Skip to main content
SpringerLink
Log in

Approximating Characteristics of the Nikol’skii–Besov Classes \( {S}_{1,\theta}^rB\left({\mathrm{\mathbb{R}}}^d\right) \)

  • Published:
Ukrainian Mathematical Journal Aims and scope

We establish the exact-order estimates for the approximation of the classes \( {S}_{1,\theta}^rB\left({\mathrm{\mathbb{R}}}^d\right) \) by entire functions of exponential type such that the supports of their Fourier transforms lie in a step hyperbolic cross. The error of approximation is estimated in the metric of the Lebesgue space Lq(ℝd), 1 < q ≤  ∞ .

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. H. Wang and Y. Sun, “Approximation of multivariate functions with certain mixed smoothness by entire functions,” Northeast. Math. J., 11, No. 4, 454–466 (1995).

    MathSciNet  MATH  Google Scholar 

  2. H. Wang and Y. Sun, “Approximation of functions in \( \tilde{S_1^rL},{S}_1^rH \) by entire functions,” Approxim. Theory Appl., 15, No. 4, 88–93 (1999).

    Google Scholar 

  3. S. Ya. Yanchenko, “Approximation of the classes \( {S}_{1,\theta}^rB\left({\mathrm{\mathbb{R}}}^d\right) \) of functions of many variables by entire functions of a special form,” Ukr. Mat. Zh., 62, No. 8, 1124–1138 (2010); English translation:Ukr. Math. J., 62, No. 8, 1307–1325 (2011).

  4. S. Ya. Yanchenko, “Approximation of functions from the classes \( {S}_{p,\theta}^rB \) in the uniform metric,” Ukr. Mat. Zh., 65, No. 5, 698–705 (2013); English translation:Ukr. Math. J., 65, No. 5, 771–779 (2013).

  5. S. Ya. Yanchenko, “Estimates for the approximating characteristics of the classes of functions \( {S}_{1,\theta}^rB\left({\mathrm{\mathbb{R}}}^d\right) \) in the uniform metric,” in: “Approximation Theory of Functions and Related Problems,” Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences [in Ukrainian], Kyiv, 10, No. 1 (2013), pp. 328–340.

  6. S. M. Nikol’skii, “Functions with predominant mixed derivative satisfying the multiple H¨older condition,” Sib. Mat. Zh., 4, No. 6, 1342–1364 (1963).

    Google Scholar 

  7. T. I. Amanov, “Theorems on representation and embedding for the function spaces \( {S}_{p,\theta}^rB\left({\mathrm{\mathbb{R}}}_n\right) \) and \( {S_{p,\theta}^{(r)}}^{\ast }B\left(0\le {x}_j\le 2\pi, j=1,\dots, n\right), \)” Tr. Mat. Inst. Akad. Nauk SSSR, 77, 5–34 (1965).

    Google Scholar 

  8. 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).

    Google Scholar 

  9. P. I. Lizorkin, “Generalized Liouville differentiation and the method of multiplicators in the theory of embeddings of classes of differentiable functions,” Tr. Mat. Inst. Akad. Nauk SSSR, 105, 89–167 (1969).

    Google Scholar 

  10. V. S. Vladimirov, Equations of Mathematical Physics [in Russian], Nauka, Moscow (1981).

    Google Scholar 

  11. Y. M. Berezans’kyi, H. F. Us, and Z. G. Sheftel’, Functional Analysis [in Ukrainian], Chyzhykov, Lviv (2014).

    Google Scholar 

  12. H. Wang, “Quadrature formulas for classes of functions with bounded mixed derivative or difference,” Sci. China (Ser. A), 40, No. 5, 449–458 (1997).

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

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

    MathSciNet  Google Scholar 

  15. G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities [Russian translation], Inostrannaya Literatura, Moscow (1948).

    Google Scholar 

  16. H. Wang, “Representation and approximation of multivariate function with bounded mixed smoothness by hyperbolic wavelets,” J. Math. Anal. Appl., 291, 698–715 (2004).

    Article  MathSciNet  Google Scholar 

  17. H.-J. Schmeisser and W. Sickel, “Spaces of functions of mixed smoothness and approximation from hyperbolic crosses,” J. Approxim. Theory, 128, 115–150 (2004).

    Article  MathSciNet  Google Scholar 

  18. V. N. Temlyakov, “Estimation of the asymptotic characteristics of classes of functions with bounded mixed derivative,” Tr. Mat. Inst. Akad. Nauk SSSR, 189, 138–168 (1988).

    Google Scholar 

  19. A. S. Romanyuk, “Approximation of the Besov classes of periodic functions of several variables in a space Lq,” Ukr. Mat. Zh., 43, No. 10, 1398–1408 (1991); English translation:Ukr. Math. J., 43, No. 10, 1297–1306 (1991).

  20. A. S. Romanyuk, “Approximation of the classes \( {B}_{p,\theta}^r \) of periodic multivariable functions by linear methods and the best approximations,” Mat. Sb., 195, No. 2, 91–116 (2004).

    Article  Google Scholar 

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

    Article  Google Scholar 

  22. É. M. Galeev, “Approximation of the classes of periodic functions of several variables by kernel operators,” Mat. Zametki, 47, No. 3, 32–41 (1990).

    MathSciNet  Google Scholar 

  23. A. S. Romanyuk, Approximating Characteristics of the Classes of Periodic Multivariate Functions [in Ukrainian], Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv (2012).

    Google Scholar 

  24. D. Dũng, V. N. Temlyakov, and T. Ullrich, Hyperbolic Cross Approximation, Preprint arXiv:1601.03978v3 [math.NA] (2017).

  25. N. N. Pustovoitov, “Representation and approximation of periodic multivariable functions with given mixed modulus of continuity,” Anal. Math., 20, 35–48 (1994).

    Article  MathSciNet  Google Scholar 

  26. N. N. Pustovoitov, “Approximation of multidimensional functions with given majorant of the mixed moduli of continuity,” Mat. Zametki, 65, No. 1, 107–117 (1999).

    Article  MathSciNet  Google Scholar 

  27. 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).

    MathSciNet  Google Scholar 

  28. S. A. Stasyuk, “Best approximations and Kolmogorov and trigonometric widths of the classes \( {B}_{p,\theta}^{\varOmega } \) 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).

  29. S. A. Stasyuk and S. Ya. Yanchenko, “Approximation of functions from Nikolskii–Besov type classes of generalized mixed smoothness,” Anal. Math., 41, 311–334 (2015).

    Article  MathSciNet  Google Scholar 

  30. S. Ya. Yanchenko, “Approximations of classes \( {B}_{p,\theta}^{\varOmega } \) of functions of many variables by entire functions in the space Lq(ℝd),” Ukr. Mat. Zh., 62, No. 1, 123–135 (2010); English translation:Ukr. Math. J., 62, No. 1, 136–150 (2010).

Download references

Author information

Authors and Affiliations

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

    S. Ya. Yanchenko

  2. Hnatyuk Ternopil’ National Pedagogic University, Ternopil’, Ukraine

    O. Ya. Radchenko

Authors
  1. S. Ya. Yanchenko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. O. Ya. Radchenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Ya. Yanchenko.

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 71, No. 10, pp. 1405–1421, October, 2019.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yanchenko, S.Y., Radchenko, O.Y. Approximating Characteristics of the Nikol’skii–Besov Classes \( {S}_{1,\theta}^rB\left({\mathrm{\mathbb{R}}}^d\right) \). Ukr Math J 71, 1608–1626 (2020). https://doi.org/10.1007/s11253-020-01734-9

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-020-01734-9

Search

Navigation