2022/02/13 by Frederick, Christina, Mayeli, Azita
#FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2202.06343
A Gabor system generated by a window function g∈ L2(ℝd) and a separable set Λ× Γ⊂ ℝ2d is the collection of time-frequency shifts of g given by \mathcal G(g, Λ× Γ) = \ e2πi ξ⋅ tg(t-x)\ (x,ξ)∈ Λ× Γ. One of the fundamental problems in Gabor analysis is to characterize all windows and time-frequency sets that generate a Gabor frame or Gabor orthonormal basis. The case of Gabor orthonormal bases generated by characteristic functions g=χΩ has been solved by Han and Wang. In this paper, we build on these results and obtain a full characterization of Riesz Gabor systems of the form \mathcal G(χΩ, Λ× Γ) when Ω is a tiling of ℝd with respect to Λ. Furthermore, for a certain class of lattices Λ× Γ, we prove that a necessary condition for the characteristic function of a multi-tiling set to serve as a window function for a Riesz Gabor basis is that the set must be a tiling set. To prove this, we develop new results on the zeros of the Zak transform and connect these results to Gabor frames.