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Constructive Approximation

, Volume 13, Issue 1, pp 99–152 | Cite as

Jackson and smoothness theorems for freud weights

  • Z. Ditzian
  • D. S. Lubinsky
Article

Abstract

We prove Jackson, realization, and converse theorems for Freud weights inL p, 0<p ≤ ∞. Even forp ≥ 1, our conditions on the weight in our Jackson theorems are far less restrictive than those previously imposed. Moreover, the method—of first approximating by a spline, and then by a polynomial—is new in this context, and of intrinsic interest, since it avoids the use of orthogonal polynomials for Freud weights. We establish some properties of the modulus of smoothness, valid inL p for 0<p ≤ ∞. Since theK-functional is identically zero inL p,p<1, the analysis of the modulus of continuity involves a different tool, namely, realization, which works inL p for all 0<p ≤ ∞. We deduce Marchaud-type inequalities.

AMS classification

Primary 41A10 42C05 

Key words and phrases

Freud weights Polynomial approximation Jackson theorems Moduli of continuity K-Functionals Realization 

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Copyright information

© Springer-Verlag New York Inc 1997

Authors and Affiliations

  • Z. Ditzian
    • 1
  • D. S. Lubinsky
    • 2
  1. 1.Department of MathematicsThe University of AlbertaEdmontonCanada
  2. 2.Department of MathematicsUniversity of the WitwatersrandRepublic of South Africa

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