Finite Fourier transforms computed by a modified Filon-Euler-Maclaurin method

The Filon-Euler-Maclaurin (FEM) method for computing the finite Fourier transform of a function in terms of the fast Fourier transform plus end point corrections is presented. Methods previously developed are derived within a unified framework. New methods of quadratic, quartic and quintic accuracy are given. Our formulation allows one to easily construct methods of higher order.