Quadrature processes for integrals of Cauchy type

We study questions relating to convergence of the process
$$ \int_{-1}^{+1}\rho(t)\frac{f(t)}{t-x}dt\approx\sum_{k=1}^n\alpha_{k,n}(x)f(x_k^{(n)})\qquad(-1<x<1), $$
wherein the singular integral is taken in the principal value sense. General conditions for convergence in the class of continuously differentiable functions $f$ are formulated. In the case of the weight function $\rho(t)=(\sqrt{1-t^2})^{-1}$, we investigate, under various assumptions on $f$, the convergence of a specific quadrature process.