# Sampling of Entire Functions of Several Complex Variables on a Lattice and Multivariate Gabor Frames

## TL;DR

This paper constructs entire functions in multiple complex variables that vanish on specific lattices and demonstrates that certain dense lattices do not serve as sampling sets for the Bargmann-Fock space or generate Gabor frames.

## Contribution

It introduces a general method for constructing entire functions vanishing on complex lattices and links these constructions to the failure of certain lattices to be sampling sets or Gabor frames.

## Key findings

- Lattices of density >1 can fail to be sampling sets for the Bargmann-Fock space.
- Certain dense lattices do not generate Gabor frames.
- A connection between complex analysis and time-frequency analysis is established.

## Abstract

We give a general construction of entire functions in $d$ complex variables that vanish on a lattice of the form $L = A (Z + i Z )^d$ 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 $C ^2$. By using an equivalent real-variable formulation, we show that these lattices fail to generate a Gabor frame.