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Szegö quadrature formulas for certain Jacobi-type weight functions

Authors: Leyla Daruis, Pablo González-Vera and Olav Njåstad
Journal: Math. Comp. 71 (2002), 683-701
MSC (2000): Primary 41A55, 33C45
Published electronically: October 4, 2001
MathSciNet review: 1885621
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Abstract: In this paper we are concerned with the estimation of integrals on the unit circle of the form $\int_0^{2\pi}f(e^{i\theta})\omega(\theta)d\theta$ by means of the so-called Szegö quadrature formulas, i.e., formulas of the type $\sum_{j=1}^n\lambda_jf(x_j)$ with distinct nodes on the unit circle, exactly integrating Laurent polynomials in subspaces of dimension as high as possible. When considering certain weight functions $\omega(\theta)$ related to the Jacobi functions for the interval $[-1,1],$ nodes $\{x_j\}_{j=1}^n$ and weights $\{\lambda_j\}_{j=1}^n$ in Szegö quadrature formulas are explicitly deduced. Illustrative numerical examples are also given.

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Additional Information

Leyla Daruis
Affiliation: Department of Mathematical Analysis, La Laguna University, Tenerife, Canary Islands, Spain

Pablo González-Vera
Affiliation: Corresponding author: Department of Mathematical Analysis, La Laguna University, 38271- La Laguna, Tenerife, Spain
Email: Fax: 34-922-318195

Olav Njåstad
Affiliation: Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway

Keywords: Weight functions, quadrature formulas, orthogonal polynomials, Szeg\"o polynomials, error bounds
Received by editor(s): February 11, 2000
Received by editor(s) in revised form: July 10, 2000
Published electronically: October 4, 2001
Additional Notes: The work of the first author was performed as part of a grant of the Gobierno de Canarias.
The work of the second author was supported by the Scientific Research Project of the Spanish D.G.E.S. under contract PB96-1029.
Article copyright: © Copyright 2001 American Mathematical Society