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Noninterpolatory Integration Rules for Cauchy Principal Value Integrals

P. Rabinowitz and D. S. Lubinsky
Mathematics of Computation
Vol. 53, No. 187 (Jul., 1989), pp. 279-295
DOI: 10.2307/2008361
Stable URL: http://www.jstor.org/stable/2008361
Page Count: 17
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Abstract

Let $w(x)$ be an admissible weight on $\lbrack-1, 1\rbrack$ and let $\{p_n(x)\}^\infty_0$ be its associated sequence of orthonormal polynomials. We study the convergence of noninterpolatory integration rules for approximating Cauchy principal value integrals $$I(f; \lambda):= \not\int^1_{-1} w(x) \frac{f(x)}{x - \lambda} dx,\quad \lambda \in (-1, 1).$$ This requires investigation of the convergence of the expansion $$I(f; \lambda) \sim \sum^\infty_{k = 0} (f, p_k)q_k(\lambda),\quad \lambda \in (-1, 1),$$ in terms of the functions of the second kind $\{q_k(\lambda)\}^\infty_0$ associated with $w$, where $$(f, p_k):= \int^1_{-1} w(x)f(x)p_k(x) dx\quad\text{and}\quad q_k(\lambda):= \not{\int}^1_{-1} w(x) \frac{p_k(x)}{x - \lambda} dx,\\ k = 0, 1, 2,\ldots, \lambda \in (-1, 1).$$

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