Convergence properties of a class of product formulas for weakly singular integral equations

Abstract: We examine the convergence of product quadrature formulas of interpolatory type, based on the zeros of certain generalized Jacobi polynomials, for the discretization of integrals of the type where the kernel is weakly singular and the function has singularities only at the endpoints . In particular, when , , , and has algebraic singularities of the form , , we prove that the uniform rate of convergence of the rules is in the case of the first kernel, and if , or if , for the second, where m is the number of points in the quadrature rule.