Polynomial Approximations of Functions with Endpoint Singularities and Product Integration Formulas

Abstract

Several problems of mathematical physics lead to Fredholm integral equations of the second kind where the kernels are either weakly or strongly sin- gular and the known terms are smooth. These equations have solutions which are smooth in the whole interval of integration except at the endpoints where they have mild singularities. In this paper we derive new pointwise and uniform polynomial approximation error estimates for that type of function. These esti- mates are then used to obtain bounds for the remainder terms of interpolatory product rules, based on the zeros of classical Jacobi orthogonal polynomials, that have been proposed for the discretization of integrals of the form / k(x,y)f(x)dx, appearing in the integral equations mentioned above.