Decompositions of Rational Gabor Representations

TL;DR

This paper provides a detailed decomposition of rational Gabor representations, revealing their structure and conditions for Parseval frames, and introduces new proofs for density conditions in Gabor analysis.

Contribution

It offers a novel direct integral irreducible decomposition of rational Gabor representations and characterizes when these representations relate to the left regular representation.

Findings

Decomposition of Gabor representations into irreducible components.

Identification of conditions for subrepresentations to be equivalent to the Gabor representation.

New proof of the density condition for rational Gabor systems.

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 $L^{2}(\mathbb{R})^{d}.$