## Generalized noninterpolatory rules for Cauchy principal value integrals

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- by Philip Rabinowitz PDF
- Math. Comp.
**54**(1990), 271-279 Request permission

## Abstract:

Consider the Cauchy principal value integral \[ I(kf;\lambda ) = \oint k(x)\frac {{f(x)}}{{x - \lambda }} dx,\quad - 1 < \lambda < 1.\] If we approximate $f(x)$ by $\sum _{j = 0}^N\;{a_j}{p_j}(x;w)$ where $\{ {p_j}\}$ is a sequence of orthonormal polynomials with respect to an admissible weight function*w*and ${a_j} = (f,{p_j})$, then an approximation to $I(kf;\lambda )$ is given by $\sum _{j = 0}^N\;{a_j}I(k{p_j};\lambda )$. If, in turn, we approximate ${a_j}$ by ${a_{jm}} = \sum _{i = 1}^m\;{w_{im}}f({x_{im}}){p_j}({x_{im}})$, then we get a double sequence of approximations $\{ Q_m^N(f;\lambda )\}$ to $I(kf;\lambda )$. We study the convergence of this sequence by relating it to the sequence of approximations associated with $I(wf;\lambda )$ which has been investigated previously.

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## Additional Information

- © Copyright 1990 American Mathematical Society
- Journal: Math. Comp.
**54**(1990), 271-279 - MSC: Primary 65D30
- DOI: https://doi.org/10.1090/S0025-5718-1990-0990601-6
- MathSciNet review: 990601