A Factorization Theorem for the Fourier Transform of a Separable Locally Compact Abelian Group

Abstract

By a separable locally compact Abelian group, abbreviated SLCA group, we will mean a locally compact Abelian group whose topology comes from a separable metric. Let A be a SLCA group and let Δ be a subgroup of A with a discrete induced topology and such that A/Δ is compact. Let  be the dual group of A and let

$$T:{L^2}\left( A \right) \to {L^2}(\hat A)$$

be the Fourier transform. In this paper we will show that there is a natural factorization of 7 into the product of three unitary operators. When one specializes this factorization to the case where A = Z/r₁r₂, Z the integers one obtains the Cooley-Tukey algorithm [5]; when A is a finite Abelian group one obtains the results in [3], and for more general groups one obtains the results of A. Weil in [6,7].