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Asymptotic behaviour of Stieltjes polynomials for ultraspherical weight functions

Authors
Journal
Journal of Computational and Applied Mathematics
0377-0427
Publisher
Elsevier
Publication Date
Volume
65
Identifiers
DOI: 10.1016/0377-0427(95)00107-7
Keywords
  • Stieltjes Polynomials
  • Asymptotic Representation
  • Ultraspherical Weight Function
  • Kronrod Extensions Of Gauss Quadrature Formulae
  • Kronrod Extensions Of Lobatto Quadrature Formulae
  • Positivity Of Weights

Abstract

Abstract For the ultraspherical weight functions w λ(x) = (1 − x 2) λ − 1 2 , an asymptotic representation of the Stieltjes polynomials is proved for 1< λ⩽2, which holds uniformly in every closed subinterval of (−1, 1). This extends and completes our earlier results (for 0⩽ λ⩽1) in the sense that the problem is solved for all ultraspherical weight functions for which Stieltjes polynomials are known to have only real distinct zeros inside (−1, 1) for all n ∈ N . The main result is applied to prove positivity results for Kronrod extensions of Gauss and Lobatto quadrature formulae.

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