Skip to main content
SpringerLink
Log in

Approximation by interpolation trigonometric polynomials on classes of periodic analytic functions

  • Published:
Ukrainian Mathematical Journal Aims and scope

We establish asymptotically unimprovable interpolation analogs of Lebesgue-type inequalities on the sets \( C_{\beta}^{\psi }{L_p} \) of (ψ, β)-differentiable functions generated by sequences ψ(k) that satisfy the d’Alembert conditions. We find asymptotic equalities for the least upper bounds of approximations by interpolation trigonometric polynomials on the classes \( C_{{\beta, p}}^{\psi } \), 1 ≤ p ≤ ∞.

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.

Similar content being viewed by others

References

  1. A. I. Stepanets and A. S. Serdyuk, “Approximation of periodic analytic functions by interpolation trigonometric polynomials,” Ukr. Mat. Zh., 52, No. 12, 1689–1701 (2000); English translation: Ukr. Math. J., 52, No. 12, 1928–1941 (2000).

    Article  MathSciNet  Google Scholar 

  2. O. I. Stepanets’ and A. S. Serdyuk, “Estimate for the remainder of approximation by interpolation trigonometric polynomials on classes of infinitely differentiable functions,” in: Theory of Approximation of Functions and Its Applications [in Ukrainian], Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv (2000), pp. 446–460.

  3. A. S. Serdyuk, “On asymptotically exact estimates for the accuracy of approximation of functions of higher smoothness by interpolation trigonometric polynomials,” Dopov. Nats. Akad. Nauk Ukr., No. 8, 29–33 (1999).

  4. A. S. Serdyuk, “Approximation of infinitely differentiable periodic functions by interpolation trigonometric polynomials,” Ukr. Mat. Zh., 56, No. 4, 495–505 (2004); English translation: Ukr. Math. J., 56, No. 4, 601–613 (2004).

    Article  MathSciNet  Google Scholar 

  5. A. I. Stepanets, Classification and Approximation of Periodic Functions [in Russian], Naukova Dumka, Kiev (1987).

    Google Scholar 

  6. A. I. Stepanets, Methods of Approximation Theory [in Russian], Part 1, Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev (2002).

    Google Scholar 

  7. A. S. Serdyuk, “Approximation of classes of analytic functions by Fourier sums in uniform metric,” Ukr. Mat. Zh., 57, No. 8, 1079–1096 (2005); English translation: Ukr. Math. J., 57, No. 8, 1275–1296 (2005).

    Article  MathSciNet  Google Scholar 

  8. A. S. Serdyuk and V. A. Voitovych, “Approximation of classes of entire functions by interpolation analogs of de la Vall´ee Poussin sums,” in: Theory of Approximation of Functions and Related Problems [in Ukrainian], Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv (2010), pp. 274–297.

  9. M. F. Timan, Approximation and Properties of Periodic Functions [in Russian], Naukova Dumka, Kiev (2009).

    Google Scholar 

  10. A. S. Serdyuk and S. O. Chaichenko, “Approximation of classes of analytic functions by a linear method of special form,” Ukr. Mat. Zh., 63, No. 1, 102–109 (2011); English translation: Ukr. Math. J., 63, No. 1, 125–133 (2011).

    Article  MATH  Google Scholar 

  11. A. S. Serdyuk and I. V. Sokolenko, “Asymptotic behavior of best approximations of classes of periodic analytic functions defined by moduli of continuity,” in: Proceedings of the “Mathematical Analysis, Differential Equations, and Their Applications” Bulgarian–Turkish–Ukrainian Scientific Conference (Sunny Beach, September 15–20, 2010), “Prof. Marin Drinov,” Sofia (2011), pp. 173–182.

  12. A. I. Stepanets and A. S. Serdyuk, “Approximation by Fourier sums and best approximations on classes of analytic functions,” Ukr. Mat. Zh., 52, No. 3, 375–395 (2000); English translation: Ukr. Math. J., 52, No. 3, 433–456 (2000).

    Article  MathSciNet  Google Scholar 

  13. A. S. Serdyuk, “Approximation of classes of analytic functions by Fourier sums in the metric of the space L p ,” Ukr. Mat. Zh., 57, No. 10, 1395–1408 (2005); English translation: Ukr. Math. J., 57, No. 10, 1635–1651 (2005).

    Article  MathSciNet  Google Scholar 

  14. A. S. Serdyuk and A. P. Musienko, “Lebesgue-type inequalities for de la Vallée Poussin sums in the approximation of Poisson integrals,” in: Theory of Approximation of Functions and Related Problems [in Ukrainian], Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv (2010), pp. 298–316.

  15. N. P. Korneichuk, Exact Constants in Approximation Theory [in Russian], Nauka, Moscow (1987).

    Google Scholar 

  16. S. A. Telyakovskii, “Approximation of functions of higher smoothness by Fourier sums,” Ukr. Mat. Zh., 41, No. 4, 510–518 (1989); English translation: Ukr. Math. J., 41, No. 4, 444–451 (1989).

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv, Ukraine

    A. S. Serdyuk

Authors
  1. A. S. Serdyuk
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 64, No. 5, pp. 698–712, May, 2012.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Serdyuk, A.S. Approximation by interpolation trigonometric polynomials on classes of periodic analytic functions. Ukr Math J 64, 797–815 (2012). https://doi.org/10.1007/s11253-012-0679-0

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-012-0679-0

Keywords

Search

Navigation