Maximum of the modulus of kernels in Gauss-Turán quadratures
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- by Gradimir V. Milovanović, Miodrag M. Spalević and Miroslav S. Pranić PDF
- Math. Comp. 77 (2008), 985-994 Request permission
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 $|K_{n,s}(z)|$ 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.References
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Additional Information
- Gradimir V. Milovanović
- 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. Spalević
- Affiliation: Department of Mathematics and Informatics, University of Kragujevac, Faculty of Science, P.O. Box 60, 34000 Kragujevac, Serbia
- MR Author ID: 600543
- Email: spale@kg.ac.yu
- Miroslav S. Pranić
- 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
- 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))
- © Copyright 2007 American Mathematical Society
- 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
- MathSciNet review: 2373188