The Gabor wave front set (Q2447941)

The authors study a global notion of wave front set \(\mathrm{WF}_G(u)\), which they call the Gabor wave front set, first introduced by Hörmander. Its definition is based on the global pseudodifferential of Shubin and adapted to study the singularities of tempered distributions \(u\). The authors prove that this notion of wave front set may also be equivalently characterized by decay in some cones on the short time Fourier transform \(V_\phi(u)\). They prove that it is enough to probe this decay on a lattice in the time-frequency plane whose lattice parameters are small enough such that the time-frequency shifts of \(\phi\) constitute a Gabor frame. Furthermore, the authors establish microellipticity and microlocality properties for pseudodifferential operators, and in particular microlocality of operators with Symbols in the global Hörmander class \(S_{0,0}^0\). Lastly, they discuss the Gabor wave front set examples of elementary distributions and compare these with their corresponding \(\mathcal{S}\)-wave front sets, defined by Coriasco and Maniccia.