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

作者：Debrouwere Andreas; Vindas Jasson^*

来源：Journal of Mathematical Analysis and Applications, 2016, 438(2): 889-908.

DOI：10.1016/j.jmaa.2016.02.034

• Wave front sets
• Fourier series
• Ultradifferentiable functions
• Quasianalytic classes
• Fourier-Lebesgue spaces
• Ultradistributions

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 R-d. In particular, we prove the following discrete characterization of the analytic wave front set of a distribution f is an element of D'(Omega). Let Lambda be a lattice in R-d and let U be an open convex neighborhood of the origin such that U boolean AND Lambda* = {0}. The analytic wave front set WFA(f) coincides with the complement in Omega x (R-d \ {0}) of the set of points (x(0), xi(0)) for which there are an open neighborhood V subset of Omega boolean AND (x(0) + U) of xo, an open conic neighborhood Gamma of xi(0), and a bounded sequence (f(p))(p is an element of N) in epsilon'(Omega boolean AND (x(0) + U)) with f(p) = f on V such that for some h > 0
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• 出版日期2016-6-15