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Positive quadrature formulas III: asymptotics of weights

Author: Franz Peherstorfer
Journal: Math. Comp. 77 (2008), 2241-2259
MSC (2000): Primary 65D32; Secondary 42C05
Published electronically: May 1, 2008
MathSciNet review: 2429883
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Abstract: First we discuss briefly our former characterization theorem for positive interpolation quadrature formulas (abbreviated qf), provide an equivalent characterization in terms of Jacobi matrices, and give links and applications to other qf, in particular to Gauss-Kronrod quadratures and recent rediscoveries. Then for any polynomial $ t_n$ which generates a positive qf, a weight function (depending on $ n$) is given with respect to which $ t_n$ is orthogonal to $ \mathbb{P}_{n-1}$. With the help of this result an asymptotic representation of the quadrature weights is derived. In general the asymptotic behaviour is different from that of the Gaussian weights. Only under additional conditions do the quadrature weights satisfy the so-called circle law. Corresponding results are obtained for positive qf of Radau and Lobatto type.

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Franz Peherstorfer
Affiliation: Abteilung für Dynamische Systeme und Approximationstheorie, Institut für Analysis, J.K. Universität Linz, Altenberger Strasse 69, 4040 Linz, Austria

Received by editor(s): June 6, 2007
Received by editor(s) in revised form: September 4, 2007
Published electronically: May 1, 2008
Additional Notes: The author was supported by the Austrian Science Fund FWF, project no. P20413-N18.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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