Skip to main content
SpringerLink
Log in

Best approximation methods and widths for some classes of functions in H q,ρ , 1 ≤ q ≤ ∞, 0 < ρ ≤ 1

  • Published:
Siberian Mathematical Journal Aims and scope Submit manuscript

Abstract

We compute the exact values of widths for various widths for the classes W (r) q,a (Φ, μ), μ ≥ 1, of analytic functions in the disk belonging to the Hardy space H q , q ≥ 1, whose averaged moduli of continuity of the boundary values of the derivatives with respect to the argument f (r) a , r ∈ N, are dominated by a given function Φ. For calculating the linear and Gelfand n-widths, we use best linear approximation for these functions.

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. Tikhomirov V. M., “The widths of sets in function spaces and best approximation theory,” Uspekhi Mat. Nauk, 15, No. 3, 81–120 (1960).

    Google Scholar 

  2. Pinkus A., n-Widths in Approximation Theory, Springer-Verlag, Berlin (1985).

    Book  MATH  Google Scholar 

  3. Babenko K. I., “On best approximations of a class of functions,” Izv. Akad. Nauk SSSR Ser. Mat., 22, No. 5, 631–640 (1958).

    MathSciNet  Google Scholar 

  4. Taikov L. V., “On the best linear methods of approximating functions of classes Br and Hr,” Uspekhi Mat. Nauk, 18, No. 4, 183–189 (1963).

    MathSciNet  MATH  Google Scholar 

  5. Scheick J. T., “Polynomial approximation of functions analytic in a disk,” Proc. Amer. Math. Soc., 17, 1238–1243 (1966).

    Article  MathSciNet  MATH  Google Scholar 

  6. Taikov L. V., “On the best approximation in the mean of certain classes of analytic functions,” Math. Notes, 1, No. 2, 104–109 (1967).

    Article  MathSciNet  MATH  Google Scholar 

  7. Taikov L. V., “Diameters of certain classes of analytic functions,” Math. Notes, 22, No. 2, 650–656 (1977).

    Article  MathSciNet  MATH  Google Scholar 

  8. Dveĭrin M. Z., “On approximation of functions analytic in the unit disk,” in: Metric Problems of the Theory of Functions and Mappings [in Russian], Naukova Dumka, Kiev, 1975, pp. 41–54.

    Google Scholar 

  9. Dveĭrin M. Z. and Chebanenko I. V., “Polynomial approximation in the Banach spaces of analytic functions,” in: Theory of Mappings and Approximation of Functions [in Russian], Naukova Dumka, Kiev, 1983, pp. 62–73.

    Google Scholar 

  10. Ainulloev N. and Taikov L. V., “Best approximation in the sense of Kolmogorov of classes of functions analytic in the unit disc,” Math. Notes, 40, No. 3, 699–705 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  11. Vakarchuk S. B., “Best linear methods of approximation and widths of classes of analytic functions in a disk,” Math. Notes, 57, No. 1, 21–27 (1995).

    Article  MathSciNet  Google Scholar 

  12. Vakarchuk S. B., “On the best linear approximation methods and the widths of certain classes of analytic functions,” Math. Notes, 65, No. 2, 153–158 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  13. Shabozov M. Sh. and Shabozov O. Sh., “Widths of some classes of analytic functions in the Hardy space H2,” Math. Notes, 68, No. 5, 675–579 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  14. Osipenko K. Yu., “On optimal recovery methods in Hardy–Sobolev spaces,” Sb. Math., 192, No. 2, 225–244 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  15. Shabozov M. Sh., “The widths of some classes of analytic functions in the Bergman space,” Dokl. Akad. Nauk, 383, No. 2, 171–174 (2002).

    MathSciNet  Google Scholar 

  16. Shabozov M. Sh. and Yusupov G. A., “The best approximation and values of widths for some classes of analytic functions,” Dokl. Akad. Nauk, 382, No. 6, 747–749 (2002).

    MathSciNet  MATH  Google Scholar 

  17. Vakarchuk S. B., “Exact values of widths of classes of analytic functions on the disk and best linear approximation methods,” Math. Notes, 72, No. 5, 615–619 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  18. Vakarchuk S. B., “On some extremal problems of approximation theory on the complex plane,” Ukr. Mat. Zh., 56, No. 9, 1155–1171 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  19. Shvedenko S. V., “Hardy classes and related spaces of analytic functions in the unit disk, polydisk, and ball,” J. Soviet Math., 39, No. 6, 3011–3087 (1987).

    Article  MATH  Google Scholar 

  20. Shabozov M. Sh. and Shabozov O. Sh., “On the best approximation of some classes of analytic functions in weighted Bergman spaces B2,γ,” Dokl. Math., 75, No. 1, 97–100 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  21. Vakarchuk S. B. and Zabutnaya V. I., “Best linear approximation methods for functions of Taikov classes in the Hardy spaces H q,ρ, q ≥ 1, 0 < ρ ≤ 1,” Math. Notes, 85, No. 3, 322–327 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  22. Vakarchuk S. B. and Shabozov M. Sh., “The widths of classes of analytic functions in a disc,” Sb. Math., 201, No. 8, 1091–1110 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  23. Shabozov M. Sh. and Langarshoev M. R., “The best linear methods and values of widths for some classes of analytic functions in the Bergman weight space,” Dokl. Math., 87, No. 3, 338–341 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  24. Yusupov G. A., “The best linear methods for approximation of functions in the Hardy spaces H q,R, 0 < R < 1,” Dokl. AN Republic of Tajikistan, 56, No. 12, 946–952 (2013).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Juraev Institute of Mathematics, Dushanbe, Tajikistan

    M. Sh. Shabozov

  2. Tajik National University, Dushanbe, Tajikistan

    G. A. Yusupov

Authors
  1. M. Sh. Shabozov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. A. Yusupov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Sh. Shabozov.

Additional information

Dushanbe. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 57, No. 2, pp. 469–478, March–April, 2016; DOI: 10.17377/smzh.2016.57.219.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shabozov, M.S., Yusupov, G.A. Best approximation methods and widths for some classes of functions in H q,ρ , 1 ≤ q ≤ ∞, 0 < ρ ≤ 1. Sib Math J 57, 369–376 (2016). https://doi.org/10.1134/S0037446616020191

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0037446616020191

Keywords

Search

Navigation