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Mathematics of Computation

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The remainder term for analytic functions of symmetric Gaussian quadratures

Author: Thomas Schira
Journal: Math. Comp. 66 (1997), 297-310
MSC (1991): Primary 41A55; Secondary 65D30, 65D32
MathSciNet review: 1372009
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Abstract: For analytic functions the remainder term of Gaussian quadrature rules can be expressed as a contour integral with kernel $K_n$. In this paper the kernel is studied on elliptic contours for a great variety of symmetric weight functions including especially Gegenbauer weight functions. First a new series representation of the kernel is developed and analyzed. Then the location of the maximum modulus of the kernel on suitable ellipses is determined. Depending on the weight function the maximum modulus is attained at the intersection point of the ellipse with either the real or imaginary axis. Finally, a detailed discussion for some special weight functions is given.

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Thomas Schira
Affiliation: Institut für Praktische Mathematik, Universität Karlsruhe, D–76128 Karlsruhe, Germany

Keywords: Gaussian quadrature, remainder term for analytic functions, contour integral representation, kernel function
Received by editor(s): February 12, 1995
Received by editor(s) in revised form: January 26, 1996
Article copyright: © Copyright 1997 American Mathematical Society