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

Zeros of the Zak transform of totally positive functions