Fast Fourier transform for discontinuous functions

In computational electromagnetics and other areas of computational science and engineering, Fourier transforms of discontinuous functions are often required. We present a fast algorithm for the evaluation of the Fourier transform of piecewise smooth functions with uniformly or nonuniformly sampled data by using a double interpolation procedure combined with the fast Fourier transform (FFT) algorithm. We call this the discontinuous FFT algorithm. For N sample points, the complexity of the algorithm is O(νNp + νN log (N)) where p is the interpolation order and ν is the oversampling factor. The method also provides a new nonuniform FFT algorithm for continuous functions. Numerical experiments demonstrate the high efficiency and accuracy of this discontinuous FFT algorithm.