Reconstruction inversion formulas for the Laguerre Gabor transform

In this paper, we define and study the Gabor transform in the context of the Laguerre hypergroup. We prove some of its basic properties, such as Plancherel theorem, inversion formula and Calder´on’s reproducing inversion formula. Next, using the harmonic analysis related to Laguerre hypergroup, we examine spaces of Sobolev type for which we make explicit kernels reproducing. Exploiting the aforesaid theory, we introduce and study the extremal function associated with the Gabor transform. Finally, by utilizing the reproducing kernels we establishimportant estimates for this extremal function.