A spline collocation method for singular integral equations with piecewise continuous coefficients

We consider the collocation method with piecewise linear trial functions for systems of singular integral equations with Cauchy kernel and piecewise continuous coefficients. Necessary and sufficient conditions for the stability in L² are given. The results are obtained in the case of a closed Ljapunov curve as well as in the case of an interval. The proof of the main theorem is based on a modification of the Banach algebra technique established in the local principle by Gohberg and Krupnik 2]. Our results extend those obtained by Prößdorf and Schmidt 9, 10] from the case of continuous coefficients and unit circle to the case of piecewise continuous coefficients.