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Anti-Szego quadrature rules

Authors: Sun-Mi Kim and Lothar Reichel
Journal: Math. Comp. 76 (2007), 795-810
MSC (2000): Primary 65D32, 42A10; Secondary 30E20
Published electronically: November 28, 2006
MathSciNet review: 2291837
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Abstract: Szego quadrature rules are discretization methods for approximating integrals of the form $ \int_{-\pi}^{\pi} f(e^{it}) d\mu(t)$. This paper presents a new class of discretization methods, which we refer to as anti-Szego quadrature rules. Anti-Szego rules can be used to estimate the error in Szego quadrature rules: under suitable conditions, pairs of associated Szego and anti-Szego quadrature rules provide upper and lower bounds for the value of the given integral. The construction of anti-Szego quadrature rules is almost identical to that of Szego quadrature rules in that pairs of associated Szego and anti-Szego rules differ only in the choice of a parameter of unit modulus. Several examples of Szego and anti-Szego quadrature rule pairs are presented.

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Additional Information

Sun-Mi Kim
Affiliation: Department of Mathematical Sciences, Kent State University, Kent, Ohio 44242

Lothar Reichel
Affiliation: Department of Mathematical Sciences, Kent State University, Kent, Ohio 44242

Keywords: Numerical integration, error estimation, Szeg\H{o} polynomials, trigonometric polynomials
Received by editor(s): October 12, 2004
Received by editor(s) in revised form: September 23, 2005
Published electronically: November 28, 2006
Additional Notes: Research supported in part by NSF grant DMS-0107858.
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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