The abc problem for Gabor systems
One of fundamental problems in Gabor analysis is to identify window functions and time-frequency shift lattices such that the corresponding Gabor system is a Gabor frame for the space of all square-integrable functions on the real line. The range of density parameters for shift lattices is fully known surprisingly only for few window functions, including the Gaussian window and totally positive windows. For window functions, especially outside Feichtinger algebra, the range of density parameters could be arbitrary complicated, as Janssen's tie implies. In this talk, we provide a full classification of triples (a,b,c) for which the Gabor system generated by the ideal window function on an interval I of length c on the time-frequency lattice aZ x bZ is a Gabor frame.