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Rational Gauss-Chebyshev quadrature formulas for complex poles outside $ [-1,1]$


Authors: Karl Deckers, Joris Van Deun and Adhemar Bultheel
Journal: Math. Comp. 77 (2008), 967-983
MSC (2000): Primary 42C05, 65D32
DOI: https://doi.org/10.1090/S0025-5718-07-01982-5
Published electronically: September 28, 2007
MathSciNet review: 2373187
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Abstract: In this paper we provide an extension of the Chebyshev orthogonal rational functions with arbitrary real poles outside $ [-1,1]$ to arbitrary complex poles outside $ [-1,1]$. The zeros of these orthogonal rational functions are not necessarily real anymore. By using the related para-orthogonal functions, however, we obtain an expression for the nodes and weights for rational Gauss-Chebyshev quadrature formulas integrating exactly in spaces of rational functions with arbitrary complex poles outside $ [-1,1]$.


References [Enhancements On Off] (What's this?)

  • 1. A. Bultheel, P. González-Vera, E. Hendriksen, and O. Njåstad.
    Orthogonal Rational Functions, volume 5 of Cambridge Monographs on Applied and Computational Mathematics.
    Cambridge University Press, 1999. MR 1676258 (2000c:33001)
  • 2. L. Daruis, P. González-Vera, and O. Njåstad.
    Szego quadrature formulas for certain Jacobi-type weight functions.
    Math. Comp., 71:683-701, 2001. MR 1885621 (2002k:41043)
  • 3. A. A. Gonchar and E. A. Rakhmanov.
    Equilibrium measure and the distribution of zeros of extremal polynomials.
    Math. USSR Sbornik, 53:119-130, 1986.
  • 4. E. B. Saff and V. Totik.
    Logarithmic potentials with external fields, volume 316 of Grundlehren der mathematischen Wissenschaften.
    Springer, Berlin Heidelberg, 1997. MR 1485778 (99h:31001)
  • 5. W. Van Assche and I. Vanherwegen.
    Quadrature formulas based on rational interpolation.
    Math. Comp., 61(204):765-783, 1993. MR 1195424 (94a:65014)
  • 6. J. Van Deun and A. Bultheel.
    Orthogonal rational functions and quadrature on an interval.
    J. Comput. Appl. Math., 153(1-2):487-495, 2003. MR 1985717 (2004e:42043)
  • 7. J. Van Deun, A. Bultheel, and P. González Vera.
    On computing rational Gauss-Chebyshev quadrature formulas.
    Math. Comp., 75:307-326, 2006. MR 2176401 (2006e:41060)
  • 8. P. Van gucht and A. Bultheel.
    A relation between orthogonal rational functions on the unit circle and the interval $ [-1,1]$.
    Comm. Anal. Th. Continued Fractions, 8:170-182, 2000. MR 1789681 (2001h:42037)
  • 9. J.A.C. Weideman and D.P. Laurie.
    Quadrature rules based on partial fraction expansions.
    Numerical Algorithms, 24:159-178, 2000. MR 1784997 (2001f:65032)

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Additional Information

Karl Deckers
Affiliation: Department of Computer Science, K. U. Leuven, B-3001 Heverlee, Belgium
Email: karl.deckers@cs.kuleuven.be

Joris Van Deun
Affiliation: Department of Computer Science, K. U. Leuven, B-3001 Heverlee, Belgium
Address at time of publication: Department of Mathematics and Computer Science, Universiteit Antwerpen, B-2020 Antwerpen, Belgium
Email: joris.vandeun@ua.ac.be

Adhemar Bultheel
Affiliation: Department of Computer Science, K. U. Leuven, B-3001 Heverlee, Belgium
Email: adhemar.bultheel@cs.kuleuven.be

DOI: https://doi.org/10.1090/S0025-5718-07-01982-5
Keywords: Quadrature formulas, orthogonal rational functions
Received by editor(s): February 9, 2006
Published electronically: September 28, 2007
Additional Notes: The work of the first two authors was partially supported by the Fund for Scientific Research (FWO), projects ‘CORFU: Constructive study of orthogonal functions’, grant #G.0184.02 and, ‘RAM: Rational modelling: optimal conditioning and stable algorithms’, grant #G.0423.05, and by the Belgian Programme on Interuniversity Attraction Poles, initiated by the Belgian Federal Science Policy Office. The scientific responsibility rests with the authors.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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