  We characterize the wave front set $WF^P_\ast(u)$ with respect to the
iterates of a linear partial differential operator with constant coefficients
of a classical distribution $u\in{\mathcal D}'(\Omega)$, $\Omega$ an open
subset in ${\mathbb R}^n$. We use recent Paley-Wiener theorems for generalized
ultradifferentiable classes in the sense of Braun, Meise and Taylor. We also
give several examples and applications to the regularity of operators with
variable coefficients and constant strength. Finally, we construct a
distribution with prescribed wave front set of this type.
