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Stieltjes Polynomials and Related Quadrature Formulae for a Class of Weight Functions

Authors: Walter Gautschi and Sotirios E. Notaris
Journal: Math. Comp. 65 (1996), 1257-1268
MSC (1991): Primary 33C45, 65D32
MathSciNet review: 1344614
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Abstract: Consider a (nonnegative) measure $d \sigma $ with support in the interval $[a,b]$ such that the respective orthogonal polynomials, above a specific index $\ell $, satisfy a three-term recurrence relation with constant coefficients. We show that the corresponding Stieltjes polynomials, above the index $2\ell -1$, have a very simple and useful representation in terms of the orthogonal polynomials. As a result of this, the Gauss-Kronrod quadrature formulae for $d \sigma $ have all the desirable properties, namely, the interlacing of nodes, their inclusion in the closed interval $[a,b]$ (under an additional assumption on $d \sigma $), and the positivity of all weights. Furthermore, the interpolatory quadrature formulae based on the zeros of the Stieltjes polynomials have positive weights, and both of these quadrature formulae have elevated degrees of exactness.

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Additional Information

Walter Gautschi
Affiliation: Department of Computer Sciences, Purdue University, West Lafayette, Indiana 47907-1398

Sotirios E. Notaris
Affiliation: Department of Communications and Mass Media, National and Capodistrian University of Athens, GR-10562, Athens, Greece

Keywords: Stieltjes polynomials, Gauss-Kronrod quadrature formulae, interpolatory quadrature formulae
Received by editor(s): November 15, 1994
Article copyright: © Copyright 1996 American Mathematical Society