Abstract

With the help of a new representation of the Stieltjes polynomial it is shown by using Bessel functions that the Stieltjes polynomial with respect to the ultraspherical weight function with parameter $\lambda$ has only few real zeros for $\lambda > 3$ and sufficiently large n. Since the nodes of the Gauss--Kronrod quadrature formulae subdivide into the zeros of the Stieltjes polynomial and the Gaussian nodes, it follows immediately that Gauss--Kronrod quadrature is not possible for $\lambda > 3$. On the other hand, for $\lambda = 3$ and sufficiently large n, even partially positive Gauss--Kronrod quadrature is possible.

MSC codes

  1. 33C10
  2. 33C45
  3. 42C05
  4. 65D32

Keywords

  1. Gauss--Kronrod quadrature
  2. Stieltjes polynomials
  3. orthogonal polynomials
  4. Bessel functions

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Published In

cover image SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Pages: 927 - 948
ISSN (online): 1095-7170

History

Published online: 25 July 2006

MSC codes

  1. 33C10
  2. 33C45
  3. 42C05
  4. 65D32

Keywords

  1. Gauss--Kronrod quadrature
  2. Stieltjes polynomials
  3. orthogonal polynomials
  4. Bessel functions

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