Decompositions of Rational Gabor Representations
Let Γ=⟨Tk ,Ml :k ∈ ℤd ,l ∈ Bℤd ⟩ be a group of unitary operators where Tk is a translation operator and Ml is a modulation operator acting on L2(ℝ)d. 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 π : (ℤm ×Bℤd) ⋊ ℤd → Γ where π(θ,l,k) = e2πiθMlTk . We also show that the left regular representation of (ℤm ×Bℤd) ⋊ ℤd which is identified with Γ via π is unitarily equivalent to a direct sum of card ([Γ,Γ]) many disjoint subrepresentations: L0, L1, ⋯ , Lcard([Γ,Γ])−1. It is shown that for any k ≠ 1 the subrepresentation Lk of the left regular representation is disjoint from the Gabor representation. Furthermore, we prove that there is a subrepresentation L1 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 L2(ℝ)d such that π(Γ)f is a Parseval frame in L2(ℝ)d.
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