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Convergence results for piecewise linear quadratures for Cauchy principal value integrals

Author: Philip Rabinowitz
Journal: Math. Comp. 51 (1988), 741-747
MSC: Primary 65D30; Secondary 41A55
MathSciNet review: 958639
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Abstract: Conditions on k and f are given for the pointwise and uniform convergence to the Cauchy principal value integral

$\displaystyle \int {\frac{{k(x)f(x)}}{{x - \lambda }}\,dx} ,\quad - 1 < \lambda < 1,$

of a sequence of integrals of piecewise linear approximations to $ f(x)$ or $ {g_\lambda }(x) = (f(x) = f(\lambda ))/(x - \lambda )$. The important special case, $ k(x) = {(1 - x)^\alpha }{(1 + x)^\beta }$, is considered in detail.

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Keywords: Cauchy principal value integrals, piecewise linear approximation, product integration, Jacobi weight function
Article copyright: © Copyright 1988 American Mathematical Society

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