Skip to main content
SpringerLink
Log in

Approximation of the Classes of Periodic Functions of One and Many Variables from the Nikol’skii–Besov and Sobolev Spaces

  • Published:
Ukrainian Mathematical Journal Aims and scope

We establish the exact-order estimates for the best orthogonal trigonometric approximations of the Nikol’skii–Besov classes \( {B}_{1,\theta}^r \)(\( \mathbbm{T} \)d), 1 ≤ θ ≤ ∞, of periodic functions of one and many variables with predominant mixed derivative in the space Bθ, 1 (\( \mathbbm{T} \)d). In the multidimensional case, d ≥ 2, we establish the exact-order estimates for the approximations of the indicated classes of functions by their step-hyperbolic Fourier sums and determine the orders of orthoprojection widths in the same space. The behaviors of the corresponding approximation characteristics of the Sobolev classes \( {W}_{1,\alpha}^r \)(\( \mathbbm{T} \)d) with d ∈ {1, 2} are also investigated.

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. E. S. Belinsky, “Estimates of entropy numbers and Gaussian measures for classes of functions with bounded mixed derivative,” J. Approx. Theory, 93, 114–127 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  2. A. S. Romanyuk and V. S. Romanyuk, “Approximating characteristics of the classes of periodic multivariate functions in the space B∞,1,” Ukr. Mat. Zh., 71, No. 2, 271–282 (2019); English translation: Ukr. Math. J., 71, No. 2, 308–321 (2019).

  3. A. S. Romanyuk and V. S. Romanyuk, “Estimation of some approximating characteristics of the classes of periodic functions of one and many variables,” Ukr. Mat. Zh., 71, No. 8, 1102–1115 (2019); English translation: Ukr. Math. J., 71, No. 8, 1257–1272 (2020).

  4. M. V. Hembarskyi, S. B. Hembarska, and K. V. Solich, “The best approximations and widths of the classes of periodical functions of one and several variables in the space B∞,1,” Mat. Stud., 51, No. 1, 74–85 (2019).

    Article  MathSciNet  Google Scholar 

  5. A. S. Romanyuk and V. S. Romanyuk, “Approximating characteristics and properties of operators of the best approximation of classes of functions from the Sobolev and Nikol’skii–Besov spaces,” Ukr. Mat. Visn., 17, No. 3, 372–395 (2020).

    Google Scholar 

  6. T. I. Amanov, “Representation and embedding theorems for the function spaces \( {S}_{p,\theta}^{(r)} \) B(ℝn) and \( {S}_{p,\theta}^{(r)} \)* B (0 ≤ xj ≤ 2π; j = 1, . . . ,n),” Tr. Mat. Inst. Akad. Nauk SSSR, 77, 5–34 (1965).

  7. 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).

  8. É. S. Belinskii, “Approximation by a ‘floating’ system of exponents on classes of periodic functions with bounded mixed derivative,” in: Investigations in the Theory of Functions of Many Real Variables [in Russian], Yaroslavl. Univ. (1990), pp. 16–33.

  9. A. S. Romanyuk, “Approximation of classes of functions of many variables by their orthogonal projections onto subspaces of trigonometric polynomials,” Ukr. Mat. Zh., 48, No. 1, 80–89 (1996); English translation: Ukr. Math. J., 48, No. 1, 90–100 (1996).

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

    MathSciNet  MATH  Google Scholar 

  11. 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).

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

    Google Scholar 

  13. A. S. Romanyuk, 1 Approximating Characteristics of Classes of Periodic Functions of Many Variables [in Russian], Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv (2012).

  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. V. N. Temlyakov, 1 Approximation of Periodic Functions, Nova Science Publishers, New York (1993).

    MATH  Google Scholar 

  16. D. Dung, “Approximation of functions of many variables on a torus by trigonometric polynomials,” Mat. Sb., 131(173), No. 2, 251–271 (1986).

  17. 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).

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

  19. V. N. Temlyakov, “Widths of some classes of functions of several variables,” Dokl. Akad. Nauk SSSR, 267, No. 2, 314–317 (1982).

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  21. A. V. Andrianov and V. N. Temlyakov, “On two methods of extension of the properties of systems of functions of one variable to their tensor product,” Tr. Mat. Inst. Ros. Akad. Nauk, 219, 32–43 (1997).

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  23. A. S. Romanyuk, “Best approximations and widths of classes of periodic functions of many variables,” Mat. Sb., 199, No. 2, 93–114 (2008).

    Google Scholar 

  24. A. S. Romanyuk, “Widths and the best approximation of the classes \( {B}_{p,\theta}^r \) of periodic functions of many variables,” Anal. Math., 37, 181–213 (2011).

  25. D. B. Bazarkhanov, “Estimates of the Fourier widths of Niklo’skii–Besov-type and Lizorkin–Triebel-type classes of periodic functions of many variables,” Mat. Zametki, 87, No. 2, 305–308 (2010).

    Google Scholar 

  26. D. B. Bazarkhanov, “Approximation by splashes and Fourier widths of classes of periodic functions of many variables. II,” Anal. Math., 38, No. 4, 249–289 (2012).

    MathSciNet  MATH  Google Scholar 

  27. A. S. Romanyuk, “Estimates for approximation characteristics of the Besov classes \( {B}_{p,\theta}^r \) of periodic functions of many variables in the space Lq. I,” Ukr. Mat. Zh., 53, No. 9, 1224–1231 (2001); English translation: Ukr. Math. J., 53, No. 9, 1473–1482 (2001).

  28. A. S. Romanyuk, “Estimates for approximation characteristics of the Besov classes \( {B}_{p,\theta}^r \) of periodic functions of many variables in the space Lq. II,” Ukr. Mat. Zh., 53, No. 10, 1402–1408 (2001); English translation: Ukr. Math. J., 53, No. 10, 1703–1711 (2001).

  29. A. S. Romanyuk and S. Ya. Yanchenko, “Estimates of approximating characteristics and the properties of operators of best approximation for the classes of periodic functions in the space B1,1,” Ukr. Mat. Zh., 73, No. 8, 1102–1119 (2021); English translation: Ukr. Math. J., 73, No. 8, 1278–1298 (2022).

  30. D. Dũng, V. N. Temlyakov, and T. Ullrich, Hyperbolic Cross Approximation, Birkhäuser, CRM Barcelona (2018).

Download references

Author information

Authors and Affiliations

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

    A. S. Romanyuk & S. Ya. Yanchenko

Authors
  1. A. S. Romanyuk
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Ya. Yanchenko
    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. 74, No. 6, pp. 844–855, June, 2022. Ukrainian DOI: https://doi.org/10.37863/umzh.v74i6.7141.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Romanyuk, A.S., Yanchenko, S.Y. Approximation of the Classes of Periodic Functions of One and Many Variables from the Nikol’skii–Besov and Sobolev Spaces. Ukr Math J 74, 967–980 (2022). https://doi.org/10.1007/s11253-022-02110-5

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-022-02110-5

Search

Navigation