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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

On Quadrature Convergence of Extended Lagrange Interpolation


Authors: Walter Gautschi and Shikang Li
Journal: Math. Comp. 65 (1996), 1249-1256
MSC (1991): Primary 41A05, 65D32; Secondary 33C45
MathSciNet review: 1344613
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Abstract: Quadrature convergence of the extended Lagrange interpolant $L_{2n+1}f$ for any continuous function $f$ is studied, where the interpolation nodes are the $n$ zeros $\tau _i$ of an orthogonal polynomial of degree $n$ and the $n+1$ zeros $\hat {\tau }_j$ of the corresponding ``induced'' orthogonal polynomial of degree $n+1$. It is found that, unlike convergence in the mean, quadrature convergence does hold for all four Chebyshev weight functions. This is shown by establishing the positivity of the underlying quadrature rule, whose weights are obtained explicitly. Necessary and sufficient conditions for positivity are also obtained in cases where the nodes $\tau _i$ and $\hat {\tau }_j$ interlace, and the conditions are checked numerically for the Jacobi weight function with parameters $\alpha $ and $\beta $. It is conjectured, in this case, that quadrature convergence holds for $| \alpha | \leq \frac {1}{2}, ~ | \beta | \leq \frac {1}{2}$.


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Additional Information

Walter Gautschi
Affiliation: Department of Computer Sciences, Purdue University, West Lafayette, Indiana 47907-1398
Email: wxg@cs.purdue.edu

Shikang Li
Affiliation: Department of Mathematics, Southeastern Louisiana University, Hammond, Louisiana 70402
Email: kli@selu.edu

DOI: http://dx.doi.org/10.1090/S0025-5718-96-00731-4
PII: S 0025-5718(96)00731-4
Received by editor(s): April 20, 1995
Article copyright: © Copyright 1996 American Mathematical Society