Sampling and Interpolation in Bargmann-fock Spaces of Polyanalytic Functions

We give a complete characterization of all lattice sampling and interpolating sequences in the Fock space of polyanalytic functions (polyFock spaces), displaying a ”Nyquist rate” which increases with the degree of polyanaliticity. This is done introducing a unitary mapping between vector valued Hilbert spaces and poly-Fock spaces. This mapping extends Bargmann ́s theory to polyanalytic spaces. Then we connect the mapping to Gabor transforms with Hermite windows and apply duality principles from time-frequency analysis in order to reduce the problem to a ”purely holomorphic” situation.