Time-Frequency Analysis and Representations of the Discrete Heisenberg Group

Abstract

The operators [ϱ _ω (j, k, l)f](t) = e ^2πiωl e ^2πiωkt f(t + j) on $L^{2}(\mathbb{R})$ constitute a representation of the discrete Heisenberg group. We investigate how this representation decomposes as a direct integral of irreducible representations. The answer is quite different depending on whether ω is rational or irrational, and in the latter case it provides illustrations of some interesting pathological phenomena.