The approximation of cauchy singular integrals and their limiting values at the endpoints of the curve of integration

We examine a specific approximating process for the singular integral $$S^* (f;x) equiv frac{1}{pi }int_{ - 1}^{ + 1} {frac{{f(t)}}{{sqrt {1 - l^2 } (t - x)}}} dt( - 1< x< 1)$$ taken in the principal value sense. We study the influence of some local properties of the functionf on the convergence of the approximations. Next, assuming that (S^* (f;c) equiv mathop {lim }limits_{x to c} S^* (f;x)) , where c is an arbitrary one of the endpoints ?1 and 1, we show that the conditions which guarantee the existence of the limiting values S*(f; c) (c=±1) and, moreover, the convergence of the process at an arbitrary point x∈ (?1, 1) are not always sufficient for convergence of the approximations at the endpoints.