Gabor frames for rational functions

Abstract

We study the frame properties of the Gabor systemsIn particular, we prove that for Herglotz windows g such systems always form a frame for L 2 (R) if α, β > 0, αβ ≤ 1. For general rational windows g ∈ L 2 (R) we prove that G(g; α, β) is a frame for L 2 (R) if 0 < α, β, αβ < 1, αβ ∈ Q and ĝ(ξ) = 0, ξ > 0, thus confirming Daubechies conjecture for this class of functions. We also discuss some related questions, in particular sampling in shift-invariant subspaces of L 2 (R).

Gaussian Gabor frames, Seshadri constants and generalized Buser–Sarnak invariants

“…Namely, that {π λ g iI } λ∈Λ is a Gabor frame if and only if the covolume |Λ| < 1, [Lyu92,Sei92,SW92]. Recent progress on the description of the frame set of a Gabor atom has been made for totally positive functions GAFA [GS13,GRS18] and for rational functions [BKL21]. Note that all aforementioned results on frame sets for Gabor systems are for uniformly discrete point sets in the plane.…”

Gaussian Gabor frames, Seshadri constants and generalized Buser--Sarnak invariants

“…Namely, that (g iI , Λ) is a Gabor frame if and only if the covolume |Λ| < 1, [39,49,50]. Recent progress on the description of the frame set of a Gabor atom has been made for totally positive functions [26,25] and for rational functions [3]. Note that all aforementioned results on frame sets for Gabor systems are for uniformly discrete point sets in the plane R 2 .…”