Advertisement

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

  • K. M. Zhyhallo
  • T. V. ZhyhalloEmail author
Article

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].

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 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).MathSciNetGoogle Scholar
  2. 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. 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).Google Scholar
  4. 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).Google Scholar
  5. 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).Google Scholar
  6. 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).Google Scholar
  7. 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).Google Scholar
  8. 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).Google Scholar
  9. 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).CrossRefGoogle Scholar
  10. 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).MathSciNetGoogle Scholar
  11. 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).MathSciNetGoogle Scholar
  12. 12.
    I. A. Shevchuk, “On the uniform approximation of functions on a segment,” Mat. Zametki, 40, No. 1, 36–48 (1986).MathSciNetGoogle Scholar
  13. 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).Google Scholar
  14. 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).Google Scholar
  15. 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).MathSciNetzbMATHGoogle Scholar
  16. 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).MathSciNetzbMATHGoogle Scholar
  17. 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).Google Scholar
  18. 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).Google Scholar
  19. 19.
    I. S. Gradshtein and I. M. Ryzhik, Tables of Integrals, Sums, Series, and Products [in Russian], Fizmatiz, Moscow (1963).Google Scholar

Copyright information

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

Authors and Affiliations

  1. 1.L. Ukrainka East-European National UniversityLutskUkraine

Personalised recommendations