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Maximum of the modulus of kernels in Gauss-Turán quadratures


Authors: Gradimir V. Milovanovic, Miodrag M. Spalevic and Miroslav S. Pranic
Journal: Math. Comp. 77 (2008), 985-994
MSC (1991): Primary 41A55; Secondary 65D30, 65D32
DOI: https://doi.org/10.1090/S0025-5718-07-02032-7
Published electronically: November 14, 2007
MathSciNet review: 2373188
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Abstract: We study the kernels $ K_{n,s}(z)$ in the remainder terms $ R_{n,s}(f)$ of the Gauss-Turán quadrature formulae for analytic functions on elliptical contours with foci at $ \pm 1$, when the weight $ \omega$ is a generalized Chebyshev weight function. For the generalized Chebyshev weight of the first (third) kind, it is shown that the modulus of the kernel $ \vert K_{n,s}(z)\vert$ attains its maximum on the real axis (positive real semi-axis) for each $ n\geq n_0, n_0=n_0(\rho,s)$. It was stated as a conjecture in [Math. Comp. 72 (2003), 1855-1872]. For the generalized Chebyshev weight of the second kind, in the case when the number of the nodes $ n$ in the corresponding Gauss-Turán quadrature formula is even, it is shown that the modulus of the kernel attains its maximum on the imaginary axis for each $ n\geq n_0, n_0=n_0(\rho,s)$. Numerical examples are included.


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Additional Information

Gradimir V. Milovanovic
Affiliation: Department of Mathematics, University of Niš, Faculty of Electronic Engineering, P.O. Box 73, 18000 Niš, Serbia
Email: grade@elfak.ni.ac.yu

Miodrag M. Spalevic
Affiliation: Department of Mathematics and Informatics, University of Kragujevac, Faculty of Science, P.O. Box 60, 34000 Kragujevac, Serbia
Email: spale@kg.ac.yu

Miroslav S. Pranic
Affiliation: Department of Mathematics and Informatics, University of Banja Luka, Faculty of Science, M. Stojanovića 2, 51000 Banja Luka, Bosnia and Herzegovina
Email: pranic77m@yahoo.com

DOI: https://doi.org/10.1090/S0025-5718-07-02032-7
Keywords: Gauss-Tur\'an quadrature, Chebyshev weight functions, remainder term for analytic functions, error estimate, contour integral representation, confocal ellipses, kernel.
Received by editor(s): August 15, 2006
Received by editor(s) in revised form: December 4, 2006
Published electronically: November 14, 2007
Additional Notes: The authors were supported in part by the Swiss National Science Foundation (SCOPES Joint Research Project No. IB7320-111079 “New Methods for Quadrature”) and the Serbian Ministry of Science (Research Projects: “Approximation of linear operators” (No. #144005) & “Orthogonal systems and applications” (No. #144004C))
Article copyright: © Copyright 2007 American Mathematical Society