Gabor frames and totally positive functions

Let g be a totally positive function of finite type, that is, ĝ(ξ)=∏ν=1M(1+2πiδνξ)−1 for δν∈R, δν≠0, and M≥2, and let α,β>0. Then the set {e2πiβltg(t−αk):k,l∈Z} is a frame for L2(R) if and only if αβ<1. This result is a first positive contribution to a conjecture of Daubechies from 1990. Until now, the complete characterization of lattice parameters α, β that generate a frame has been known for only six window functions g. Our main result now yields an uncountable class of window functions. As a by-product of the proof method, we also derive new sampling theorems in shift-invariant spaces and obtain the correct Nyquist rate.