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Stieltjes polynomials and Lagrange interpolation

Authors: Sven Ehrich and Giuseppe Mastroianni
Journal: Math. Comp. 66 (1997), 311-331
MSC (1991): Primary 42A05, 65D05
MathSciNet review: 1388888
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Abstract: Bounds are proved for the Stieltjes polynomial $E_{n+1}$, and lower bounds are proved for the distances of consecutive zeros of the Stieltjes polynomials and the Legendre polynomials $P_n$. This sharpens a known interlacing result of Szegö. As a byproduct, bounds are obtained for the Geronimus polynomials $G_n$. Applying these results, convergence theorems are proved for the Lagrange interpolation process with respect to the zeros of $E_{n+1}$, and for the extended Lagrange interpolation process with respect to the zeros of $P_n E_{n+1}$ in the uniform and weighted $L^p$ norms. The corresponding Lebesgue constants are of optimal order.

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Sven Ehrich
Affiliation: Universität Hildesheim, Institut für Mathematik, D–31141 Hildesheim, Germany

Giuseppe Mastroianni
Affiliation: Università degli Studi della Basilicata, Dipartimento di Matematica, I–85100 Potenza, Italy

Keywords: Stieltjes polynomials, Lagrange interpolation, extended Lagrange interpolation, convergence
Received by editor(s): June 20, 1995
Received by editor(s) in revised form: December 4, 1995
Article copyright: © Copyright 1997 American Mathematical Society