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Stieltjes-type polynomials on the unit circle

Authors: B. de la Calle Ysern, G. López Lagomasino and L. Reichel
Journal: Math. Comp. 78 (2009), 969-997
MSC (2000): Primary 65D32, 42A10, 42C05; Secondary 30E20
Published electronically: October 27, 2008
MathSciNet review: 2476567
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Abstract: Stieltjes-type polynomials corresponding to measures supported on the unit circle $ \mathbb{T}$ are introduced and their asymptotic properties away from $ \mathbb{T}$ are studied for general classes of measures. As an application, we prove the convergence of an associated sequence of interpolating rational functions to the corresponding Carathéodory function. In turn, this is used to give an estimate of the rate of convergence of certain quadrature formulae that resemble the Gauss-Kronrod rule, provided that the integrand is analytic in a neighborhood of $ \mathbb{T}$.

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B. de la Calle Ysern
Affiliation: Departamento de Matemática Aplicada, E. T. S. de Ingenieros Industriales, Universidad Politécnica de Madrid, José G. Abascal 2, 28006 Madrid, Spain

G. López Lagomasino
Affiliation: Departamento de Matemáticas, Escuela Politécnica Superior, Universidad Carlos III de Madrid, Universidad 30, 28911 Leganés, Spain

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

Keywords: Quadrature rules on the unit circle, para-orthogonal polynomials, Stieltjes polynomials, Gauss-Kronrod quadrature
Received by editor(s): October 3, 2007
Received by editor(s) in revised form: April 25, 2008
Published electronically: October 27, 2008
Additional Notes: The work of B. de la Calle received support from Dirección General de Investigación (DGI), Ministerio de Educación y Ciencia, under grants MTM2006-13000-C03-02 and MTM2006-07186 and from UPM-CAM under grants CCG07-UPM/000-1652 and CCG07-UPM/ESP-1896
The work of G. López was supported by DGI under grant MTM2006-13000-C03-02 and by UC3M-CAM through CCG06-UC3M/ESP-0690
The work of L. Reichel was supported by an OBR Research Challenge Grant.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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