On the structure of Gabor and super Gabor spaces

Abstract

We study the structure of Gabor and super Gabor spaces as subspaces of L2(R2d) and specialize the results to the case where the spaces are generated by vectors of Hermite functions. We then show that such spaces are isometrically isomorphic to Fock spaces of polyanalytic functions and obtain structure theorems and orthogonal projections for both spaces at once. In particular we recover a structure result obtained by N. Vasilevskii using complex analysis and special functions. In contrast, our methods use only time-frequency analysis, exploring a link between time-frequency analysis and the theory of polyanalytic functions, provided by the polyanalytic part of the Gabor transform with a Hermite window, the polyanalytic Bargmann transform.