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Estimation of Some Approximating Characteristics of the Classes of Periodic Functions of One and Many Variables

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Ukrainian Mathematical Journal Aims and scope

We establish the exact-order estimates for some approximating characteristics of the classes \( {\mathbbm{W}}_{p,\alpha}^r \) and \( {\mathbbm{B}}_{p,\theta}^r \) of periodic functions of one and many variables in the norm of the space B∞,1.

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References

  1. 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).

  2. D. Ding, V. N. Temlyakov, and T. Ullrich, Hyperbolic Cross Approximation, Preprint arXiv: 1601.03978 v 3 [math. NA] 21 Apr. (2017).

  3. V. S. Romanyuk, “Kolmogorov widths and entropy numbers in the Orlicz spaces with Luxembourg norm,” Ukr. Mat. Zh., 69, No. 5, 682–694 (2017); English translation:Ukr. Math. J., 69, No. 5, 796–810 (2017).

  4. 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 

  5. O. V. Besov, “Investigations of one family of function spaces in connection with embedding and extension theorems,” Tr. Mat. Inst. Akad. Nauk SSSR, 60, 42–61 (1961).

    Google Scholar 

  6. S. M. Nikol’skii, “Inequalities for entire functions of finite degree and their application to the theory of differentiable functions of many variables,” Tr. Mat. Inst. Akad. Nauk SSSR, 38, 244–278 (1951).

    Google Scholar 

  7. T. I. Amanov, “Representation and embedding theorems for the function spaces \( {S}_{p,\theta}^{(r)}B\left({\mathbb{R}}^n\right) \) and \( {S}_{p,\theta}^{(r)}B\left(0\le {x}_j\le 2\uppi; j=1,\dots, n\right) \),” Tr. Mat. Inst. Akad. Nauk SSSR, 77, 5–34 (1965).

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

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

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

    MATH  Google Scholar 

  11. A. S. Romanyuk, 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).

  12. 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 

  13. A. S. Romanyuk, “Entropy numbers and widths for the classes \( {B}_{p,\theta}^r \) of periodic functions of many variables,” Ukr. Mat. Zh., 68, No. 10, 1403–1417 (2016); English translation:Ukr. Math. J., 68, No. 10, 1620–1636 (2017).

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

  15. B. Carl, “Entropy numbers, s-numbers, and eigenvalue problems,” J. Funct. Anal., 41, 290–306 (1981).

    Article  MathSciNet  Google Scholar 

  16. B. S. Kashin and V. N. Temlyakov, “On the estimate for approximating characteristics of classes of functions with bounded mixed derivative,” Mat. Zametki, 58, No. 6, 922–925 (1995).

    Google Scholar 

  17. V. N. Temlyakov, “An inequality for trigonometric polynomials and its application for estimating the Kolmogorov widths,” East J. Approxim., 2, No. 1, 89–98 (1996).

    MathSciNet  MATH  Google Scholar 

  18. 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).

    Article  Google Scholar 

  19. A. S. Romanyuk, “Approximations 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).

    Article  Google Scholar 

  20. 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).

  21. R. M. Trigub and E. S. Belinsky, Fourier Analysis and Approximation of Functions, Kluwer Academic Publishers, Dordrecht (2004).

    Book  Google Scholar 

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Authors and Affiliations

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

    A. S. Romanyuk

  2. Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv, Ukraine

    V. S. Romanyuk

Authors
  1. A. S. Romanyuk
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  2. V. S. Romanyuk
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Correspondence to A. S. Romanyuk.

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 71, No. 8, pp. 1102–1115, August, 2019.

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Romanyuk, A.S., Romanyuk, V.S. Estimation of Some Approximating Characteristics of the Classes of Periodic Functions of One and Many Variables. Ukr Math J 71, 1257–1272 (2020). https://doi.org/10.1007/s11253-019-01711-x

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  • DOI: https://doi.org/10.1007/s11253-019-01711-x

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