Skip to main content
Log in

Asymptotic estimates for the best trigonometric and bilinear approximations of classes of functions of several variables

  • Published:
Ukrainian Mathematical Journal Aims and scope

We obtain exact order estimates for the best M-term trigonometric approximations of the Besov classes B ∞,θ r in the space L q . We also determine the exact orders of the best bilinear approximations of the classes of functions of 2d variables generated by functions of d variables from the classes B p,θ r with the use of translation of arguments.

This is a preview of subscription content, log in via an institution to check access.

Access this article

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. 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 

  2. O. V. Besov, “On some family of functional spaces. Imbedding and extension theorems,” Dokl. Akad. Nauk SSSR, 126, No. 6, 1163–1165 (1959).

    MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

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

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  6. A. S. Romanyuk, “Best trigonometric approximations of classes of periodic functions of many variables in the uniform metric,” Mat. Zametki, 82, No. 2, 247–261 (2007).

    MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  8. A. S. Romanyuk, “Approximation of classes of periodic functions of many variables,” Mat. Zametki, 71, No. 1, 109–121 (2002).

    MathSciNet  Google Scholar 

  9. A. S. Romanyuk, “Bilinear and trigonometric approximations of the Besov classes B p,θ r of periodic functions of many variables,” Izv. Ros. Akad. Nauk, Ser. Mat., 70, No. 2, 69–98 (2006).

    MathSciNet  Google Scholar 

  10. B. S. Kashin and V. N. Temlyakov, ”On the best M-term approximations and entropy of sets in the space L 1,” Mat. Zametki, 56, No. 5, 57–86 (1994).

    MathSciNet  Google Scholar 

  11. R. A. DeVore and V. N. Temlyakov, “Nonlinear approximation by trigonometric sums,” J. Fourier Anal. Appl., 2, No. 1, 29–48 (1995).

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  13. V. N. Temlyakov, “Bilinear approximation and applications,” Tr. Mat. Inst. Akad. Nauk SSSR, 187, 194–215 (1989).

    MathSciNet  Google Scholar 

  14. V. N. Temlyakov, “Estimates for asymptotic characteristics of classes of functions with bounded mixed derivative or difference,” Tr. Mat. Inst. Akad. Nauk SSSR, 189, 138–168 (1989).

    MathSciNet  Google Scholar 

  15. V. N. Temlyakov, “Estimates for the best bilinear approximations of functions of two variables and some their applications,” Mat. Sb., 176, No. 1, 93–107 (1987).

    Google Scholar 

  16. S. M. Nikol’skii, “Inequalities for entire functions of finite degree and their application in the theory of differentiable functions of many variables,” Tr. Mat. Inst. Akad. Nauk SSSR, 38, 244–278 (1951).

    Google Scholar 

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

    MATH  Google Scholar 

  18. A. S. Romanyuk, “Kolmogorov widths and trigonometric widths of the Besov classes B p,θ r of periodic functions of many variables,” Mat. Sb., 197, No. 1, 71–96 (2006).

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 62, No. 4, pp. 536–551, April, 2010.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Romanyuk, A.S., Romanyuk, V.S. Asymptotic estimates for the best trigonometric and bilinear approximations of classes of functions of several variables. Ukr Math J 62, 612–629 (2010). https://doi.org/10.1007/s11253-010-0375-x

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-010-0375-x

Keywords

Navigation