  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 B_m of order m, we
determine a large set of lattice parameters a,b>0 such that the Gabor family of
time-frequency shifts is a frame for L^2(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 ab<1. For even two-sided exponentials
and the related exponential B-spline of order 2, we find lower frame-bounds A,
which show the asymptotically linear decay A (1-ab) as the density ab of the
time-frequency lattice tends to the critical density ab=1.
