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Mathematics of Computation

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Quadrature formulas based on rational interpolation

Authors: Walter Van Assche and Ingrid Vanherwegen
Journal: Math. Comp. 61 (1993), 765-783
MSC: Primary 65D32; Secondary 41A05, 41A55, 42C05
MathSciNet review: 1195424
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Abstract: We consider quadrature formulas based on interpolation using the basis functions $ 1/(1 + {t_k}x)\quad (k = 1,2,3, \ldots )$ on $ [ - 1,1]$, where $ {t_k}$ are parameters on the interval $ ( - 1,1)$. We investigate two types of quadratures: quadrature formulas of maximum accuracy which correctly integrate as many basis functions as possible (Gaussian quadrature), and quadrature formulas whose nodes are the zeros of the orthogonal functions obtained by orthogonalizing the system of basis functions (orthogonal quadrature). We show that both approaches involve orthogonal polynomials with modified weights which depend on the number of quadrature nodes. The asymptotic distribution of the nodes is obtained as well as various interlacing properties and monotonicity results for the nodes.

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Keywords: Quadrature, rational interpolation, orthogonal polynomials
Article copyright: © Copyright 1993 American Mathematical Society

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