Advertisement

Approximate characteristics of the classes \( {B}_{p,\theta}^{\Omega} \) of periodic functions of one variable and many ones

  • Mykhailo V. Hembars’kyiEmail author
  • Svitlana B. Hembars’ka
Article
  • 1 Downloads

Abstract

We obtained the exact-by-order estimates of some approximate characteristics of classes of the Nikol’skii–Besov type of periodic functions of one variable and many ones in the space B∞,1 such that the norm in it is not weaker than the L-norm.

Keywords

Classes of the Nikol’skii–Besov type best orthogonal trigonometric approximation graduated hyperbolic cross 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. S. Romanyuk, “Entropic numbers and widths of the classes \( {B}_{p,\theta}^r \) of periodic functions of many variables,” Ukr. Mat. Zh., 68, No. 10, 1403–1417 (2016).Google Scholar
  2. 2.
    M. V. Hembars’kyi and S. B. Hembars’ka, “Widths of the classes \( {B}_{p,\theta}^{\Omega} \) of periodic functions of many variables in the space B1,1,” Ukr. Mat. Visn., 15, No. 1, 43–57 (2018).Google Scholar
  3. 3.
    A. S. Romanyuk and V. S. Romanyuk, “Approximate characteristics of the classes of periodic functions of many variables in the space B∞,1,” Ukr. Mat. Zh., 71, No. 2, 271–282 (2019).Google Scholar
  4. 4.
    M. V. Hembars’kyi, S. B. Hembars’ka, and K. V. Solich, “The best approximations and widths of classes of periodic functions of one variable and many ones in the space B∞,1,” Matem. Stud., (in press).Google Scholar
  5. 5.
    Dinh Dung, V. N. Temlyakov, and T. Ullrich, Hyperbolic Cross Approximation, arXiv: 1601.03978 v3 [math. NA] 21 Apr. 2017.Google Scholar
  6. 6.
    S. N. Bernstein, Collection of Works, vol. II. Constructional Theory of Functions (1931 – 1953) [in Russian], AS of the USSR, Moscow, 1954.Google Scholar
  7. 7.
    S. B. Stechkin, “On the order of the best approximations of continuous functions,” Izv. Akad. Nauk SSSR. Ser. Mat., 15, 219–242 (1951).MathSciNetGoogle Scholar
  8. 8.
    N. K. Bari and S. B. Stechkin, “The best approximations and differential properties of two conjugate functions,” Trudy Mosk. Mat. Obshch., 5, 483–522 (1956).MathSciNetGoogle Scholar
  9. 9.
    Sun Yongsheng and Wang Heping, “Representation and approximation of multivariate periodic functions with bounded mixed moduli of smoothness,” Trudy MIAN SSSR, 219, 356–377 (1997).MathSciNetzbMATHGoogle Scholar
  10. 10.
    T. I. Amanov, “The theorems of representation and embedding for the functional spaces \( {S}_{p,\theta}^{(r)}B\left({\mathrm{\mathbb{R}}}_n\right) \) and \( {S}_{p,\theta}^{(r)\ast }, \) (0 ≤ x j ≤ 2 π; j = 1, . . . , n),” Trudy MIAN SSSR, 77, 5–34 (1965).Google Scholar
  11. 11.
    P. I. Lizorkin and S. M. Nikol’skii, “The spaces of functions with mixed smoothness from the decomposition viewpoint,” Trudy MIAN SSSR, 187, 143–161 (1989).Google Scholar
  12. 12.
    S. M. Nikol’skii, “Functions with dominating mixed derivative satisfying the multiple H’ólder condition,” Sibir. Mat. Zh., 4, No. 6, 1342–1364 (1963).Google Scholar
  13. 13.
    N. N. Pustovoitov, “The representation and approximation of periodic functions of many variables with given mixed modulus of continuity,” Anal. Math., 20, 35–48 (1994).MathSciNetCrossRefGoogle Scholar
  14. 14.
    S. A. Stasyuk and O. V. Fedunik, “Approximative characteristics of the classes \( {B}_{p,\theta}^{\Omega} \) of periodic functions of many variables,” Ukr. Mat. Zh., 58, No. 5, 692–704 (2006).Google Scholar
  15. 15.
    V. N. Temlyakov, Approximation of Periodic Functions, Nova Sci., New York, 1993.zbMATHGoogle Scholar
  16. 16.
    A. S. Romanyuk, Approximative Characteristics of the Classes of Periodic Functions of Many Variables [in Russian], Institute of Mathematics of the NAS of Ukraine, Kiev (2012).Google Scholar
  17. 17.
    A. S. Romanyuk, “The best trigonometric approximations of the classes of periodic functions of many variables in a uniform metric,” Mat. Zametki, 82, No. 2, 247–261 (2007).CrossRefGoogle Scholar
  18. 18.
    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. Sborn., 195, No. 2, 91–116 (2004).Google Scholar
  19. 19.
    A. S. Romanyuk, “Widths and the best approximations of the classes \( {B}_{p,\theta}^r \) of periodic functions of many variables,” Anal. Math., 37, 181–213 (2011).Google Scholar
  20. 20.
    S. A. Stasyuk, “Approximations of the classes \( {B}_{p,\theta}^{\Omega} \) of periodic functions of many variables in a uniform metric,” Ukr. Mat. Zh., 54, No. 11, 1551–1559 (2002).Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Mykhailo V. Hembars’kyi
    • 1
    Email author
  • Svitlana B. Hembars’ka
    • 1
  1. 1.Lesya Ukrainka East-European National UniversityLutskUkraine

Personalised recommendations