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Error bounds for Gauss-Turán quadrature formulae of analytic functions

Authors: Gradimir V. Milovanovic and Miodrag M. Spalevic
Journal: Math. Comp. 72 (2003), 1855-1872
MSC (2000): Primary 41A55; Secondary 65D30, 65D32
Published electronically: May 30, 2003
MathSciNet review: 1986808
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Abstract: We study the kernels of the remainder term $R_{n,s}(f)$ of Gauss-Turán quadrature formulas

\begin{displaymath}\int_{-1}^1f(t)w(t)\,dt=\sum_{\nu=1}^n \sum_{i=0}^{2s}A_{i,\n... ...au_\nu) +R_{n,s}(f)\qquad(n\in \mathbb{N};\, s\in\mathbb{N}_0)\end{displaymath}

for classes of analytic functions on elliptical contours with foci at $\pm1$, when the weight $w$ is one of the special Jacobi weights $w^{(\alpha,\beta)}(t)=(1-t)^\alpha(1+t)^\beta$ $(\alpha=\beta=-1/2$; $\alpha=\beta=1/2+s$; $\alpha=-1/2$, $\beta=1/2+s$; $\alpha=1/2+s$, $\beta=-1/2)$. We investigate the location on the contour where the modulus of the kernel attains its maximum value. Some 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

Miodrag M. Spalevic
Affiliation: Faculty of Science, Department of Mathematics and Informatics, P.O. Box 60, 34000 Kragujevac, Serbia

Keywords: Gauss-Tur\'an quadrature, $s$-orthogonality, zeros, multiple nodes, weight, measure, degree of exactness, remainder term for analytic functions, error estimate, contour integral representation, kernel function
Received by editor(s): February 7, 2002
Received by editor(s) in revised form: April 21, 2002
Published electronically: May 30, 2003
Additional Notes: The authors were supported in part by the Serbian Ministry of Science, Technology and Development (Project: Applied Orthogonal Systems, Constructive Approximation and Numerical Methods).
Dedicated: This paper is dedicated to Professor Walter Gautschi on the occasion of his 75th birthday
Article copyright: © Copyright 2003 American Mathematical Society