Zeros of the Zak Transform of averaged totally positive functions

Let α>0 and let g∈L1(R) be a continuous function, whose Fourier transform is ĝ(ω)=Ce−γω2e−2πiδω∏ν=1∞e2πiδνω1+2πiδνω∏j=1meλj−2πiαω−1λj−2πiαω,where C>0, γ⩾0, δ,δν,λj∈R, ∑ν=1∞δν2<∞, m∈Z+. We prove that its Zak transform Zαg(x,ω)=∑k∈Zg(x+αk)e−2πikαω has only one zero (x∗,12α) in the fundamental domain [0,α)×0,1α. In particular, the result is valid for totally positive functions. Earlier it was known for such functions without the factor e−γω2. We also establish simplicity of the zero with respect to each variable and give the applications to Gabor analysis. The described class of functions is closed under convolution.