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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Noninterpolatory integration rules for Cauchy principal value integrals
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by P. Rabinowitz and D. S. Lubinsky PDF
Math. Comp. 53 (1989), 279-295 Request permission

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
  • © Copyright 1989 American Mathematical Society
  • 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