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 $\mathrm{\Lambda }=A(\mathbb{Z}+i\mathbb{Z}{)}^d$ 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 ${\mathbb{C}}^2$. By using an equivalent real-variable formulation, we show that these lattices fail to generate a Gabor frame.

KEYWORDS:

• Gabor frame
• Gauss function
• lattice
• Weierstrass sigma function
• interpolating function
• entire functions of several variables

AMS SUBJECT CLASSIFICATIONS:

• 42C15
• 33C90
• 32A30
• 94A12

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 We always use the ‘real’ inner product: $z\cdot w=\sum _{j=1}^d{z}_j{w}_j$ for $z=\left(z_j\right),w=\left(w_j\right)\in {\mathbb{C}}^d$.

2 Note that over $\mathbb{Z}\left[i\right]$ the greatest common divisor is only determined up to multiplication with $\pm 1,\pm i$. We refer the reader to [32] for the facts on division in $\mathbb{Z}\left[i\right]$.

Additional information

Funding

K. G. was supported in part by the project 31887-N32 of the Austrian Science Fund (FWF).