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Constructive sparse trigonometric approximations for the functions with generalized mixed smoothness

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Abstract

The order bounds (in the case of uniform metrics) and exact order bounds (in the case of integral metrics) for the best m-term trigonometric approximation of periodic functions with generalized mixed smoothness from classes close to the Nikol’skii–Besov-type ones are obtained. Every of the upper bounds is realized by a constructive method based on greedy algorithms.

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References

  1. N. K. Bari and S. B. Stechkin, “The best approximations and differential properties of two conjugate functions,” Trudy Mosk. Mat. Obshch., 2, 489–523 (1952).

    Google Scholar 

  2. S. N. Bernshtein, Collection of Works, Vol. 2, Constructive Theory of Functions [in Russian], Izdat. Akad. Nauk SSSR, Moscow, 1954.

  3. M. K. Potapov, B. V. Simonov, and S. Yu. Tikhonov, “Mixed moduli of smoothness in L p : a survey,” Surveys in Approximation Theory, 8, 1–57 (2013).

    MathSciNet  MATH  Google Scholar 

  4. N. N. Pustovoitov, “Representation and approximation of periodic multivariate functions with given mixed modulus of continuity,” Anal. Math., 20, No. 1, 35–48 (1994).

    Article  MathSciNet  Google Scholar 

  5. Y. Sun and H. Wang, “Representation and approximation of multivariate periodic functions with bounded mixed moduli of smoothness,” Trudy Mat. Inst. im. V.A. Steklova, 219, 356–377 (1997).

    MathSciNet  Google Scholar 

  6. V. N. Temlyakov, “Approximation of functions with bounded mixed derivative,” Trudy MIAN SSSR, 178, 1–112 (1986).

    MathSciNet  MATH  Google Scholar 

  7. P. I. Lizorkin and S. M. Nikol’skii, “Spaces of functions with mixed smoothness from the decomposition viewpoint,” Trudy MIAN SSSR, 187, 143–161 (1989).

    Google Scholar 

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

    Google Scholar 

  9. G. A. Akishev, “Estimates of the Kolmogorov widths of the classes Nikol’skii–Besov–Amanov in the Lorentz space,” Trudy IMM UrO RAN, 21, No. 4, 3–13 (2015).

  10. N. N. Pustovoitov, “Approximation of mulltidimensional functions with given majorant of mixed moduli of continuity,” Mat. Zamet., 65, No. 1, 107–117 (1999).

    Article  Google Scholar 

  11. S. A. Stasyuk, “The best approximations of periodic multivariate functions from the classes \( {B}_{p,\theta}^{\Omega}, \)” Mat. Zamet., 87, No. 1, 108–121 (2010).

  12. S. A. Stasyuk, “Approximation by Fourier sums and Kolmogorov widths of the classes \( {\mathbf{MB}}_{p,\theta}^{\Omega} \) of periodic functions of several variables,” Trudy IMM UrO RAN, 20, No. 1, 247–257 (2014).

  13. S. A. Stasyuk, “Approximation of the classes \( {\mathbf{MB}}_{p,\theta}^{\Omega} \) by de la Vallée–Poussin sums in a uniform metric,” in: Mathematical Problems of Mechanics and Computational Mathematics [in Russian], Institute of Mathematics of the NAS of Ukraine, Kyiv, 2014, pp. 308–317.

  14. N. V. Derev’yanko, “Approximation of the classes \( {H}_p^{\Omega} \) of periodic multivariate functions in the space L q ,” Ukr. Mat. Zh., 66, No. 5, 634–644 (2014).

  15. Sh. A. Balgimbaeva and T. I. Smirnov, “Estimates of the Fourier widths of the classes of periodic functions with mixed modulus of smoothness,” Trudy IMM UrO RAN, 21, No. 4, 78–94 (2015).

  16. S. A. Stasyuk and S. Ya. Yanchenko, “Approximation of functions from the Nikol’skii–Besov-type classes with generalized mixed smoothness,” Anal. Math., 41, No. 4, 311–334 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  17. V. N. Temlyakov, “Constructive sparse trigonometric approximations and other problems for functions with mixed smoothness,” Mat. Sborn., 206, No. 11, 131–160 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  18. V. N. Temlyakov, “Constructive sparse trigonometric approximation for functions with small mixed smoothness,” arXiv: 1503.00282v1 [math.NA], March 1, 2015, 1–30.

  19. D. Dũng, V. N. Temlyakov, and T. Ullrich, “Hyperbolic cross approximation,” 2016, arXiv: 1601.03978v1 [math.NA], Jan. 15, 2016, 1–154.

  20. S. A. Stasyuk, “The best m-term trigonometric approximation for periodic functions with small mixed smoothness from classes of the Nikol’skii–Besov-type,” Ukr. Mat. Zh., 68, No. 7, 983–1003 (2016).

    Article  MathSciNet  Google Scholar 

  21. S. A. Stasyuk, “Approximation of some smooth classes of periodic multivariate functions by polynomials according to the Haar tensor system,” Trudy IMM UrO RAN, 21, No. 4, 251–260 (2015).

  22. A. S. Romanyuk, Approximative Characteristics of the Classes of Periodic Multivariate Functions [in Ukrainian], Institute of Mathematics of the NAS of Ukraine, Kiev, 2012.

    Google Scholar 

  23. S. A. Stasyuk, “Best m-term trigonometric approximation of periodic functions of several variables from the Nikol’skii–Besov classes for small smoothness,” J. Approx. Theory, 177, 1–16 (2014).

    Article  MathSciNet  Google Scholar 

  24. S. A. Stasyuk, “The best M-term trigonometric approximations of the classes of multivariate functions \( {B}_{p,\theta}^{\Omega}, \)” Ukr. Mat. Zh., 54, No. 3, 381–394 (2002).

  25. S. A. Stasyuk, “The approximation of the classes \( {B}_{p,\theta}^{\Omega} \) of periodic multivariate functions in a uniform metric,” Ukr. Mat. Zh., 54, No. 11, 1551–1559 (2002).

  26. A. F. Konograi and S. A. Stasyuk, “The best M-term trigonometric approximations of the classes \( {B}_{p,\theta}^{\Omega} \) of periodic multivariate functions in the space L q ,” Ukr. Mat. Zh., 60, No. 9, 1206–1224 (2008).

  27. É. S. Belinskii, “Approximation of a floating system of exponents on the classes of periodic functions with bounded mixed derivative,” in: Research on the Theory of Functions of Many Real Variables [in Russian], Yaroslavl’ Univ., Yaroslavl’, 1988, pp. 16–33.

  28. É. S. Belinskii, “Approximation of functions of several variables by trigonometric polynomials with given number of harmonics and estimates of ε-entropy,” Anal. Math., 15, No. 2, 67–74 (1989).

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics of the NAS of Ukraine, Kiev, Ukraine

    Sergei A. Stasyuk

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  1. Sergei A. Stasyuk
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Correspondence to Sergei A. Stasyuk.

Additional information

Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 13, No. 3, pp. 408–420 July–September, 2016.

The work is performed under the partial support by FP7-People-2011-IRSES (project No. 295164 (EUMLS: EU{Ukrainian Mathematicians for Life Sciences)).

Translated from Russian by V.V. Kukhtin

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Stasyuk, S.A. Constructive sparse trigonometric approximations for the functions with generalized mixed smoothness . J Math Sci 222, 787–796 (2017). https://doi.org/10.1007/s10958-017-3332-5

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  • DOI: https://doi.org/10.1007/s10958-017-3332-5

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