The remainder term for analytic functions of symmetric Gaussian quadratures
The remainder term for analytic functions of symmetric Gaussian quadratures
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DOI：
10.1090/s0025-5718-97-00798-9
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发表时间：
1997
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影响因子：
--
通讯作者：
Thomas Schira
中科院分区：
文献类型：
--
作者：
Thomas Schira
对于解析函数，高斯正交规则的余项可以表示为核K n的轮廓积分。本文研究了椭圆轮廓上各种对称权函数特别是Gegenbauer权函数的核问题。首先，提出并分析了核的一种新的级数表示。然后确定核的最大模在合适的椭圆上的位置。根据权重函数，最大模量在椭圆与实轴或虚轴的交点处获得。最后，对一些特殊的权函数进行了详细的讨论。
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.