# Decompositions of Rational Gabor Representations

Research paper by **Vignon Oussa**

Indexed on: **12 Jan '15**Published on: **12 Jan '15**Published in: **Mathematics - Representation Theory**

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#### Abstract

Let $\Gamma=\langle T_{k},M_{l}:k\in\mathbb{Z}^{d},l\in B\mathbb{Z}%
^{d}\rangle $ 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 $\pi:\left( \mathbb{Z}_{m}\times B\mathbb{Z}^{d}\right)
\rtimes\mathbb{Z}^{d}\rightarrow\Gamma$ 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 $\Gamma$ 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 $k\neq 1$ 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 $\Gamma$ which has a
subrepresentation equivalent to $\pi$ if and only if $\left\vert \det
B\right\vert \leq1.$ 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 $L^{2}(\mathbb{R})^{d}.$