The advent of the fast Fourier transform method has greatly extended our ability to implement Fourier methods on digital computers. A description of the alogorithm and its programming is given here and followed by a theorem relating its operands, the finite sample sequences, to the continuous functions they often are intended to approximate. An analysis of the error due to discrete sampling over finite ranges is given in terms of aliasing. Procedures for computing Fourier integrals, convolutions and lagged products are outlined. Copyright © 1969 by The Institute of Electrical and Electronics Engineers, Inc.