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Error bounds for Gauss-Turán quadrature formulae of analytic functions

Authors:	Gradimir V Milovanovic Miodrag M Spalevic

Institution:	Department of Mathematics, University of Nis, Faculty of Electronic Engineering, P.O. Box 73, 18000 Nis, Serbia ; Faculty of Science, Department of Mathematics and Informatics, P.O. Box 60, 34000 Kragujevac, Serbia

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 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.

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

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