Sampling and interpolation in Bargmann–Fock spaces of polyanalytic functions

Abstract

Using Gabor analysis, we give a complete characterization of all lattice sampling and interpolating sequences in the Fock space of polyanalytic functions, displaying a “Nyquist rate” which increases with n, the degree of polyanaliticity of the space. Such conditions are equivalent to sharp lattice density conditions for certain vector-valued Gabor systems, namely superframes and Gabor super-Riesz sequences with Hermite windows, and in the case of superframes they were studied recently by Gröchenig and Lyubarskii. The proofs of our main results use variations of the Janssen–Ron–Shen duality principle and reveal a duality between sampling and interpolation in polyanalytic spaces, and multiple interpolation and sampling in analytic spaces. To connect these topics we introduce the polyanalytic Bargmann transform, a unitary mapping between vector-valued Hilbert spaces and polyanalytic Fock spaces, which extends the Bargmann transform to polyanalytic spaces. Motivated by this connection, we discuss a vector-valued version of the Gabor transform. These ideas have natural applications in the context of multiplexing of signals. We also point out that a recent result of Balan, Casazza and Landau, concerning density of Gabor frames, has important consequences for the Gröchenig–Lyubarskii conjecture on the density of Gabor frames with Hermite windows.