Completeness of Gabor systems

We investigate the completeness of Gabor systems with respect to several classes of window functions on rational lattices. Our main results show that the time–frequency shifts of every finite linear combination of Hermite functions with respect to a rational lattice are complete in L2(R)L2(R), thus generalizing a remark of von Neumann (and proved by Bargmann, Perelomov et al.). An analogous result is proven for functions that factor into certain rational functions and the Gaussian. The results are also interesting from a conceptual point of view since they show a vast difference between the completeness and the frame property of a Gabor system. In the terminology of physics we prove new results about the completeness of coherent state subsystems.