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

Abstract

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.