Advertisement

Constructive Approximation

, Volume 42, Issue 1, pp 129–159 | Cite as

New Moduli of Smoothness: Weighted DT Moduli Revisited and Applied

  • K. A. Kopotun
  • D. LeviatanEmail author
  • I. A. Shevchuk
Article

Abstract

We introduce new moduli of smoothness for functions \(f\in L_p[-1,1]\cap C^{r-1}(-1,1)\), \(1\le p\le \infty \), \(r\ge 1\), that have an \((r-1)\)st locally absolutely continuous derivative in \((-1,1)\), and such that \(\varphi ^rf^{(r)}\) is in \(L_p[-1,1]\), where \(\varphi (x)=(1-x^2)^{1/2}\). These moduli are equivalent to certain weighted Ditzian–Totik (DT) moduli, but our definition is more transparent and simpler. In addition, instead of applying these weighted moduli to weighted approximation, which was the purpose of the original DT moduli, we apply these moduli to obtain Jackson-type estimates on the approximation of functions in \(L_p[-1,1]\) (no weight), by means of algebraic polynomials. Moreover, we also prove matching inverse theorems, thus obtaining constructive characterization of various smoothness classes of functions via the degree of their approximation by algebraic polynomials.

Keywords

Approximation by polynomials in the \(L_p\)-norm Degree of approximation Jackson-type estimates Moduli of smoothness 

Mathematics Subject Classification

41A10 41A17 41A25 

References

  1. 1.
    DeVore, Ronald A., Lorentz, George G.: Constructive Approximation, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 303. Springer, Berlin (1993)Google Scholar
  2. 2.
    Ditzian, Z.: Polynomial approximation and \(\omega ^r_\phi (f, t)\) twenty years later. Surv. Approx. Theory 3, 106–151 (2007)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Ditzian, Z., Totik, V.: Moduli of smoothness. Springer Series in Computational Mathematics, vol. 9. Springer, New York (1987)Google Scholar
  4. 4.
    Dzyadyk, V.K., Shevchuk, I.A.: Theory of Uniform Approximation of Functions by Polynomials. Walter de Gruyter, Berlin (2008)zbMATHGoogle Scholar
  5. 5.
    Kopotun, K.A., Leviatan, D., Shevchuk, I.A.: Are the degrees of the best (co)convex and unconstrained polynomial approximations the same? II. Ukrainian Math. J. 62(3), 420–440 (2010)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Shevchuk, I.A.: Polynomial Approximation and Traces of Functions Continuous on a Segment. Naukova Dumka, Kiev (1992). (Russian)Google Scholar
  7. 7.
    Timan, A. F.: Theory of approximation of functions of a real variable. Dover Publications, New York (1994) Translated from the Russian by J. Berry; Translation edited and with a preface by J. Cossar; Reprint of the 1963 English translationGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ManitobaWinnipegCanada
  2. 2.Raymond and Beverly Sackler School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael
  3. 3.Faculty of Mechanics and MathematicsNational Taras Shevchenko University of KyivKyivUkraine

Personalised recommendations