Time-frequency analysis for the multidimensional Gabor Transform

The main crux of this paper is to introduce a new integral transform called the multidimensional Hankel–Gabor transform and to give some new results related to this transform as Plancherel’s, Parseval’s, inversion and Calderón’s reproducing formulas. Next, we analyse the concentration of this transform on sets of finite measure and we give uncertainty principle for orthonormal sequences and Donoho–Stark’s type uncertainty principle. Last, we introduce a new class of pseudo-differential operator \({\mathscr {L}}_{u,v}(\sigma )\) ${\mathcal{L}}_{u,v}(\sigma )$ called localization operator which depend on a symbol \(\sigma \) $\sigma$ and two functions u and v, we give a criteria in terms of the symbol \(\sigma \) $\sigma$ for its boundedness and compactness, we also show that this operator belongs to the Schatten-Von Neumann classes \(S^p\) $S^p$ for all \(p \in [1; +\infty ]\) $p\in [1;+\infty ]$ and we give a trace formula.