Approximations of singular integral equations on Lyapunov contours in Banach spaces

A general approximation method is investigated for the numerical evaluation of the singular integral equations on Lyapunov contours, defined in Banach spaces. The method consists in the application of the Faber polynomials and the Faber-Laurent expansion. First, some theorems are proved for the approximation of functions in a complex domain, while these are defined in the Banach space, Hγ(Γ), (0 < γ ≤ 1), where Γ denotes a closed Lyapunov contour. These results are further used in order to prove the existence and uniqueness of the solutions for the systems on which the singular integral equations are reduced.