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Sampling of entire functions of several complex variables on a lattice and multivariate Gabor frames

ABSTRACT We give a general construction of entire functions in d complex variables that vanish on a lattice of the form for an invertible complex-valued matrix. As an application, we exhibit a class of lattices of density >1 that fail to be a sampling set for the Bargmann–Fock space in . By using an equivalent real-variable formulation, we show that these lattices fail to generate a Gabor frame.