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Trigonometric and Linear Widths for the Classes of Periodic Multivariate Functions

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Ukrainian Mathematical Journal Aims and scope

We establish the exact-order estimates for the trigonometric widths of Nikol’skii–Besov \( {B}_{\infty, \theta}^r \) and Sobolev \( {W}_{\infty, \alpha}^r \) classes of periodic multivariate functions in the space L q , 1 < q < 1. The behavior of the linear widths of Nikol’skii–Besov \( {B}_{p,\theta}^r \) classes in the space L q is investigated for some relations between the parameters p and q.

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References

  1. A. S. Romanyuk, “Kolmogorov and trigonometric widths of the Besov classes \( {B}_{p,\theta}^r \) of periodic functions of many variables,” Mat. Sb., 197, No. 1, 71–96 (2006).

    Article  MathSciNet  Google Scholar 

  2. A. S. Romanyuk, “Best approximations and widths for the classes of periodic functions of many variables,” Mat. Sb., 199, No. 2, 93–114 (2008).

    Article  Google Scholar 

  3. 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  MATH  MathSciNet  Google Scholar 

  4. A. S. Romanyuk, “On the problem of linear widths of the classes \( {B}_{p,\theta}^r \) of periodic functions of many variables,” Ukr. Mat. Zh., 66, No. 7, 970–982 (2014); English translation: Ukr. Math. J., 66, No. 7, 1085–1098 (2014).

  5. 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).

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

    Google Scholar 

  7. O. V. Besov, V. P. Il’in, and S. M. Nikol’skii, Integral Representations of Functions and Embedding Theorems [in Russian], Nauka, Moscow (1975).

    MATH  Google Scholar 

  8. T. I. Amanov, Spaces of Differentiable Functions with Predominant Mixed Derivative [in Russian], Nauka, Alma-Ata (1976).

    Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  10. A. S. Romanyuk, Approximating Characteristics of the Classes of Periodic Functions of Many Variables [in Russian], Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv (2012).

  11. D. Ding, V. N. Temlyakov, and T. Ullrich, Hyperbolic Cross Approximation, Preprint arXiv: 1601.03978 v 2[math. NA] 2 Dec. (2016).

  12. 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 

  13. P. S. Ismagilov, “Widths of sets in linear normalized spaces and approximation of functions by trigonometric polynomials,” Usp. Mat. Nauk, 29, No. 3, 161–178 (1974).

    Google Scholar 

  14. S. B. Stechkin, “On the absolute convergence of orthogonal series,” Dokl. Akad. Nauk SSSR, 102, No. 1, 37–40 (1955).

    MATH  MathSciNet  Google Scholar 

  15. A. S. Romanyuk, “Best M-term trigonometric approximations of the Besov classes of periodic functions of many variables,” Izv. Ros. Akad. Nauk, Ser. Mat., 67, No. 2, 61–100 (2003).

    Article  MATH  Google Scholar 

  16. A. Kolmogoroff, “Über die beste Annäherung von Functionen einer gegeben Functionenclasse,” Ann. Math., 37, 107–111 (1936).

    Article  MATH  MathSciNet  Google Scholar 

  17. V. M. Tikhomirov, “Widths of sets in function spaces and the theory of best approximations,” Usp. Mat. Nauk, 15, No. 3, 81–120 (1960).

    Google Scholar 

  18. É. M. Galeev, “Linear widths of the Hölder–Nikol’skii classes of periodic functions of many variables,” Mat. Zametki, 59, No. 2, 189–199 (1996).

    Article  MathSciNet  Google Scholar 

  19. A. S. Romanyuk, “Approximation of the Besov classes of periodic functions of several variables in a space L q ,Ukr. Mat. Zh., 43, No. 10, 1398–1408 (1991); English translation: Ukr. Math. J., 43, No. 10, 1297–1306 (1991).

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

    MathSciNet  Google Scholar 

  21. 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).

    MATH  MathSciNet  Google Scholar 

  22. A. S. Romanyuk, “Approximation 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 

  23. R. A. de Vore and V. N. Temlyakov, “Nonlinear approximation by trigonometric sums,” Fourier Anal. Appl., 2, No. 1, 29–48 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  24. Yu. V. Malykhin and K. S. Ryutin, “Product of octahedra is badly approximated in the metric of l 2,1 ,Mat. Zametki, 101, No. 1, 85–90 (2017).

    Article  MATH  MathSciNet  Google Scholar 

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 69, No. 5, pp. 670–681, May, 2017.

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Romanyuk, A.S. Trigonometric and Linear Widths for the Classes of Periodic Multivariate Functions. Ukr Math J 69, 782–795 (2017). https://doi.org/10.1007/s11253-017-1395-6

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  • DOI: https://doi.org/10.1007/s11253-017-1395-6

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