We study the Zak transform of totally positive (TP) functions. We use the
convergence of the Zak transform of TP functions of finite type to prove that
the Zak transforms of all TP functions without Gaussian factor in the Fourier
transform have only one zero in their fundamental domain of quasi-periodicity.
Our proof is based on complex analysis, especially the Theorem of Hurwitz and
some real analytic arguments, where we use the connection of TP functions of
finite type and exponential B-splines.