Sampling and interpolation in the Bargmann-Fock space of polyanalytic functions

Abstract

We give a complete characterization of all lattice sampling and interpolating sequences in the Fock space of polyanalytic functions (poly-Fock spaces), displaying a ”Nyquist rate” which increases with n, the degree of polyanaliticity of the space: A sequence of lattice points is sampling if and only if its density is strictly larger than n, and it is interpolating if and only if its density is strictly smaller than n. In our method of proof we introduce a unitary mapping between vector valued Hilbert spaces and poly-Fock spaces which allows the extension of Bargmann´s theory to polyanalytic spaces. Then we connect this 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.