Approximate wave function from approximate non-representable Wigner distributions

We have recently derived a phase space approximation for wave propagation, and we have shown that this approximation is accurate especially compared to the stationary phase approximation. The method gives an approximate Wigner distribution, which raises the question, is the approximate Wigner distribution representable? If it is, then we can invert it to obtain the corresponding wave function. If it is not, can we obtain an approximate wave function from it, and how good is the approximation? Although the answer to the first question is no, we show that from the Wigner approximation one can recover the exact spatial spectrum magnitude and derivative of the phase. We also consider other methods for obtaining approximate wave functions from the Wigner approximation.