Fast and stable contour integration for high order divided differences via elliptic functions

In this paper, we will present a new method for evaluating high order divided differences for certain classes of analytic, possibly, operator valued functions. This is a classical problem in numerical mathematics but also arises in new applications such as, e.g., the use of generalized convolution quadrature to solve retarded potential integral equations. The functions which we will consider are allowed to grow exponentially to the left complex half plane, polynomially to the right half plane and have an oscillatory behaviour with increasing imaginary part. The interpolation points are scattered in a large real interval. Our approach is based on the representation of divided differences as contour integral and we will employ a subtle parameterization of the contour in combination with a quadrature approximation by the trapezoidal rule.