The paper is devoted to the foundation of approximation methods for integral equations of the form (aI+SbI+K)f=g, where S is the Cauchy singular integral operator on (-1,1) and K is a weakly singular integral operator. Here a,b,g are given functions on (-1,1) and the unknown function f on (-1,1) is looked for. It is assumed that a and b are real-valued and Hölder continuous functions on [-1,1] without common zeros and that g belongs to some weighted space of Hölder continuous functions. In particular, g may have a finite number of singularities. Based on known spectral properties of Cauchy singular integral operators approximation methods for the numerical solution of the above equation are constructed, where both aspects the theoretical convergence and the numerical practicability are taken into account. The weighted uniform convergence of these methods is studied using a general approach based on the theory of approximation spaces. With the help of this approach it is possible to prove simultaneously the stability, the convergence and results on the order of convergence of the approximation methods under consideration.

The paper is devoted to the foundation of approximation methods for integral equations of the form (aI+SbI+K)f=g, where S is the Cauchy singular integral operator on (-1,1) and K is a weakly singular integral operator. Here a,b,g are given functions on (-1,1) and the unknown function f on (-1,1) is looked for. It is assumed that a and b are real-valued and Hölder continuous functions on [-1,1] without common zeros and that g belongs to some weighted space of Hölder continuous functions. In particular, g may have a finite number of singularities. Based on known spectral properties of Cauchy singular integral operators approximation methods for the numerical solution of the above equation are constructed, where both aspects the theoretical convergence and the numerical practicability are taken into account. The weighted uniform convergence of these methods is studied using a general approach based on the theory of approximation spaces. With the help of this approach it is possible to prove simultaneously the stability, the convergence and results on the order of convergence of the approximation methods under consideration.