Computing with the Finite Fourier Transform Is Mathematics!

In November of 1979, there appeared a paper in the Bulletin of the AMS by L. Auslander and R. Tolimieri with the delightful title of “Is Computing with the Finite Fourier Transform Pure or Applied Mathematics?” The question was of course rhetorical and one of the main goals of the paper was to show that the finite Fourier transform, and the family of efficient algorithms used to compute it, the “Fast Fourier Transform” (FFT), pillars of the world of digital signal processing, were of interest to both pure and applied mathematicians. Auslander had come of age as an applied mathematician at a time when pure and applied mathematicians still received much of the same training. Whether one then turned one’s skills towards more or less applied areas became a matter of taste. As Tolimieri retells is, Auslander had become distressed at the development of a separate discipline of applied mathematics which had grown apart from much of core mathematics. The effect of this was detrimental on both sides. On the one hand applied mathematicians had fewer tools to bring to problems, and conversely, pure mathematicians were often turning a blind eye to the potentially rich bed of inspiration provided by “real world” problems. Auslander had hoped that their paper might be one which helped to show that really in the end, it was all just mathematics, and in so doing help mend a perceived growing rift in the mathematics community. This paper is intended as both a companion to the earlier paper [2], as well as an update. Primarily, it aims to demonstrate that the investigation of the finite and fast Fourier transform continues to be a vibrant and interesting direction of mathematical research. Auslander and Tolimieri concentrated on its relations to nilpotent harmonic analysis and theta functions. In this paper, connections between the famous Cooley-Tukey FFT and group representation theory are stressed. However, it is the further hope that this paper, taken together with its predecessor, continues to give some indication of the rich interplay that can be found at the nexus (whereever that may be!) of pure and applied mathematics, and possibly motivate others to investigate the challenges and problems that lie there.