Quadrature processes for integrals of Cauchy type

Abstract

We study questions relating to convergence of the process $$\int_{ - 1}^{ + 1} \rho (t)\frac{{f(t)}}{{t - x}}dt \approx \sum\nolimits_{k = 1}^n {\alpha _{k,n} } (x)f(x_k^{(n)} )( - 1< x1)$$ wherein the singular integral is taken in the principal value sense. General conditions for convergence in the class of continuously differentiable functionsf are formulated. In the case of the weight function ρ(t)=(√1-t2)−1, we investigate, under various assumptions onf, the convergence of a specific quadrature process.