Ukrainian Mathematical Journal

, Volume 68, Issue 12, pp 1975–1985 | Cite as

Order Estimates for the Approximating Characteristics of Functions from the Classes \( {S}_{p,\theta}^{\Omega}B\left({\mathbb{R}}^d\right) \) with a Given Majorant of Mixed Modules of Continuity in the Uniform Metric

  • S. Ya. Yanchenko
Article
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We establish the exact-order estimates for the approximation of functions of several variables from the classes \( {S}_{p,\theta}^{\Omega}B \) defined on d in the norm of L ( d ) by entire functions of the exponential type with supports of their Fourier transforms in the sets generated by the level surfaces of a function Ω.

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References

  1. 1.
    S. A. Stasyuk and S. Ya. Yanchenko, “Approximation of functions from Nikolskii–Besov-type classes of generalized mixed smoothness,” Anal. Math., 41, 311–334 (2015).MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    P. I. Lizorkin and C. 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
  3. 3.
    N. K. Bari and S. B. Stechkin, “Best approximations and differential properties of two conjugate functions,” Tr. Mosk. Mat. Obshch., 5, 483–522 (1956).MathSciNetGoogle Scholar
  4. 4.
    P. I. Lizorkin, “Generalized Liouville differentiation and the method of multiplicators in the embedding theory of classes of differentiable functions,” Tr. Mat. Inst. Akad. Nauk SSSR, 105, 89–167 (1969).MathSciNetGoogle Scholar
  5. 5.
    S. M. Nikol’skii, “Functions with predominant mixed derivative satisfying the Hölder multiple condition,” Sib. Mat. Zh., 4, No. 6, 1342–1364 (1963).Google Scholar
  6. 6.
    T. I. Amanov, “Theorems of representation and embedding for the function spaces \( {S}_{p,\theta}^{(r)}B\left({\mathbb{R}}_n\right) \) and \( {S}_{p,\theta}^{(r)_{\ast }}B\left(0\le {x}_j\le 2\pi; j=1,\dots, n\right) \),” Tr. Mat. Inst. Akad. Nauk SSSR, 77, 5–34 (1965).Google Scholar
  7. 7.
    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).Google Scholar
  8. 8.
    A. S. Romanyuk, “Approximation of classes of periodic functions of several variables,” Ukr. Mat. Zh., 44, No. 5, 662–672 (1992); English translation: Ukr. Math. J., 44, No. 5, 596–606 (1992).Google Scholar
  9. 9.
    N. N. Pustovoitov, “Approximation of multidimensional functions with given majorant of mixed modules of continuity,” Mat. Zametki, 65, No. 1, 107–117 (1999).MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    S. A. Stasyuk, “Best approximations of periodic functions of several variables from the classes \( {B}_{p,\theta}^{\varOmega } \) ,Mat. Zametki, 87, No. 1, 108–121 (2010).MathSciNetCrossRefGoogle Scholar
  11. 11.
    S. A. Stasyuk, “Approximation of the classes \( {\mathrm{MB}}_{p,\theta}^{\Omega} \) by de la Vallée-Poussin sums in the uniform metric,” in: Collection of Works “Mathematical Problems in Mechanics and Numerical Mathematics” [in Ukrainian], Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv, 11, No. 4 (2014), pp. 308–317.Google Scholar
  12. 12.
    S. M. Nikol’skii, Approximation of Functions of Several Variables and Embedding Theorems [in Russian], Nauka, Moscow (1969).MATHGoogle Scholar
  13. 13.
    H. Wang and Y. Sun, “Approximation of multivariate functions with certain mixed smoothness by entire functions,” Northeast. Math. J., 11, No. 4, 454–466 (1995).MathSciNetMATHGoogle Scholar
  14. 14.
    S. Ya. Yanchenko, “Estimates for approximating characteristics of the classes of functions \( {S}_{p,\theta}^rB\left({\mathbb{R}}^d\right) \) in the uniform metric,” in: Collection of Works “Approximation Theory of Functions and Related Problems” [in Ukrainian], Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv, 10, No. 1 (2013), pp. 328–340.Google Scholar
  15. 15.
    S. Ya. Yanchenko, “Order estimates for approximating characteristics of functions from generalized classes of mixed smoothness of the Nikol’ski–Besov type,” in: Collection of Works “Approximation Theory of Functions and Related Problems” [in Ukrainian], Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv, 11, No. 3 (2014), pp. 330–343.Google Scholar
  16. 16.
    S. Ya. Yanchenko, “Approximation of functions from the classes \( {S}_{p,\theta}^rB \) in the uniform metric,” Ukr. Mat. Zh., 65, No. 5, 698–705 (2013); English translation: Ukr. Math. J., 65, No. 5, 771–779 (2013).Google Scholar
  17. 17.
    A. S. Romanyuk, “Approximation of the classes \( {B}_{p,\theta}^r \) of periodic functions of several variables by linear methods and the best approximations,” Mat. Sb., 195, No. 2, 91–116 (2004).MathSciNetCrossRefGoogle Scholar

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© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  • S. Ya. Yanchenko
    • 1
  1. 1.Institute of MathematicsUkrainian National Academy of SciencesKyivUkraine

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