  We study the question under which conditions the zero set of a (cross-)
Wigner distribution W (f, g) or a short-time Fourier transform is empty. This
is the case when both f and g are generalized Gaussians, but we will construct
less obvious examples consisting of exponential functions and their
convolutions. The results require elements from the theory of totally positive
functions, Bessel functions, and Hurwitz polynomials. The question of zero-free
Wigner distributions is also related to Hudson's theorem for the positivity of
the Wigner distribution and to Hardy's uncertainty principle. We then construct
a class of step functions S so that the Wigner distribution W (f, 1 (0,1))
always possesses a zero f $\in$ S $\cap$ L p for p < $\infty$, but may be
zero-free for f $\in$ S $\cap$ L $\infty$. The examples show that the question
of zeros of the Wigner distribution may be quite subtle and relate to several
branches of analysis.
