Soto and Claverie give an elegant proof of this characterization, using the Segal–Bargmann transform. The reasoning is as follows. The Husimi Q function of ψ ψ may be computed as the squared magnitude of the Segal–Bargmann transform of ψ ψ , multiplied by a Gaussian. Meanwhile, the Husimi Q function is the convolution of the Wigner function with a Gaussian. If the Wigner function of ψ ψ is non-negative everywhere on phase space, then the Husimi Q function will be strictly positive everywhere on phase space. Thus, the Segal–Bargmann transform F ( x + i p ) of ψ ψ will be nowhere zero. Thus, by a standard result from complex analysis, we have