Skip to main content
SpringerLink
Log in

Approximation of functions from Nikolskii–Besov type classes of generalized mixed smoothness

Приближѳниѳ функций из классов типа Никольского-Бесова обобщeнной смешаннои гладкости

  • Published:
Analysis Mathematica Aims and scope Submit manuscript

Abstract

We obtain order estimates of approximation of the classes S Ω p,θ B of functions of several variables defined on ℝd, in the L q norm, by entire functions of exponential type with supports of their Fourier transforms in sets that generated by the level surfaces of a function Ω(t).

Резюме

Получены порядковые оценки приближения классов S Ω p,θ B функций мпогих переменпых, определенных на ℝd, в метрике пространства L q с помощью целых функций экспоненциального типа с носителями их преобразований Фурье на множествах, которые порождаются поверхностями уровня функции Q(t).

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. N. K. Bari and S. B. Stechkin, Best approximations and differential properties of two conjugate functions, Tr. Mosk. Mat. Obshch., 5(1956), 483–522 (in Russian).

    MATH  Google Scholar 

  2. S. A. Stasyuk and S. Ya. Yanchenko, Best approximations of the classes BO p,? of functions of many variables in the space L p(Rd), Zb. Pr. Inst. Mat. NAN Ukr., 5(1)(2008), 367–384 (in Ukrainian).

    MATH  Google Scholar 

  3. S. M. Nikolskii, Functions with dominant mixed derivative, satisfying a multiple Hölder condition, Sibirsk. Mat. Zh., 4(1963), 1342–1364 (in Russian).

    Google Scholar 

  4. T. I. Amanov, Representation and imbedding theorems for the function spaces S (r) p,θ B(Rn) and S (r)* p,θ B, (0 ≥ xj ≥ 2π, j = 1,..., n), Trudy Mat. Inst. Steklov, 77(1965), 5–34 (in Russian).

    MathSciNet  MATH  Google Scholar 

  5. P. I. Lizorkin, Generalized Liouville differentiation and the multiplier method in the theory of imbeddings of classes of differentiable functions, Trudy Mat. Inst. Steklov, 105(1969), 89–167 (in Russian).

    MathSciNet  MATH  Google Scholar 

  6. V. S. Vladimirov, Equations of mathematical physics, Nauka (Moscow, 1967).

    Google Scholar 

  7. S. M. Nikolskii, Approximation of functions of several variables and imbedding theorems, Springer-Verlag (New York–Heidelberg, 1975).

    Book  Google Scholar 

  8. P. I. Lizorkin, A Littlewood–Paley type theorem for multiple Fourier integrals, Trudy Mat. Inst. Steklov, 89(1967), 214–230 (in Russian).

    MathSciNet  MATH  Google Scholar 

  9. P. I. Lizorkin, Properties of functions in the spaces Λr p,θ , Trudy Mat. Inst. Steklov, 131(1974), 158–181 (in Russian).

    MathSciNet  Google Scholar 

  10. P. I. Lizorkin and S. M. Nikol’skii, Function spaces of mixed smoothness from the decomposition point of view, Proc. Steklov Inst. Math., 187(1990), 163–184.

    MATH  Google Scholar 

  11. Sun Yongsheng and Wang Heping, Representation and approximation of multivariate periodic functions with bounded mixed moduli of smoothness, Proc. Steklov Inst. Math., 219(1997), 350–371.

    Google Scholar 

  12. N. N. Pustovoitov, Representation and approximation of periodic functions of several variables with given mixed modulus of continuity, Analysis Math., 20(1994), 35–48 (in Russian).

    Article  MathSciNet  Google Scholar 

  13. A. S. Romanyuk, Approximation of classes of periodic functions of several variables, Ukrainian Math. J., 44(1992), 596–606.

    Article  MathSciNet  MATH  Google Scholar 

  14. N. N. Pustovoitov, Approximation of multidimensional functions with a given majorant of mixed moduli of continuity, Math. Notes, 65(1999), 89–98.

    Article  MathSciNet  MATH  Google Scholar 

  15. S. A. Stasyuk, Best approximations of periodic functions of several variables from the classes B Ω p,θ , Math. Notes, 87(2010), 102–114.

    Article  MathSciNet  MATH  Google Scholar 

  16. S. A. Stasyuk, Approximation by Fourier sums and Kolmogorov widths for classes MBΩ p,θ of periodic functions of several variables, Trudy Inst. Mat. Mekh., 20(2014), 247–257 (in Russian).

    MathSciNet  Google Scholar 

  17. G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Russian translation, Izd. Inostr. Literatury (Moscow,1948).

    Google Scholar 

  18. Wang Heping and Sun Yongsheng, Approximation of multivariate functions with certain mixed smoothness by entire functions, Northeast. Math. J., 11(1995), 454–466.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, National Academy of Sciences of Ukraine, Tereshchenkivska St. 3, 01601, Kyiv, Ukraine

    S. A. Stasyuk & S. Ya. Yanchenko

Authors
  1. S. A. Stasyuk
    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. A. Stasyuk.

Additional information

The first author was partially supported by the FP7-People-2011-IRSES project number 295164 (EUMLS: EU-Ukrainian Mathematicians for Life Sciences)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Stasyuk, S.A., Yanchenko, S.Y. Approximation of functions from Nikolskii–Besov type classes of generalized mixed smoothness. Anal Math 41, 311–334 (2015). https://doi.org/10.1007/s10476-015-0305-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10476-015-0305-0

Keywords

Search

Navigation