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. © 2009 Elsevier Inc. All rights reserved.
Abreu, L. D. (2010). Sampling and interpolation in Bargmann-Fock spaces of polyanalytic functions. Applied and Computational Harmonic Analysis, 29(3), 287–302. https://doi.org/10.1016/j.acha.2009.11.004