Discrete characterizations of wave front sets of Fourier-Lebesgue and quasianalytic type

We obtain discrete characterizations of wave front sets of Fourier-Lebesgue and quasianalytic type. It is shown that the microlocal properties of an ultradistribution can be obtained by sampling the Fourier transforms of its localizations over a lattice in ${\mathbb{R}}^d$. In particular, we prove the following discrete characterization of the analytic wave front set of a distribution $f\in {\mathcal{D}}^{\mathrm{\prime }}(\mathrm{\Omega })$. Let $\mathrm{\Lambda }$ be a lattice in ${\mathbb{R}}^d$ and let $U$ be an open convex neighborhood of the origin such that $U\cap {\mathrm{\Lambda }}^{\ast }=\{0\}$. The analytic wave front set $W{F}_A(f)$ coincides with the complement in $\mathrm{\Omega }\times ({\mathbb{R}}^d\setminus \{0\})$ of the set of points $(x_0,{\xi }_0)$ for which there are an open neighborhood $V\subset \mathrm{\Omega }\cap (x_0+U)$ of $x_0$, an open conic neighborhood $\mathrm{\Gamma }$ of ${\xi }_0$, and a bounded sequence $(f_p{)}_{p\in \mathbb{N}}$ in ${\mathcal{E}}^{\mathrm{\prime }}(\mathrm{\Omega }\cap (x_0+U))$ with $f_p=f$ on $V$ such that for some $h > 0$ $$\underset{\mu \in \mathrm{\Gamma }\cap \mathrm{\Lambda }}{\sup }\mathrm{\mid }\widehat{f_p}(\mu )\mathrm{\mid }\mathrm{\mid }\mu {\mathrm{\mid }}^p\le h^{p+1}p!,\mathrm{\forall }p\in \mathbb{N}\mathrm{.}$$