Skip to main content
SpringerLink
Log in

Best bilinear approximations of the classes \( S_{p,\theta }^\Omega B \) of periodic functions of many variables

  • Published:
Ukrainian Mathematical Journal Aims and scope

We obtain exact-order estimates for the best bilinear approximations of the classes \( S_{p,\theta }^\Omega B \) of periodic functions of many variables in the space L q under certain restrictions on the parameters p, q, and θ.

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. S. N. Bernshtein, Constructive Theory of Functions (1931–1953) [in Russian], Vol. 2, Academy of Sciences of the USSR, Moscow (1954).

    Google Scholar 

  2. N. K. Bari and S. B. Stechkin, “Best approximations and differential properties of two conjugate functions,” Tr. Mosk. Mat. Obshch., 5, 483–522 (1956).

    MATH  Google Scholar 

  3. Y. Sun and H. Wang, “Representation and approximation of multivariate periodic functions with bounded mixed moduli of smoothness,” Tr. Mat. Inst. Ros. Akad. Nauk, 219, 356–377 (1997).

    Google Scholar 

  4. N. N. Pustovoitov, “Representation and approximation of periodic functions of many variables with given mixed modulus of continuity,” Anal. Math., 20, No. 1, 35–48 (1994).

    Article  MathSciNet  Google Scholar 

  5. P. I. Lizorkin and S. M. Nikol’skii, “Spaces of functions of mixed smoothness from the decomposition point of view,” Tr. Mat. Inst. Akad. Nauk SSSR, 187, 143–161 (1989).

    MathSciNet  Google Scholar 

  6. S. M. Nikol’skii, Approximation of Functions of Many Variables and Imbedding Theorems [in Russian], Nauka, Moscow (1969).

    Google Scholar 

  7. S. A. Stasyuk and O. V. Fedunyk, “Approximation characteristics of the classes \( B_{p,\theta }^\Omega \) of periodic functions of many variables,” Ukr. Mat. Zh., 58, No. 5, 692–704 (2006); English translation: Ukr. Math. J., 58, No. 5, 779–793 (2006).

    Article  Google Scholar 

  8. S. B. Stechkin, “On the absolute convergence of orthogonal series,” Dokl. Akad. Nauk SSSR, 102, No. 1, 37–40 (1955).

    MathSciNet  MATH  Google Scholar 

  9. V. N. Temlyakov, “Approximation of functions with bounded mixed derivative,” Tr. Mat. Inst. Akad. Nauk SSSR, 178, 1–112 (1986).

    MathSciNet  Google Scholar 

  10. A. S. Romanyuk, “Best M-term trigonometric approximation of the Besov classes of periodic functions of many variables,” Izv. Ros. Akad. Nauk, Ser. Mat., 67, No. 2, 61–100 (2003).

    MathSciNet  Google Scholar 

  11. S. A. Stasyuk, “Approximation of functions of many variables of the classes \( H_p^\Omega \) by polynomials in the Haar system,” Anal. Math., 35, 257–271 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  12. A. F. Konohrai and S. A. Stasyuk, “Best M-term trigonometric approximations of the classes \( B_{p,\theta }^\Omega \) of periodic functions of many variables in the space L q ,” Ukr. Mat. Zh., 60, No. 9, 1196–1214 (2008); English translation: Ukr. Math. J., 60, No. 9, 1396–1417 (2008).

    Article  Google Scholar 

  13. S. A. Stasyuk, “Best M-term trigonometric approximations of the classes \( B_{p,\theta }^\Omega \) of functions of many variables,” Ukr. Mat. Zh., 54, No. 3, 381–394 (2002); English translation: Ukr. Math. J., 54, No. 3, 470–486 (2002).

    Article  MathSciNet  Google Scholar 

  14. V. N. Temlyakov, Approximation of Periodic Functions, Nova Science, New York (1993).

    MATH  Google Scholar 

  15. S. A. Stasyuk, “Best M-term orthogonal trigonometric approximations of the classes \( B_{p,\theta }^\Omega \) of periodic functions of many variables,” Ukr. Mat. Zh., 60, No. 5, 647–656 (2008); English translation: Ukr. Math. J., 60, No. 5, 744–757 (2008).

    Article  MathSciNet  Google Scholar 

  16. A. S. Romanyuk and V. S. Romanyuk, “Asymptotic estimates for the best trigonometric and bilinear approximations of classes of functions of several variables,” Ukr. Mat. Zh., 62, No. 4, 536–551 (2010); English translation: Ukr. Math. J., 62, No. 4, 612–629 (2010).

    Article  MathSciNet  Google Scholar 

  17. E. Schmidt, “Zur Theorie der linearen und nichtlinearen Integralgleichungen. I,” Math. Ann., 63, 433–476 (1907).

    Article  MathSciNet  Google Scholar 

  18. V. N. Temlyakov, “Bilinear approximation and related problems,” Tr. Mat. Inst. Ros. Akad. Nauk, 194, 229–248 (1991).

    Google Scholar 

  19. V. N. Temlyakov, “Approximation of periodic functions of many variables by combinations of functions of a smaller number of variables,” Tr. Mat. Inst. Akad. Nauk SSSR, 173, 243–252 (1986).

    MathSciNet  Google Scholar 

  20. V. N. Temlyakov, “Estimates for the best bilinear approximations of functions of two variables and their applications,” Mat. Sb., 176, No. 1, 16–33 (1987).

    MathSciNet  Google Scholar 

  21. A. S. Romanyuk, “Bilinear and trigonometric approximations of the Besov classes \( B_{p,\theta }^r \) of periodic functions of many variables,” Izv. Ros. Akad. Nauk, Ser. Mat., 70, No. 2, 69–98 (2006).

    MathSciNet  Google Scholar 

  22. S. B. Kashin and A. A. Saakyan, Orthogonal Series [in Russian], Nauka, Moscow (1984).

    Google Scholar 

  23. A. F. Konohrai, “Widths of the classes \( B_{p,\theta }^\Omega \) of periodic functions of many variables,” Mat. Stud., 29, No. 2, 192–206 (2008).

    MathSciNet  Google Scholar 

  24. S. A. Stasyuk, “Best approximations and Kolmogorov and trigonometric widths of the classes \( B_{p,\theta }^\Omega \) of periodic functions of many variables,” Ukr. Mat. Zh., 56, No. 11, 1557–1568 (2004); English translation: Ukr. Math. J., 56, No. 11, 1849–1863 (2004).

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    K. V. Solich

Authors
  1. K. V. Solich
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 63, No. 6, pp. 809–826, June, 2011.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Solich, K.V. Best bilinear approximations of the classes \( S_{p,\theta }^\Omega B \) of periodic functions of many variables. Ukr Math J 63, 940–961 (2011). https://doi.org/10.1007/s11253-011-0554-4

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-011-0554-4

Keywords

Search

Navigation