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)


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 .


Weight Function Multivariate Setting Bold Face Letter Converse Theorem Weighted Polynomial 
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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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