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The Gabor wave front set

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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

$$\begin{aligned} WF_G(a^w(x,D) u)\subseteq WF_G(u), \quad u \in \fancyscript{S}'({\mathbb {R}}^{d}),\ a \in S_{0,0}^0, \end{aligned}$$

where \(S_{0,0}^0\) 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 \(\fancyscript{S}\)-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.

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Acknowledgments

We thank one of the referees for pointing out the generalization, expressed in Remark 3.6, of the Gabor frame lattice in the definition of the Gabor wave front set.

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Correspondence to Patrik Wahlberg.

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Communicated by K. Gröchenig.

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Rodino, L., Wahlberg, P. The Gabor wave front set. Monatsh Math 173, 625–655 (2014). https://doi.org/10.1007/s00605-013-0592-0

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