On Partition of Unities Generated by Entire Functions and Gabor Frames in $L^2({\mathbb R}^d)$ and $\ell ^2({\mathbb Z}^d)$

Abstract

We characterize the entire functions P of d variables, $d\ge 2,$ for which the ${\mathbb Z}^d$-translates of $P\chi _{[0,N]^d}$ satisfy the partition of unity for some $N\in \mathbb N.$ In contrast to the one-dimensional case, these entire functions are not necessarily periodic. In the case where P is a trigonometric polynomial, we characterize the maximal smoothness of $P\chi _{[0,N]^d},$ as well as the function that achieves it. A number of especially attractive constructions are achieved, e.g., of trigonometric polynomials leading to any desired (finite) regularity for a fixed support size. As an application we obtain easy constructions of matrix-generated Gabor frames in $L^2({\mathbb R}^d) ,$ with small support and high smoothness. By sampling this yields dual pairs of finite Gabor frames in $\ell ^2({\mathbb Z}^d).$