Collocation for singular integral equations with fixed singularities of particular Mellin type

This paper is concerned with the stability of collocation methods for Cauchy singular integral equations with fixed singularities on the interval [-1, 1]. The operator in these equations is supposed to be of the form aI+bS+B± with piecewise continuous functions a and b. The operator S is the Cauchy singular integral operator and B± is a finite sum of integral operators with fixed singularities at the points ±1 of special kind. The collocation methods search for approximate solutions of the form v(x)pn(x) or μ(x)pn(x) with Chebyshev weights (Formula presented.), respectively, and collocation with respect to Chebyshev nodes of first and third or fourth kind is considered. For the stability of collocation methods in a weighted L2-space, we derive necessary and sufficient conditions