#### Title

Decompositions of Rational Gabor Representations

#### Publication Date

2014

#### Document Type

Article

#### Abstract

Let Γ=⟨*T** _{k}* ,

*M*

*:*

_{l}*k*∈ ℤ

*,*

^{d}*l*∈

*B*ℤ

*⟩ be a group of unitary operators where*

^{d}*T*

*is a translation operator and*

_{k}*M*

*is a modulation operator acting on*

_{l}*L*

^{2}(ℝ)

*. Assuming that*

^{d}*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

*π*: (ℤ

*×*

_{m}*B*ℤ

*) ⋊ ℤ*

^{d}*→ Γ where*

^{d}*π*(

*θ*,

*l*,

*k*) =

*e*

^{2}

^{πiθ}*M*

_{l}*T*

*. We also show that the left regular representation of (ℤ*

_{k}*×*

_{m}*B*ℤ

*) ⋊ ℤ*

^{d}*which is identified with Γ via*

^{d}*π*is unitarily equivalent to a direct sum of card ([Γ,Γ]) many disjoint subrepresentations:

*L*

_{0},

*L*

_{1}, ⋯ ,

*L*

_{card}

_{([}

_{Γ}

_{,}

_{Γ}

_{])}

_{−}

_{1}. It is shown that for any

*k*≠ 1 the subrepresentation

*L*

*of the left regular representation is disjoint from the Gabor representation. Furthermore, we prove that there is a subrepresentation*

_{k}*L*

_{1}of the left regular representation of Γ which has a subrepresentation equivalent to

*π*if and only if |det

*B*| ≤ 1. Using a central decomposition of the representation

*π*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}(ℝ)

*such that*

^{d}*π*(Γ)

*f*is a Parseval frame in

*L*

^{2}(ℝ)

*.*

^{d}#### Original Citation

Oussa, V. (2014). Decompositions of Rational Gabor Representations. arXiv:1408.2024 [math.RT].

#### Virtual Commons Citation

Oussa, Vignon (2014). Decompositions of Rational Gabor Representations. In *Mathematics Faculty Publications.* Paper 42.

Available at: https://vc.bridgew.edu/math_fac/42

## Comments

This is the pre-print version harvested from ArXiv. (http://arxiv.org/abs/1408.2024).

Published version available:

Oussa, V. (2015). Decompositions of Rational Gabor Representations. In Jens G. Christensen, Susanna Dann, Axita Mayeli, Gesture Olafsson (Eds.),

Contemporary Mathematics: Trends in Harmonic Analysis and Its Applications. AMS Special Session on Harmonic Analysis and Its Applications(pp. 37-54). Baltimore, MD: American Mathematical Society. doi: 10.1090/conm/650.