Skip to main content

Inequalities of the type of the generalized Jackson theorem for the best approximations

Very simple methods for deriving upper estimates of the type of the generalized Jackson theorem for the best approximations of periodic functions are proposed, The results obtained are similar to the generalized Jackson theorem, providing estimates in terms of the moduli of continuity. Bibliography: 12 titles.

References

  1. V. K. Dzyadyk, Introduction to the Theory of Uniform Approximation of Functions by Polynomials [in Russian], Moscow (1977).

  2. R. A. De Vore and G. G. Lorentz, Constructive Approximation, Berlin–New York (1993).

  3. A. F. Timan, Theory of Approximation of Functions of a Real Variable [in Russian], Moscow (1960).

  4. N. I. Akhiezer, Lectures in Approximation Theory [in Russian]. Moscow (1965).

  5. Yu. A. Brudnyi. “On a theorem of local best approximation,” in: Functional Analysis and Function Theory [in Russian], Kazan’ (1964), pp. 43–49.

  6. V. V. Zhuk, Approximation of Periodic Functions [in Russian]. Leningrad (1982).

  7. V. V. Zhuk, Structural Properties of Functions and Approximation Accuracy [in Russian], Leningrad (1984).

  8. V. V. Zhuk and V. F. Kuzyutin, Function Approximation and Numerical Integration [in Russian], St. Petersburg (1995).

  9. S. Foucart, Y. Kryakin, and A. Shadrin, “On the exact constant in the Jackson–Stechkin inequality for the uniform metric,” Constr. Approx., 29, 157–179 (2009).

    MathSciNet  MATH  Article  Google Scholar 

  10. O. L. Vinogradov and V. V. Zhuk, “Estimates for functionals with a known moment sequence in terms of deviation of Steklov type averages,” Zap. Nauchn. Semin. POMI, 383, 5–32 (2010).

    Google Scholar 

  11. O. L. Vinogradov and V. V. Zhuk, “The rate of decrease of constants in Jackson type inequalities in dependence of the order of modulus of continuity,” Zap. Nauchn. Semin. POMI, 383, 33–52 (2010).

    Google Scholar 

  12. O. L. Vinogradov and V. V. Zhuk, “Estimates for functionals with a known finite set of moments in terms of deviations of operators based on the Steklov averages and finite differences,” Zap. Nauchn. Semin. POMI, 392, 32–66 (2011).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to V. V. Zhuk.

Additional information

Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 404, 2012, pp. 135–156.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Zhuk, V.V. Inequalities of the type of the generalized Jackson theorem for the best approximations. J Math Sci 193, 75–88 (2013). https://doi.org/10.1007/s10958-013-1435-1

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-013-1435-1

Keywords

  • Periodic Function
  • Jackson Theorem