Skip to main content
SpringerLink
Log in

Best Trigonometric and Bilinear Approximations for the Classes of (ψ, β)-Differentiable Periodic Functions

  • Published:
Ukrainian Mathematical Journal Aims and scope

We establish exact-order estimates for the best m-term trigonometric approximations of the classes L ψ β,1 in the space L q , 2 < q < ∞. We also determine the exact orders of the best bilinear approximations for the classes of functions of two variables generated by functions of a single variable from the class L ψ β,p by shifts of the argument in the space L q1 , q2 , 1 ≤ q 1 , q 2 ≤ ∞.

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. A. I. Stepanets, Classification and Approximation of Periodic Functions [in Russian], Naukova Dumka, Kiev (1987).

    MATH  Google Scholar 

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

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

  4. É. S. Belinskii, “Approximation by a ‘floating’ system of exponents on classes of smooth periodic functions,” Mat. Sb., 132, No. 1, 20–27 (1987).

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

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

    Article  MathSciNet  MATH  Google Scholar 

  7. A. Zygmund, Trigonometric Series, Vols. 1, 2, Cambridge Univ. Press, Cambridge (1959).

  8. U. Z. Hrabova and A. S. Serdyuk, “Order estimates for the best approximations and approximations by Fourier sums of the classes of (ψ, β)-differential functions,” Ukr. Mat. Zh., 65, No. 9, 1186–1197 (2013); English translation: Ukr. Math. J., 65, No. 9, 1319–1331 (2014).

  9. N. P. Korneichuk, Exact Constants in the Theory of Approximation [in Russian], Nauka, Moscow (1987).

    Google Scholar 

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

    MATH  Google Scholar 

  11. A. S. Fedorenko, “Best m-term trigonometric approximations of the classes of (ψ, β)-differentiable functions of one variable,” Ukr. Mat. Zh., 52, No. 6, 850–855 (2000); English translation: Ukr. Math. J., 52, No. 6, 974–980 (2000).

  12. V. V. Shkapa, “Best orthogonal trigonometric approximations of functions from the classes L ψ β,1 ,” in: Approximation Theory of Functions and Related Problems [in Ukrainian], Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv, 11, No. 3 (2014), pp. 315–329.

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

    Article  MathSciNet  Google Scholar 

  14. V. N. Temlyakov, “Estimates for the best bilinear approximations of functions of two variables and some their applications,” Mat. Sb., 134, No. 1, 93–107 (1987).

    MathSciNet  Google Scholar 

  15. V. N. Temlyakov, “Bilinear approximation and applications,” Tr. Mat. Inst. Akad. Nauk SSSR, 187, 191–215 (1989).

    MathSciNet  Google Scholar 

  16. É. S. Belinskii, “Approximation by a ‘floating’ system of exponents on classes of periodic functions with bounded mixed derivative,” in: Investigation in the Theory of Functions of Many Real Variables [in Russian], Yaroslavl’ Universitet, Yaroslavl’ (1988), pp. 16–33.

  17. A. S. Romanyuk, “Best trigonometric and bilinear approximations of functions of several variables from the classes B r p,θ . I,” Ukr. Mat. Zh., 44, No. 11, 1535–1547 (1992); English translation: Ukr. Math. J., 44, No. 11, 1411–1424 (1992).

  18. A. S. Romanyuk, “The best trigonometric and bilinear approximations of functions of many variables from the classes B r p,θ . II,” Ukr. Mat. Zh., 45, No. 10, 1411–1423 (1993); English translation: Ukr. Math. J., 45, No. 10, 1583–1597 (1993).

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

  20. 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).

  21. K. V. Solich, “Estimates for bilinear approximations of the classes S Ω p,θ B of periodic functions of two variables,” Ukr. Mat. Zh., 64,

  22. No. 8, 1106–1120 (2012); English translation: Ukr. Math. J., 64, No. 8, 1260–1276 (2012).

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

    Google Scholar 

  24. A. S. Romanyuk, “Inequalities for the L p -norms of (ψ, β)-derivatives and Kolmogorov widths of the classes of functions of many variables L ψ β,p ,” in: Investigations in the Approximation Theory of Functions [in Russian], Proc. of the Institute of Mathematics, Academy of Sciences of Ukr. SSR, Kiev (1987), pp. 92–105.

  25. A. S. Fedorenko, “On the bestm-term trigonometric approximations of the classes of (ψ, β)-differentiable functions of one variable,” in: Boundary-Value Problems for Differential Equations, Issue 3 (1998), pp. 250–258.

  26. V. V. Shkapa, “Best approximations of the Bernoulli kernels and classes of (ψ, β)-differentiable periodic functions,” in: Mathematical Problems of Mechanics and Numerical Mathematics [in Ukrainian], Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv, 11, No. 4 (2014), pp. 413–424.

  27. V. V. Shkapa, “Estimates for the best m-term and orthogonal trigonometric approximations of functions from the classes L ψ β,p in the uniform metric,” in: Differential Equations and Related Problems [in Ukrainian], Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv, 11, No. 2 (2014), pp. 305–317.

  28. V. V. Shkapa, “Approximating characteristics of the classes L ψ β,p of periodic functions in the space L q ,” Ukr. Mat. Zh., 67, No. 8, 1139–1150 (2015); English translation: Ukr. Math. J., 67, No. 8, 1283–1291 (2015).

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

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    V. V. Shkapa

Authors
  1. V. V. Shkapa
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 68, No. 3, pp. 387–400, March, 2016.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shkapa, V.V. Best Trigonometric and Bilinear Approximations for the Classes of (ψ, β)-Differentiable Periodic Functions. Ukr Math J 68, 433–447 (2016). https://doi.org/10.1007/s11253-016-1232-3

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-016-1232-3

Search

Navigation