Zak transforms and Gabor frames of totally positive functions and exponential B-splines

We study totally positive (TP) functions of finite type and exponential B-splines as window functions for Gabor frames. We establish the connection of the Zak transform of these two classes of functions and prove that the Zak transforms have only one zero in their fundamental domain of quasi-periodicity. Our proof is based on the variation-diminishing property of shifts of exponential B-splines. For the exponential B-spline BmBm of order mm, we determine a set of lattice parameters α,β>0α,β>0 such that the Gabor family G(Bm,α,β)G(Bm,α,β) of time–frequency shifts e2πilβBm(⋅−kα)e2πilβBm(⋅−kα), k,l∈Zk,l∈Z, is a frame for L2(R)L2(R). By the connection of its Zak transform to the Zak transform of TP functions of finite type, our result provides an alternative proof that TP functions of finite type provide Gabor frames for all lattice parameters with αβ<1αβ<1. For even two-sided exponentials g(x)=λ2e−λ∣x∣ we find lower frame-bounds AA, which show the asymptotically linear decay A∼(1−αβ)A∼(1−αβ) as the density αβαβ of the time–frequency lattice tends to the critical density αβ=1αβ=1.