Rapid computation of the discrete fourier transform

Algorithms for the rapid computation of the forward and inverse discrete Fourier transform for points which are nonequispaced or whose number is unrestricted are presented. The computational procedure is based on approximation using a local Taylor series expansion and the fast Fourier transform (FFT). The forward transform for nonequispaced points is computed as the solution of a linear system involving the inverse Fourier transform. This latter system is solved using the iterative method GMRES with preconditioning. Numerical results are given to confirm the efficiency of the algorithms.