On Gabor frames with Hermite functions: polyanalytic spaces from the Heisenberg group

Abstract

Gabor frames with Hermite functions are equivalent to Fock frames with monomials windows and to sampling sequences in true poly-Fock spaces. In the L2 case, such an equivalence results from the unitarity of the so-called true poly- Bargmann transform. We will extend the equivalence to Banach spaces, applying Feichtinger-Gr¨ochenig coorbit theory to the Fock representation of the Heisenberg group. This task requires Lp estimates for the true poly-Bargmann transform which are obtained using the theory of modulation spaces. In the L2 case we will also revisit the complex variables approach and obtain an explicit formula for the interpolation problem in true poly-Fock spaces, which yields Gabor frames with Hermite functions by a duality argument.