Skip to main content
SpringerLink
Log in

Estimates of the approximation characteristics of the classes B Ω p,θ of periodic functions of several variables with given majorant of mixed moduli of continuity

  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

We obtain order-sharp estimates of the orthogonal projection widths of the classes B Ω p,θ of periodic functions of several variables whose majorant of the mixed moduli of continuity contains both exponential and logarithmic multipliers.

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. F. Konograi, “Estimates of the approximation characteristics of the classes B Ω p,θ of periodic functions of two variables with a given majorant of mixedmoduli of continuity,” Ukrain.Mat. Zh. 63(2), 176–186 (2011).

    Google Scholar 

  2. A. F. Konograi, “Estimates of the approximation characteristics of the classes B Ω p,θ of periodic functions of two variables with a given majorant of mixed moduli of continuity in L ∞,” in Zb. Prats. Inst. Mat. NAN Ukrain., Vol. 8: Theory of Approximation of Functions and Related Questions (Inst.Mat. NAN Ukrain., Kiev, 2011), pp. 97–110 [in Ukrainian].

    Google Scholar 

  3. N. K. Bari and S. B. Stechkin, “Best approximations and differential properties of two conjugate functions,” in Trudy Moskov. Mat. Obshch. (GITTL, Moscow, 1956), Vol. 5, pp. 483–522 [in Russian].

    Google Scholar 

  4. S. Yongsheng and W. Heping, “Representation and Approximation Multivariate Periodic Functions with Bounded Mixed Moduli of Smoothness,” in Trudy Mat. Inst. Steklov Vol. 219: Approximation Theory. Harmonic Analysis (Nauka, Moscow, 1997), pp. 356–377 [Proc. Steklov Inst.Math. 219, 350–371 (1997)].

    Google Scholar 

  5. N. N. Pustovoitov, “Representation and approximation of periodic functions of several variables with given mixed modulus of continuity,” Anal. Math. 20(1), 35–48 (1994).

    Article  MathSciNet  Google Scholar 

  6. P. I. Lizorkin and S.M. Nikol’skii, “Spaces of functions ofmixed smoothness from the decomposition point of view,” in Trudy Mat. Inst. Steklov Vol. 187: Studies in the Theory of Differentiable Functions of Several Variables and Its Applications. Part 13 (Nauka, Moscow, 1989), pp. 143–161 [Proc. Steklov Inst. Math. 187, 163–184 (1990)].

    Google Scholar 

  7. S. A. Stasyuk and O. V. Fedunik, “Approximation characteristics of the classes B Ω p,θ of periodic functions of many variables,” Ukrain.Mat. Zh. 58(5), 692–704 (2006) [Ukrainian Math. J. 58 (5), 779–793 (2006)].

    Article  MATH  MathSciNet  Google Scholar 

  8. V. N. Temlyakov, “Diameters of some classes of functions of several variables,” Dokl. Akad. Nauk SSSR 267(2), 314–317 (1982).

    MathSciNet  Google Scholar 

  9. Din’ Zung, “Approximation by trigonometric polynomials of functions of several variables on the torus,” Mat. Sb. 131(2), 251–271 (1986) [Math. USSR–Sb. 59 (1), 247–267 (1988)].

    Google Scholar 

  10. V. N. Temlyakov, “Estimates of the asymptotic characteristics of classes of functions with bounded mixed derivative or difference,” in Trudy Mat. Inst. Steklov Vol. 189: Collection of Works of the All-Union Workshop in Function Theory (Nauka, Moscow, 1989), pp. 138–168 [Proc. Steklov Inst. Math. 189, 161–197 (1990)].

    Google Scholar 

  11. É. M. Galeev, “Orders of the orthoprojection widths of classes of periodic functions of one and of several variables,” Mat. Zametki 43(2), 197–211 (1988) [Math. Notes 43 (2), 110–118 (1988)].

    MATH  Google Scholar 

  12. É. M. Galeev, “Approximation of classes of periodic functions of several variables by trace-class operators,” Mat. Zametki 47(3), 32–41 (1990) [Math. Notes 47 (3), 248–254 (1990)].

    MathSciNet  Google Scholar 

  13. A. V. Andrianov and V. N. Temlyakov, “On two methods of generalization of properties of univariate function systems to their tensor product,” in Trudy Mat. Inst. Steklov Vol. 219: Approximation Theory. Harmonic Analysis (Nauka, Moscow, 1997), pp. 32–43 [Proc. Steklov Inst. Math. 219, 25–35 (1997)].

    Google Scholar 

  14. N. N. Pustovoitov, “On the widths of multivariate periodic classes of functions whose mixed moduli of continuity are bounded by a product of power- and logarithmic-type functions,” Anal. Math. 34(3), 187–224 (2008).

    Article  MATH  MathSciNet  Google Scholar 

  15. A. S. Romanyuk, “Best approximations and widths of classes of periodic functions of several variables,” Mat. Sb. 199(2), 93–114 (2008) [Sb.Math. 199 (2), 253–275 (2008)].

    Article  MathSciNet  Google Scholar 

  16. A. S. Romanyuk, “Diameters and best approximation of the classes B r p,θ of periodic functions of several variables,” Anal.Math. 37, 181–213 (2011).

    Article  MATH  MathSciNet  Google Scholar 

  17. V. N. Temlyakov, “Approximation of functions with a bounded mixed derivative,” in Trudy Mat. Inst. Steklov (Nauka, Moscow, 1986), Vol. 178, pp. 3–113 [Proc. Steklov Inst.Math. 178, 1–121 (1989)].

    Google Scholar 

  18. V. N. Temlyakov, Approximation Periodic Functions, in Comput. Math. Anal. Ser. (Nova Sci. Publ., Commack, NY, 1993).

    Google Scholar 

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

    Google Scholar 

  20. S. M. Nikol’skii, “Inequalities for entire functions of finite degree and their application in the theory of differentiable functions of several variables,” in Trudy Mat. Inst. Steklov (Izd. AN SSSR, Moscow, 1951), Vol. 38, pp. 244–278 [in Russian].

    Google Scholar 

  21. D. Jackson, “Certain problem of closest approximation,” Bull. Amer.Math. Soc. 39(12), 889–906 (1933).

    Article  MathSciNet  Google Scholar 

  22. N. N. Pustovoitov, “Approximation of multidimensional functions with a given majorant of mixed moduli of continuity,” Mat. Zametki 65(1), 107–117 (1999) [Math. Notes 65 (1), 89–98 (1999)].

    Article  MathSciNet  Google Scholar 

  23. G. Hardy, D. Littlewood, and G. Pólya, Inequalities (Cambridge Univ. Press, Cambridge, 1934; Inostr. Lit., Moscow, 1948).

    Google Scholar 

Download references

Author information

Authors and Affiliations

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

    A. F. Konograi

Authors
  1. A. F. Konograi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. F. Konograi.

Additional information

Original Russian Text © A. F. Konograi, 2014, published in Matematicheskie Zametki, 2014, Vol. 95, No. 5, pp. 734–749.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Konograi, A.F. Estimates of the approximation characteristics of the classes B Ω p,θ of periodic functions of several variables with given majorant of mixed moduli of continuity. Math Notes 95, 656–669 (2014). https://doi.org/10.1134/S0001434614050095

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0001434614050095

Keywords

Search

Navigation