The Gabor wave front set

2012/07/24 by Luigi Rodino, Rodino, Luigi, Patrik Wahlberg +1 · 1 citation
Computer Science · Mathematics · #35A18 #35S05 #42B35 #42B37 #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Image and Signal Denoising Methods #Mathematical Analysis and Transform Methods #Mathematical functions and polynomials #math.AP #math.FA #msc:35A18 #msc:35S05 #msc:42B35 #msc:42B37

paper · pdf · doi:10.48550/arxiv.1207.5628

29 pages

openalex publication_date 2012/07/24 · arxiv created 2013/01/24 · arxiv updated 2013/01/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We define the Gabor wave front set WF_G(u) of a tempered distribution u in terms of rapid decay of its Gabor coefficients in a conic subset of the phase space. We show the inclusion WF_G(a^w(x,D) u) ⊆ WF_G(u), u ∈ \mathscr S'(\mathbb R^d), a ∈ S_0,0⁰, where S_0,0⁰ denotes the Hörmander symbol class of order zero and parameter values zero. We compare our definition with other definitions in the literature, namely the classical and the global wave front sets of Hörmander, and the \cS-wave front set of Coriasco and Maniccia. In particular, we prove that the Gabor wave front set and the global wave front set of Hörmander coincide.

Citations

• The wavelet transform, time-frequency localization and signal analysis
• Pseudodifferential operators and Banach algebras in mobile communications

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• Dynamical restriction for Schrödinger equations

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