Advertisement

Ukrainian Mathematical Journal

, Volume 57, Issue 12, pp 1968–1976 | Cite as

On one extremal problem for positive series

  • O. I. Stepanets’
  • A. L. Shydlich
Article

Abstract

The approximation properties of the spaces S ϕ p introduced by Stepanets’ were studied in a series of works of Stepanets’ and his disciples. In these works, problems related to the determination of exact values of n-term approximations of q-ellipsoids in these spaces were reduced to some extremal problems for series with terms that are products of elements of two nonnegative sequences one of which is fixed and the other varies on a certain set.

Since solutions of these extremal problems may be of independent interest, in the present work we develop a new method for finding these solutions that enables one to obtain the required result in a substantially shorter and more transparent way.

Keywords

Extremal Problem Approximation Property Independent Interest Require Result Positive Series 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. I. Stepanets, “Approximation characteristics of spaces S ϕp,” Ukr. Mat. Zh., 53, No. 3, 392–416 (2001).zbMATHMathSciNetGoogle Scholar
  2. 2.
    A. I. Stepanets, “Approximation characteristics of spaces S ϕp in different metrics,” Ukr. Mat. Zh., 53, No.8, 1121–1146 (2001).zbMATHMathSciNetGoogle Scholar
  3. 3.
    V. R. Voitsekhivs’kyi, “Jackson-type inequalities in the approximation of functions from the space S p by Zygmund sums,” in: Theory of Approximation of Functions and Related Problems [in Ukrainian], Institute of Mathematics, Ukrainian Academy of Sciences, Kyiv (2002), pp. 33–46.Google Scholar
  4. 4.
    A. I. Stepanets, Methods of Approximation Theory [in Russian], Vol. 2, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (2002).Google Scholar
  5. 5.
    A. I. Stepanets, “Approximation characteristics of spaces S p,” in: Proceedings of the Ukrainian Mathematical Congress-2001 [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (2002), pp. 208–226.Google Scholar
  6. 6.
    A. I. Stepanets and A. S. Serdyuk, “Direct and inverse theorems in the theory of approximation of functions in the space S p,” Ukr. Mat. Zh., 54, No. 1, 106–124 (2002).MathSciNetzbMATHGoogle Scholar
  7. 7.
    S. B. Vakarchuk, “On some extremal problems of approximation theory in spaces S p (1 ≤ p < ∞),” in: Proceedings of the Voronezh Winter Mathematical School “Modern Methods in the Theory of Functions and Related Problems” [in Russian], (Voronezh, January 26–February 2, 2003), Voronezh University, Voronezh (2003), pp. 47–48.Google Scholar
  8. 8.
    V. R. Voitsekhivs’kyi, “Widths of some classes from the space S p,” in: Extremal Problems in the Theory of Functions [in Ukrainian], Institute of Mathematics, Ukrainian Academy of Sciences, Kyiv (2003), pp. 17–26.Google Scholar
  9. 9.
    V. I. Rukasov, “Best n-term approximations in spaces with nonsymmetric metric,” Ukr. Mat. Zh., 55, No. 6, 806–816 (2003).zbMATHMathSciNetGoogle Scholar
  10. 10.
    A. S. Serdyuk, “Widths of classes of functions defined by the moduli of continuity of their ψ-derivatives in the space S p,” in: Extremal Problems in the Theory of Functions [in Ukrainian], Institute of Mathematics, Ukrainian Academy of Sciences, Kyiv (2003), pp. 229–248.Google Scholar
  11. 11.
    A. I. Stepanets, “Extremal problems of approximation theory in linear spaces,” Ukr. Mat. Zh., 55, No. 10, 1392–1423 (2003).MathSciNetGoogle Scholar
  12. 12.
    A. I. Stepanets and V. I. Rukasov, “Spaces S p with nonsymmetric metric,” Ukr. Mat. Zh., 55, No. 2, 264–277 (2003).MathSciNetzbMATHGoogle Scholar
  13. 13.
    A. I. Stepanets and V. I. Rukasov, “Best “continuous” n-term approximations in the spaces S ϕp,” Ukr. Mat. Zh., 55, No. 5, 663–670 (2003).MathSciNetzbMATHGoogle Scholar
  14. 14.
    O. I. Stepanets’ and A. L. Shydlich, “Best n-term approximations by Λ-methods in the spaces S ϕp,” Ukr. Mat. Zh., 55, No. 8, 1107–1126 (2003).Google Scholar
  15. 15.
    A. L. Shydlich, “Best n-term approximations by Λ-methods in the spaces S ϕp, ” in: Extremal Problems in the Theory of Functions [in Ukrainian], Institute of Mathematics, Ukrainian Academy of Sciences, Kyiv (2003), pp. 283–306.Google Scholar
  16. 16.
    S. B. Vakarchuk, “Jackson-type inequalities and exact values of widths of classes of functions in the spaces S p, 1 ≤ p < ∞,” Ukr. Mat. Zh., 56, No. 5, 595–605 (2004).zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    A. I. Stepanets, “Best approximations of q-ellipsoids in spaces S ϕp,” Ukr. Mat. Zh., 56, No. 10, 1378–1383 (2004).zbMATHMathSciNetGoogle Scholar
  18. 18.
    A. L. Shydlich, “On saturation of linear summation methods for Fourier series in the spaces S ϕp,” Ukr. Mat. Zh., 56, No. 1, 133–138 (2004).Google Scholar
  19. 19.
    A. I. Stepanets, “Best n-term approximations with restrictions,” Ukr. Mat. Zh., 57, No. 4, 533–553 (2005).MathSciNetzbMATHGoogle Scholar
  20. 20.
    G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge University Press, Cambridge (1934).Google Scholar
  21. 21.
    D. S. Mitrinović, J. E. Pečarić, and A. M. Fink, Classical and New Inequalities in Analysis, Kluwer, Dordrecht (1993).zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • O. I. Stepanets’
  • A. L. Shydlich

There are no affiliations available

Personalised recommendations