Decompositions of Rational Gabor Representations

Abstract

Let $\Gamma =⟨T_k,M_l:k\in {\mathbb{Z}}^d,l\in B\mathbb{Z}$ be a group of unitary operators where $T_k$ is a translation operator and $M_l$ is a modulation operator acting on $L^2\left({\mathbb{R}}^d\right).$ Assuming that $B$ is a non-singular rational matrix of order $d,$ with at least one rational non-integral entry, we obtain a direct integral irreducible decomposition of the Gabor representation which is defined by the isomorphism $\pi :\left({\mathbb{Z}}_m\times B{\mathbb{Z}}^d\right)⋊{\mathbb{Z}}^d\to \Gamma$ where $\pi \left(\theta ,l,k\right)=e^{2\pi i\theta }{M}_l{T}_k.$ We also show that the left regular representation of $\left( \mathbb{Z}_{m}\times B\mathbb{Z}% ^{d}\right) \rtimes\mathbb{Z}^{d}$ which is identified with $\Gamma$ via $\pi$ is unitarily equivalent to a direct sum of $\mathrm{card}\left(\left[\Gamma ,\Gamma \right]\right)$ many disjoint subrepresentations: $L_0,L_1,\cdots ,L_{\mathrm{card}\left(\left[\Gamma ,\Gamma \right]\right)-1}.$ It is shown that for any $k\ne 1$ the subrepresentation $L_k$ of the left regular representation is disjoint from the Gabor representation. Furthermore, we prove that there is a subrepresentation $L_1$ of the left regular representation of $\Gamma$ which has a subrepresentation equivalent to $\pi$ if and only if $\left|\det B\right|\le 1.$ Using a central decomposition of the representation $\pi$ and a direct integral decomposition of the left regular representation, we derive some important results of Gabor theory. More precisely, a new proof for the density condition for the rational case is obtained. We also derive characteristics of vectors $f$ in $L^2(\mathbb{R}{)}^d$ such that $\pi (\Gamma )f$ is a Parseval frame in $L^2(\mathbb{R}{)}^d.$