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 $f\in {\mathcal{D}}^{\prime }(\Omega )$. Let $\Lambda$ be a lattice in ${\mathbb{R}}^d$ and let $U$ be an open convex neighborhood of the origin such that $U\cap {\Lambda }^{\ast }=\{0\}$. The analytic wave front set $W{F}_A(f)$ coincides with the complement in $\Omega \times ({\mathbb{R}}^d\setminus \{0\})$ of the set of points $(x_0,{\xi }_0)$ for which there are an open neighborhood $V\subset \Omega \cap (x_0+U)$ of $x_0$, an open conic neighborhood $\Gamma$ of ${\xi }_0$, and a bounded sequence $(f_p{)}_{p\in \mathbb{N}}$ in ${\mathcal{E}}^{\prime }(\Omega \cap (x_0+U))$ with $f_p=f$ on $V$ such that for some $h>0$ \[ \sup_{μ\in Γ\cap Λ} |\widehat{f_p} (μ)| |μ|^p \leq h^{p+1}p!\:, \qquad \forall p \in \mathbb{N}. \]

21 pages

References in corpus (1)

• A smooth introduction to the wavefront set

Cited by in corpus (2)

• Optimal embeddings of ultradistributions into differential algebras
• Beyond Wiener's Lemma: Nuclear Convolution Algebras and the Inversion of Digital Filters