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Mathematics of Computation

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Noninterpolatory integration rules for Cauchy principal value integrals

Authors: P. Rabinowitz and D. S. Lubinsky
Journal: Math. Comp. 53 (1989), 279-295
MSC: Primary 41A55; Secondary 65D30
MathSciNet review: 972372
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Abstract: Let $ w(x)$ be an admissible weight on $ [ - 1,1]$ and let $ \{ {p_n}(x)\} _0^\infty $ be its associated sequence of orthonormal polynomials. We study the convergence of noninterpolatory integration rules for approximating Cauchy principal value integrals

$\displaystyle I(f;\lambda ):=\oint w(x)\frac{{f(x)}}{{x - \lambda }}\,dx,\quad \lambda \in ( - 1,1).$

This requires investigation of the convergence of the expansion

$\displaystyle I(f;\lambda ) \sim \sum\limits_{k = 0}^\infty {(f,{p_k}){q_k}(\lambda ),\quad \lambda \in ( - 1,1),} $

in terms of the functions of the second kind $ \{ {q_k}(\lambda )\} _0^\infty $ associated with w, where

$\displaystyle (f,{p_k}):=\int_{ - 1}^1 {w(x)f(x){p_k}(x)\,dx\quad {\text{and}}\quad {q_k}(\lambda } ):=\oint w(x)\frac{{{p_k}(x)}}{{x - \lambda }}\,dx,$

$ k = 0,1,2, \ldots ,\lambda \in ( - 1,1)$.

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Keywords: Cauchy principal values, numerical integration, noninterpolatory integration rules, orthogonal polynomials, functions of the second kind
Article copyright: © Copyright 1989 American Mathematical Society

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