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 RdRd. In particular, we prove the following discrete characterization of the analytic wave front set of a distribution f∈D′(Ω)f∈D′(Ω). Let Λ be a lattice in RdRd and let U   be an open convex neighborhood of the origin such that U∩Λ⁎={0}U∩Λ⁎={0}. The analytic wave front set WFA(f)WFA(f) coincides with the complement in Ω×(Rd∖{0})Ω×(Rd∖{0}) of the set of points (x0,ξ0)(x0,ξ0) for which there are an open neighborhood V⊂Ω∩(x0+U)V⊂Ω∩(x0+U) of x0x0, an open conic neighborhood Γ of ξ0ξ0, and a bounded sequence (fp)p∈N(fp)p∈N in E′(Ω∩(x0+U))E′(Ω∩(x0+U)) with fp=ffp=f on V   such that for some h>0h>0supμ∈Γ∩Λ⁡|fpˆ(μ)||μ|p≤hp+1p!,∀p∈N.