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On the Degree of Approximation in Multivariate Weighted Approximation

  • H. N. Mhaskar
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 142)

Abstract

Let s ≥ 1 be an integer, f ∈ L P (R s ) for some p, 1 ≤ p ∞ or be a continuous function on R S vanishing at infinity. We consider the degree of approxima-tion of f by expressions of the form exp \( ( - {\text{ }}\sum\limits_{k = 1}^s {{Q_k}\left( {{x_k}} \right)} )P\left( {{x_1},...,{x_s}} \right) \) where each exp(—Q k (·)) is a Freud type weight function, and P is a polynomial of specific degrees in each coordinate. Direct and converse theorems are stated. In particular, it is shown that if each Q k = lxl α for some even, positive integer a, and the degree of approximation has a power decay, then the same property holds when the weight function is replaced by exp \( ( - \sum\limits_{k = 1}^s {{a_k}{Q_k}\left( {{x_k}} \right)} ) \) for any positive constants a k .

Keywords

Weight Function Multivariate Setting Bold Face Letter Converse Theorem Weighted Polynomial 
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.

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Copyright information

© Birkhäuser Verlag Basel 2002

Authors and Affiliations

  • H. N. Mhaskar
    • 1
  1. 1.Department of MathematicsCalifornia State UniversityLos AngelesUSA

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