FOR GAUSS-TYPE QUADRATURE RULES
FOR GAUSS-TYPE QUADRATURE RULES
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对于高斯型求积规则
DOI：
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发表时间：
2006
期刊：
影响因子：
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通讯作者：
W. Gautschi
中科院分区：
文献类型：
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作者：
W. Gautschi
. 1961年，P. J. Davis和P. Rabinowitz建立了一个关于高斯和高斯-洛巴托求积规则的漂亮的“圆定理”。他们表明，在雅可比权重函数的情况下，高斯权重，适当地归一化并绘制在高斯节点上，渐近地位于以原点为中心的单位圆的上半部分。在这里，类似的结果被证明为更一般的权函数，主要是那些在Szego类，不仅为高斯和高斯-Lobatto，而且为高斯-Radau公式。对于更严格的权函数类，圆定理甚至对高斯-克朗罗德规则也成立。在势理论中，圆定理的圆可以解释为区间[（cid：0）1;1]的平衡测度的密度的倒数。类似的定理适用于（（cid：0）1;1）的任何紧子集（cid：1）上支持的权重函数，在这种情况下，（归一化的）高斯点接近（cid：1）的平衡测度的倒数密度。许多结果都用图表说明。
. In 1961, P.J. Davis and P. Rabinowitz established a beautiful “circle theorem” for Gauss and Gauss– Lobatto quadrature rules. They showed that, in the case of Jacobi weight functions, the Gaussian weights, suitably normalized and plotted against the Gaussian nodes, lie asymptotically for large orders on the upper half of the unit circle centered at the origin. Here analogous results are proved for rather more general weight functions—essentially those in the Szeg¨o class—, not only for Gauss and Gauss–Lobatto, but also for Gauss–Radau formulae. For much more restricted classes of weight functions, the circle theorem even holds for Gauss–Kronrod rules. In terms of potential theory, the semicircle of the circle theorem can be interpreted as the reciprocal density of the equilibrium measure of the interval [(cid:0)1;1℄ . Analogous theorems hold for weight functions supported on any compact subset (cid:1) of ((cid:0)1;1) , in which case the (normalized) Gauss points approach the reciprocal density of the equilibrium measure of (cid:1) . Many of the results are illustrated graphically.