Decompositions of Rational Gabor Representations
Vignon Oussa

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 be a group of unitary operators where is a translation operator and is a modulation operator acting on Assuming that is a non-singular rational matrix of order 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 where 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 via is unitarily equivalent to a direct sum of $\mathrm{card}\left( \left[ \Gamma,\Gamma\right]…
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Taxonomy
TopicsMathematical Analysis and Transform Methods · Mathematical functions and polynomials · Optical and Acousto-Optic Technologies
