Skip to main content
SpringerLink
Log in

Approximative characteristics and properties of operators of the best approximation of classes of functions from the Sobolev and Nikol’skii–Besov spaces

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

We have obtained the exact-order estimates for some approximative characteristics of the Sobolev classes \( {\mathbbm{W}}_{p,\alpha}^r \) and Nikоl’skii–Besov classes \( {\mathbbm{B}}_{p,\theta}^r \) of periodic functions of one and several variables in the norm of the space B∞,1. Properties of the linear operators realizing the orders of the best approximation for the classes \( {\mathbbm{B}}_{\infty, \theta}^r \) in this space by trigonometric polynomials generated by a set of harmonics with “numbers” from step hyperbolic crosses are 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. A. S. Romanyuk and V. S. Romanyuk, “Estimates of some approximative characteristics of classes of periodic functions of one variable and many ones,” Ukr. Mat. Zh., 71(8), 1102–1115 (2019).

    Google Scholar 

  2. 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  Google Scholar 

  3. A. S. Romanyuk and V. S. Romanyuk, “Approximative characteristics of classes of periodic functions of many variables in the space B∞,1,” Ukr. Mat. Zh., 71(2), 271–282 (2019).

    Article  Google Scholar 

  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(1), 74–85 (2019).

  5. P. I. Lizorkin and S. M. Nikol’skii, “The spaces of functions with mixed smoothness from the decompositional viewpoint,” Trudy Mat. Inst. AN SSSR, 187, 143–161 (1989).

    Google Scholar 

  6. O. V. Besov, “Studies of a family of functional spaces in connection with the theorems of embedding and continuation,” Trudy Mat. Inst. AN SSSR, 60, 42–61 (1961).

    Google Scholar 

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

    Google Scholar 

  8. T. I. Amanov, “The theorems of representation and embedding for functional spaces \( {S}_{p,\theta}^{(r)}B\left({R}_n\right) \) and \( {S}_{p,\theta}^{\underset{\ast }{(r)}}B, \) (0 ≤ xj ≤ 2π; j = 1, . . . , n),” Trudy Mat. Inst. AN SSSR, 77, 5–34 (1965).

  9. V. N. Temlyakov, “The appproximation of functions with bounded mixed derivative,” Trudy Mat. Inst. AN SSSR, 178, 1–112 (1986).

    Google Scholar 

  10. V. N. Temlyakov, Approximation of Periodic Functions, Nova Sci., New York, 1993.

    MATH  Google Scholar 

  11. A. S. Romanyuk, Approximative Characteristics of Classes of Periodic Functions of Many Variables [in Russian], Institute of Mathematics of the NAS of Ukraine, Kiev, 2012.

  12. Dinh Düng, V. Temlyakov, and T. Ullrich, Hyperbolic Cross Approximation. Birkhäuser, Basel, 2018.

  13. V. N. Temlyakov, “The widths of some classes of functions of several variables,” Dokl. AN SSSR, 267(2), 314–317 (1982).

    Google Scholar 

  14. Dinh Düng, “The approximation of functions of many variables on a torus by trigonometric polynomials,” Matem. Sbornik, 131(173)(2), 251–271 (1986).

  15. É. M. Galeev, “Orders of orthoprojective widths of classes of periodic functions of one variable and several ones,” Mat. Zametki, 43(2), 197–211 (1988).

    MathSciNet  MATH  Google Scholar 

  16. V. N. Temlyakov, “The estimates of asymptotic characteristics of classes of functions with bounded mixed derivative or difference,” Trudy Mat. Inst. AN SSSR, 189, 138–168 (1989).

    MathSciNet  Google Scholar 

  17. É. M. Galeev, “Approximation of classes of periodic functions of several variables by operators of the trace class,” Mat. Zametki, 47(3), 32–41 (1990).

    MathSciNet  Google Scholar 

  18. A. V. Andrianov and V. N. Temlyakov, “On two methods of expansion of properties of systems of functions of one variable into their tensor product,” Trudy Mat. Inst. RAN, 219, 32–43 (1997).

    Google Scholar 

  19. A. S. Romanyuk, “The estimates of approximative characteristics of the Besov classes \( {B}_{p,\theta}^r \) of periodic functions of many variables on the space Lq, I,” Ukr. Mat. Zh., 53(9), 1224–1231 (2001).

    Article  Google Scholar 

  20. A. S. Romanyuk, “The estimates of approximative characteristics of the Besov classes \( {B}_{p,\theta}^r \) of periodic functions of many variables on the space Lq, II.” Ukr. Mat. Zh., 53(10), 1402–1408 (2001).

    Article  Google Scholar 

  21. S. A. Stasyuk and O. V. Fedunyk, “The approximative characteristics of the classes \( {B}_{p,\theta}^{\Omega} \) of periodic functions of many variables,” Ukr. Mat. Zh., 58(5), 692–704 (2006).

    Article  Google Scholar 

  22. N. N. Pustovoitov, “The orthowidths of classes of multidimensional periodic functions whose majorant of mixed moduli of continuity contains power and logarithmic factors,” Anal. Math., 34(3), 187–224 (2008).

    Article  MathSciNet  Google Scholar 

  23. G. A. Akishev, “In the orthowidths of the Nikol’skii–Besov classes in Lorentz spaces,” Izv. Vyssh. Ucheb. Zav. Mat., 2, 25–33 (2009).

    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).

    Article  MathSciNet  Google Scholar 

  25. D. B. Bazarkhanov, “Estimates of the Fourier widths of classes of the Nikol’skii–Besov and Lizorkin–Triebel types of periodic functions of many variables,” Mat. Zametki, 87(2), 305–308 (2010).

    Article  Google Scholar 

  26. D. B. Bazarkhanov, “The approximation by splashes and the Fourier widths of classes of periodic functions of many variables, II,” Anal. Math., 38(4), 249–289 (2012).

  27. K. A. Bekmaganbetov and Ye. Tolengazy, “Order of the orthoprojection widths of the anisotropic Nikol’skii–Besov classes in the anisotropic Lorentz space,” Eurasian Math. J., 7(3), 8–16 (2016).

  28. Sh. A. Balgimbaeva and T. I. Smirnov, “Estimates of the Fourier widths of classes of periodic functions with a given majorant of the modulus of smoothness,” Sibir. Mat. Zh., 59(2), 277–292 (2018).

    MathSciNet  Google Scholar 

  29. O. V. Fedunyk-Yaremchuk and S. B. Hembars’ka, “Estimates of approximative characteristics of the classes \( {B}_{p,\theta}^{\Omega} \) of periodic functions of several variables with given majorant of mixed moduli of continuity in the space Lq,” Carpathian Math. Publ., 11(2), 281–295 (2019).

    Article  MathSciNet  Google Scholar 

  30. A. S. Romanyuk, “Approximation of the classes \( {B}_{p,\theta}^r \) of periodic functions of many variables by linear methods and the best approximations,” Matem. Sbornik, 195(2), 91–116 (2004).

    Article  Google Scholar 

  31. V. N. Temlyakov, “Approximation of periodic functions of several variables by trigonometric polynomials and the widths of several classes,” Izv. AN SSSR. Ser. Mat., 49(5), 986–1030 (1985).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics of the NAS of Ukraine, Kyiv, Ukraine

    Anatolii Sergiiovych Romanyuk & Viktor Sergiiovych Romanyuk

Authors
  1. Anatolii Sergiiovych Romanyuk
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Viktor Sergiiovych Romanyuk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anatolii Sergiiovych Romanyuk.

Additional information

Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 17, No. 3, pp. 372–395 July–September, 2020.

Translated from Ukrainian by V.V. Kukhtin

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Romanyuk, A.S., Romanyuk, V.S. Approximative characteristics and properties of operators of the best approximation of classes of functions from the Sobolev and Nikol’skii–Besov spaces. J Math Sci 252, 508–525 (2021). https://doi.org/10.1007/s10958-020-05177-2

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-020-05177-2

Keywords

Search

Navigation