The Cooley-Tukey FFT and Group Theory, Volume 48, Number 10

Pure and Applied Mathematics—Two Sides of a Coin In November of 1979 there appeared in the Bulletin of the AMS a paper by L. Auslander and R. Tolimieri [3] with the delightful title “Is Computing with the Finite Fourier Transform Pure or Applied Mathematics?” This rhetorical question was answered by showing that in fact the finite Fourier transform and the family of efficient algorithms used to compute it (the Fast Fourier Transform (FFT), a pillar of the world of digital signal processing) are 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 the same training. The ends towards which these skills were then directed became a matter of taste. As Tolimieri retells it,1 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 development was detrimental to both sides. On the one hand, applied mathematicians had fewer tools to bring to problems, and, conversely, pure mathematicians were often ignoring the fertile bed of inspiration provided by real-world problems. Auslander hoped their paper would help mend a growing perceived rift in the mathematical community by showing the ultimate unity of pure and applied mathematics. We will show that investigation of finite and fast Fourier transforms continues to be a varied and interesting direction of mathematical research. Whereas Auslander and Tolimieri concentrated on relations to nilpotent harmonic analysis and theta functions, we emphasize connections between the famous Cooley-Tukey FFT and group representation theory. In this way we hope to provide further evidence of the rich interplay of ideas which can be found at the nexus of pure and applied mathematics.