Let be a totally positive function of finite type, that is, for , , and , and let . Then the set is a frame for if and only if . This result is a first positive contribution to a conjecture of Daubechies from 1990. Until now, the complete characterization of lattice parameters , that generate a frame has been known for only six window functions . 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.
"Gabor frames and totally positive functions." Duke Math. J. 162 (6) 1003 - 1031, 15 April 2013. https://doi.org/10.1215/00127094-2141944