Regular representations of time‐frequency groups

Abstract

In this paper, we study the Plancherel measure of a class of non-connected nilpotent groups which is of special interest in Gabor theory. Let G be a time-frequency group. That is Even in the case where G is not type I, we are able to obtain a decomposition of the left regular representation of G into a direct integral decomposition of irreducible representations when d = 1. Some interesting applications to Gabor theory are given as well. For example, when B is an integral matrix, we are able to obtain a direct integral decomposition of the Gabor representation of G.

Decompositions of rational Gabor representations

“…The concept of applying tools of abstract harmonic analysis to time-frequency analysis, and wavelet theory is not a new idea [1,2,3,7,11]. For example in [1], Larry Baggett gives a direct integral decomposition of the Stone-von Neumann representation of the discrete Heisenberg group acting in L 2 (R).…”

“…In Section 5.5, [7] the author obtains a characterization of tight Weyl-Heisenberg frames in L 2 (R) using the Zak transform and a precise computation of the Plancherel measure of a discrete type I group. In [11], the authors present a thorough study of the left regular representations of various subgroups of the reduced Heisenberg groups. Using well-known results of admissibility of unitary representations of locally compact groups, they were able to offer new insights on Gabor theory.…”

“…It is easily derived from the work in Section 4, [11] that if B is an integral matrix, then the Gabor representation π : BZ d × Z d → Γ ⊂ U L 2 R d defined by π (l, k) = M l T k is equivalent to a Date: January 2015.…”

“…subrepresentation of the left regular representation of Γ if and only if B is a unimodular matrix. The techniques used by the authors of [11] rely on the decompositions of the left regular representation and the Gabor representation into their irreducible components. The group generated by the operators M l and T k is a commutative group which is isomorphic to Z d × BZ d .…”

“…They are then able to compare both representations. As a result, one can derive from the work in the fourth section of [11] that the representation π is equivalent to a subrepresentation of the left regular representation if and only if B is a unimodular matrix. The main objective of this paper is to generalize these ideas to the more difficult case where B ∈ GL (d, Q) and Γ is not a commutative group.…”