  We show that if the Wigner function of a (possibly mixed) quantum state
decays toward infinity faster than any polynomial in the phase space variables
$x$ and $p$, then so do all of its derivatives, i.e., it is a Schwartz function
on phase space. This is equivalent to the condition that the Husimi function is
a Schwartz function, that the quantum state is a Schwartz operator in the sense
of Keyl et al., and, in the case of a pure state, that the wavefunction is a
Schwartz function on configuration space. We discuss the interpretation of this
constraint on Wigner functions and provide explicit bounds on Schwartz
seminorms.
