Some counterexamples in the theory of Weyl-Heisenberg frames

We present an example of a positive function g with a positive Fourier transform g/spl circ/ and reasonable smoothness and decay properties such that (-1)/sup nm/exp(/spl pi/itm)g(t-n), n, m/spl isin/Z does not constitute a frame for L/sup 2/(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.