Generalized noninterpolatory rules for Cauchy principal value integrals

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.