the approximation properties of the repeated means for piecewise smooth functions. We prove that the repeated means achieve the rate of approximation for the discontinuous piecewise smooth functions which is one or two order higher than the classical de la Vallée-Poussin means and the partial Fourier sums respectively.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Nikolski S., “Sur l’allure asymptotique du reste dans l’approximation au moyen des sommes de Fejer des fonctions vérifiant la condition de Lipschitz,” Izv. Akad. Nauk SSSR Ser. Mat., vol. 4, no. 6, 509–520 (1940).

    MATH  Google Scholar 

  2. Efimov A. V., “Approximation of periodic functions by de la Vallée-Poussin sums,” Izv. Akad. Nauk SSSR Ser. Mat., vol. 23, no. 5, 737–770 (1959).

    MathSciNet  Google Scholar 

  3. Telyakovskii S. A., “Approximations to differentiate functions by linear means of their Fourier series,” Izv. Akad. Nauk SSSR Ser. Mat., vol. 24, no. 2, 213–242 (1960).

    MathSciNet  Google Scholar 

  4. Natanson G. I., “The Gibbs phenomenon for de la Vallée-Poussin sums,” in: Studies on Contemporary Problems of the Constructive Theory of Functions [Russian], Fizmatgiz, Moscow, 1961, 206–213.

    Google Scholar 

  5. Sharapudinov I. I., “Approximation properties of de la Vallée-Poussin means for Sobolev-type spaces with variable exponent,” Vestn. Dagestan. Nauch. Tsentr RAN, no. 45, 5–13 (2012).

    Google Scholar 

  6. Sharapudinov I. I., “Approximation of smooth functions in by Vallée-Poussin means,” Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., vol. 13, no. 1, Part 1, 45–49 (2013).

    MATH  Google Scholar 

  7. Sharapudinov I. I., “Approximation of functions in by trigonometric polynomials,” Izv. Math., vol. 77, no. 2, 407–434 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  8. Timan A. F., Theory of Approximation of Functions of a Real Variable, Dover Publ., Inc., New York (1994).

    Google Scholar 

  9. Dzyadyk V. K., Introduction to the Theory of Uniform Approximation of Functions by Polynomials [Russian], Nauka, Moscow (1977).

    MATH  Google Scholar 

  10. Zhuk V. V., Approximation of Periodic Functions [Russian], Leningrad Univ., Leningrad (1982).

    MATH  Google Scholar 

  11. Zygmund A., Trigonometric Series. Vol. 1 [Russian translation], Mir, Moscow (1965).

    MATH  Google Scholar 

  12. Magomed-Kasumov M. G., “Approximation properties of de la Vallée-Poussin classical means for piecewise smooth functions,” Vestn. Dagestan. Nauch. Tsentr RAN, vol. 54, 5–12 (2014).

    Google Scholar 

  13. Magomed-Kasumov M. G., “Approximation properties of de la Vallée-Poussin means for piecewise smooth functions,” Math. Notes, vol. 100, no. 2, 229–244 (2016).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dagestan Scientific Center, Makhachkala, Russia

    I. I. Sharapudinov, T. I. Sharapudinov & M. G. Magomed-Kasumov

  2. Southern Mathematical Institute, Vladikavkaz, Russia

    I. I. Sharapudinov, T. I. Sharapudinov & M. G. Magomed-Kasumov

Authors
  1. I. I. Sharapudinov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. T. I. Sharapudinov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. G. Magomed-Kasumov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to I. I. Sharapudinov or M. G. Magomed-Kasumov.

Additional information

Russian Text © The Author(s), 2019, published in Sibirskii Matematicheskii Zhurnal, 2019, Vol. 60, No. 3, pp. 695–713.

The authors were supported by the Russian Foundation for Basic Research (Grant 16-01-00486).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sharapudinov, I.I., Sharapudinov, T.I. & Magomed-Kasumov, M.G. Approximation Properties of Repeated de la Vallée-Poussin Means for Piecewise Smooth Functions. Sib Math J 60, 542–558 (2019). https://doi.org/10.1134/S0037446619030169

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0037446619030169

Keywords

Search

Navigation