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

Abstract

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 $f\in\mathcal{D}'(Ω)$. Let $Λ$ be a lattice in $\mathbb{R}^{d}$ and let $U$ be an open convex neighborhood of the origin such that $U\capΛ^{*}=\{0\}$. The analytic wave front set $WF_{A}(f)$ coincides with the complement in $Ω\times(\mathbb{R}^{d}\setminus\{0\})$ of the set of points $(x_0,ξ_0)$ for which there are an open neighborhood $V\subset Ω\cap (x_0+U)$ of $x_0$, an open conic neighborhood $Γ$ of $ξ_0$, and a bounded sequence $(f_p)_{p \in \mathbb{N}}$ in $\mathcal{E}'(Ω\cap (x_0+U))$ with $f_p= f$ on $V$ such that for some $h > 0$ \[ \sup_{μ\in Γ\cap Λ} |\widehat{f_p} (μ)| |μ|^p \leq h^{p+1}p!\:, \qquad \forall p \in \mathbb{N}. \]