Skip to main content
SpringerLink
Log in

On The Approximation of Functions from The Hölder Class Given On a Segment by Their Biharmonic Poisson Operators

  • Published:
Ukrainian Mathematical Journal Aims and scope

We obtain the exact equality for the upper bounds of deviations of biharmonic Poisson operators on the Hölder classes of functions continuous on the segment [−1; 1].

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. Yu. I. Rusetskii, “On the approximation of functions continuous on a segment by the Abel–Poisson sums,” Sib Mat. Zh., 9, No. 1, 136–144 (1968).

    MathSciNet  Google Scholar 

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

    Google Scholar 

  3. V. I. Rukasov and S. O. Chaichenko, “Approximation of analytic periodic functions by de la Vall´ee-Poussin sums,” Ukr. Mat. Zh., 54, No. 12, 1653–1668 (2002); English translation: Ukr. Math. J., 54, No. 12, 2006–2024 (2002).

  4. V. I. Rukasov and S. O. Chaichenko, “Approximation by de la Vallée-Poussin operators on the classes of functions locally summable on the real axis,” Ukr. Mat. Zh., 62, No. 7, 968–978 (2010); English translation: Ukr. Math. J., 62, No. 7, 1126–1138 (2010).

  5. Yu. I. Kharkevych and T. V. Zhyhallo, “Approximation of (ψ β)-differentiable functions defined on the real axis by Abel–Poisson operators,” Ukr. Mat. Zh., 57, No. 8, 1097–1111 (2005); English translation: Ukr. Math. J., 57, No. 8, 1297–1315 (2005).

  6. K. M. Zhyhallo and Yu. I. Kharkevych, “Approximation of conjugate differentiable functions by their Abel–Poisson integrals,” Ukr. Mat. Zh., 61, No. 1, 73–82 (2009); English translation: Ukr. Math. J., 61, No. 1, 86–98 (2009).

  7. Yu. I. Kharkevych and T. V. Zhyhallo, “Approximation of functions from the class \( {C}_{\beta, \infty}^{\psi } \) by Poisson biharmonic operators in the uniform metric,” Ukr. Mat. Zh., 60, No. 5, 669–693 (2008); English translation: Ukr. Math. J., 60, No. 5, 769–798 (2008).

  8. T. V. Zhyhallo and Yu. I. Kharkevych, “Approximating properties of biharmonic Poisson operators in the classes \( {L}_{\beta, 1}^{\psi } \),” Ukr. Mat. Zh., 69, No. 5, 650–656 (2017); English translation: Ukr. Math. J., 69, No. 5, 757–765 (2017).

  9. T. V. Zhyhallo, “Approximation of functions holding the Lipschitz conditions on a finite segment of the real axis by the Poisson–Chebyshev integrals,” J. Automat. Inform. Sci., 50, No. 5, 34–48 (2018).

    Article  Google Scholar 

  10. S. M. Nikol’skii, “On the best approximation of functions satisfying the Lipschitz condition by polynomials,” Izv. Akad. Nauk SSSR, Ser. Mat., 27, No. 4, 295–318 (1946).

    MathSciNet  Google Scholar 

  11. A. F. Timan, “Approximation of functions satisfying the Lipschitz condition by ordinary polynomials,” Izv. Akad. Nauk SSSR, Ser. Mat., 77, No. 6, 969–972 (1951).

    MathSciNet  Google Scholar 

  12. I. A. Shevchuk, “On the uniform approximation of functions on a segment,” Mat. Zametki, 40, No. 1, 36–48 (1986).

    MathSciNet  Google Scholar 

  13. V. P. Motornyi and O. V. Motornaya, “On asymptotically exact estimates for the approximation of certain classes of functions by algebraic polynomials,” Ukr. Mat. Zh., 52, No. 1, 85–99 (2000); English translation: Ukr. Math. J., 52, No. 1, 91–107 (2000).

  14. V. P. Motornyi, “Approximation of certain classes of singular integrals by algebraic polynomials,” Ukr. Mat. Zh., 53, No 3, 331–345 (2001); English translation: Ukr. Math. J., 53, No 3, 377–394 (2001).

  15. I. V. Kal’chuk and Yu. I. Kharkevych, “Complete asymptotics of the approximation of function from the Sobolev classes by the Poisson integrals,” Acta Comment. Univ. Tartu. Math., 22, No. 1, 23–36 (2018).

    MathSciNet  MATH  Google Scholar 

  16. Yu. I. Kharkevych and K. V. Pozharska, “Asymptotics of approximation of conjugate functions by the Poisson integrals,” Acta Comment. Univ. Tartu. Math., 22, No. 2, 235–243 (2018).

    MathSciNet  MATH  Google Scholar 

  17. S. B. Hembars’ka and K. M. Zhyhallo, “Approximative properties of biharmonic Poisson integrals on H¨older classes,” Ukr. Mat. Zh., 69, No. 7, 925–932 (2017); English translation: Ukr. Math. J., 69, No. 7, 1075–1084 (2017).

  18. U. Z. Hrabova, I. V. Kal’chuk, and T. A. Stepanyuk, “On the approximation of the classes \( {\mathrm{W}}_{\beta}^r{H}^{\alpha } \) by biharmonic Poisson integrals,” Ukr. Mat. Zh., 70, No. 5, 625–634 (2018); English translation: Ukr. Math. J., 70, No. 5, 719–729 (2018).

  19. I. S. Gradshtein and I. M. Ryzhik, Tables of Integrals, Sums, Series, and Products [in Russian], Fizmatiz, Moscow (1963).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. L. Ukrainka East-European National University, Lutsk, Ukraine

    K. M. Zhyhallo & T. V. Zhyhallo

Authors
  1. K. M. Zhyhallo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. T. V. Zhyhallo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to T. V. Zhyhallo.

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 71, No. 7, pp. 915–921, July, 2019.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhyhallo, K.M., Zhyhallo, T.V. On The Approximation of Functions from The Hölder Class Given On a Segment by Their Biharmonic Poisson Operators. Ukr Math J 71, 1043–1051 (2019). https://doi.org/10.1007/s11253-019-01696-7

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-019-01696-7

Search

Navigation