Some counterexamples in the theory of Weyl-Heisenberg frames

We present an example of a positive function g with a positive Fourier transform gˆ and reasonable smoothness and decay properties such that (-1)^nmexp(πitm)g(t-n), n, m∈Z does not constitute a frame for L²(R). We also give counterexamples for the statement that one can tell (in)definiteness of a Weyl-Heisenberg frame operator from (in)definiteness of its Weyl symbol