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

DOI:
https://doi.org/10.1090/S0025-5718-1989-0972372-4

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 \[ I(f;\lambda ):=\oint w(x)\frac {{f(x)}}{{x - \lambda }} dx,\quad \lambda \in ( - 1,1).\] This requires investigation of the convergence of the expansion \[ I(f;\lambda ) \sim \sum \limits _{k = 0}^\infty {(f,{p_k}){q_k}(\lambda ),\quad \lambda \in ( - 1,1),} \] in terms of the functions of the second kind $\{ {q_k}(\lambda )\} _0^\infty$ associated with *w*, where \[ (f,{p_k}):=\int _{ - 1}^1 {w(x)f(x){p_k}(x) dx\quad {\text {and}}\quad {q_k}(\lambda } ):=\oint w(x)\frac {{{p_k}(x)}}{{x - \lambda }} dx,\] $k = 0,1,2, \ldots ,\lambda \in ( - 1,1)$.

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Additional Information

Keywords:
Cauchy principal values,
numerical integration,
noninterpolatory integration rules,
orthogonal polynomials,
functions of the second kind

Article copyright:
© Copyright 1989
American Mathematical Society