Zeros of the Zak transform of totally positive functions (Q744963)

The zeroes of the Zak transform of totally positive functions are studied in this paper. The author proves that, for integrable, totally positive functions whose Fourier transform does not contain a Gaussian factor, the Zak transform of the function contains exactly one zero in its fundamental domain. The proof relies on the relationships between totally positive functions and exponential B-splines. In addition, convergence properties of Zak transforms of a sequence of functions obtained by truncating an infinite product in the Fourier transform of a totally positive function, to the Zak transform of the totally positive function, are used. As a corollary, it is shown that certain Gabor families determined by a continuous totally positive function are Gabor frames.