Hudson's Theorem for $\tau-$Wigner Transforms

In this paper, after introducing a natural generalization of the classical Wigner transform, namely the $\tau-$Wigner transforms, depending on the parameter $\tau\in[0,1]$, we study the problem of its positivity. In particular we prove two theorems of Hudson type considering the action of the $\tau-$Wigner transforms on functions and on distributions respectively. We give then an application of our results concerning Weyl and localization pseudo-differential operators.