Time-frequency analysis associated with multidimensional Hankel-Gabor transform

The main crux of this paper is to introduce a new integral transform called the multidimensional Hankel-Gabor transform which generalizes the classical Gabor Fourier 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. Last, using the best approximations and the theory of reproducing kernels, we study the extremal functions related to this transform and we give an integral representation, best estimates of these functions on weighted Sobolev spaces