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The remainder term for analytic functions of symmetric Gaussian quadratures. (English) Zbl 0854.41025
Summary: 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.

##### MSC:
 41A55 Approximate quadratures 65D30 Numerical integration 65D32 Numerical quadrature and cubature formulas
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