Skip to main content
Log in

Estimates for the Entropy Numbers of the Classes \( {B}_{p,\theta}^{\varOmega } \) of Periodic Multivariable Functions in the Uniform Metric

  • Published:
Ukrainian Mathematical Journal Aims and scope

We establish order estimates for the entropy numbers of the classes \( {B}_{p,\theta}^{\varOmega } \) of periodic multivariable functions in the uniform metric. For the proper choice of the functions Ω, these classes coincide with the Nikol’skii–Besov classes \( {B}_{p,\theta}^r. \)

This is a preview of subscription content, log in via an institution to check access.

Access this article

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. N. Kolmogorov and V. M. Tikhomirov, “𝜀-entropy and 𝜀 -capacity of sets in function spaces,” Usp. Mat. Nauk, 14, No. 2, 3–86 (1959).

    MathSciNet  Google Scholar 

  2. K. Höllig, “Diameters of classes of smooth functions,” in: Quantitative Approximation, Academic Press, New York (1980), pp. 163–176.

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

    MathSciNet  Google Scholar 

  4. E. S. Belinskii, “Approximation of functions of several variables by trigonometric polynomials with given number of harmonics, and estimates of 𝜀 -entropy,” Anal. Math., 15, 67–74 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  5. B. S. Kashin and V. N. Temlyakov, “On the best m-term approximations and entropy of sets in the space L 1 ,Mat. Zametki, 56, No. 5, 57–86 (1994).

    MathSciNet  MATH  Google Scholar 

  6. V. N. Temlyakov, “An inequality for trigonometric polynomials and its application for estimating the entropy numbers,” J. Complexity, 11, 293–307 (1995).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  8. V. N. Temlyakov, “An inequality for the entropy numbers and its application,” J. Approxim. Theory, 173, 110–121 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  9. A. S. Romanyuk, “Estimation of the entropy numbers and Kolmogorov widths for the Nikol’skii–Besov classes of periodic functions of many variables,” Ukr. Mat. Zh., 67, No. 11, 1540–1556 (2015); English translation: Ukr. Math. J., 67, No. 11, 1739–1757 (2016).

  10. D. Dung, V. N. Temlyakov, and T. Ullrich, Hyperbolic Cross Approximation, Preprint arXiv: 1601.03978v3 (2016).

  11. V. N. Temlyakov, “On the entropy numbers of the mixed smoothness function classes,” J. Approxim. Theory, 217, 26–56 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  12. N. N. Pustovoitov, “Representation and approximation of periodic functions of many variables with given modulus of continuity,” Anal. Math., 20, No. 2, 35–48 (1994).

    MathSciNet  Google Scholar 

  13. 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 

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

  15. 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 

  16. N. K. Bari and S. B. Stechkin, “Best approximations and differential properties of two adjoint functions,” Tr. Mosk. Mat. Obshch., 5, 483–522 (1956).

    Google Scholar 

  17. S. A. Stasyuk and O. V. Fedunyk, “Approximation characteristics of the classes \( {B}_{p,\theta}^{\varOmega } \) periodic functions of many variables,” Ukr. Mat. Zh., 58, No. 5, 692–704 (2006); English translation: Ukr. Math. J., 58, No. 5, 779–793 (2006).

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

    MathSciNet  Google Scholar 

  19. S. A. Stasyuk, “Approximation of the classes \( {B}_{p,\theta}^{\varOmega } \) of periodic functions of many variables in uniform metric,” Ukr. Mat. Zh., 54, No. 11, 1551–1559 (2002); English translation: Ukr. Math. J., 54, No. 11, 1885–1896 (2002).

  20. S. M. Nikol’skii, Approximation of Functions of Many Variables and Embedding Theorems [in Russian], Nauka, Moscow (1977).

    Google Scholar 

  21. G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities [Russian translation], Inostr. Lit., Moscow (1948).

    Google Scholar 

  22. A. Zygmund, Trigonometric Series [Russian translation], Vol. 1, Mir, Moscow (1965).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 70, No. 9, pp. 1249–1263, September, 2018.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pozhars’ka, K.V. Estimates for the Entropy Numbers of the Classes \( {B}_{p,\theta}^{\varOmega } \) of Periodic Multivariable Functions in the Uniform Metric. Ukr Math J 70, 1439–1455 (2019). https://doi.org/10.1007/s11253-019-01578-y

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-019-01578-y

Navigation