Decompositions of Rational Gabor Representations

Let $\Gamma=\langle T_{k},M_{l}:k\in$\mathbb{Z}$^{d},l\in B$\mathbb{Z}$% ^{d}\rangle $beagroupofunitaryoperatorswhere$T_{k}$ is a translation operator and $M_l$ is a modulation operator acting on $L^{2}\left( $\mathbb{R}$^{d}\right) .$Assumingthat$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) \rtimes$\mathbb{Z}$^{d}\rightarrow\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 $\mathrm{\Gamma }$ via $\pi$ is unitarily equivalent to a direct sum of $\mathrm{c}\mathrm{a}\mathrm{r}\mathrm{d}\left([\mathrm{\Gamma },\mathrm{\Gamma }]\right)$ many disjoint subrepresentations: $L_{0},L_{1},\cdots,L_{$\mathrm{c}\mathrm{a}\mathrm{r}\mathrm{d}$\left( \left[ \Gamma,\Gamma\right] \right) -1}.$Itisshownthatforany$k\neq 1$ 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 $\mathrm{\Gamma }$ which has a subrepresentation equivalent to $\pi$ if and only if $\left\vert \det B\right\vert \leq1.$Usingacentraldecompositionoftherepresentation$\pi$ 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 $\pi (\mathrm{\Gamma })f$ is a Parseval frame in $L^2(\mathbb{R}{)}^d\mathrm{.}$