On partition of unities generated by entire functions and Gabor frames in $\ltrd$ and $\ell^2(\mzd)$

TL;DR

This paper characterizes entire functions that generate partitions of unity via their integer translates in multiple dimensions, explores their smoothness properties, and applies these results to construct Gabor frames with desirable features.

Contribution

It provides a comprehensive characterization of entire functions generating partitions of unity in multiple dimensions and develops new constructions for Gabor frames with controlled support and smoothness.

Findings

Characterization of entire functions generating partitions of unity in multiple dimensions.

Maximal smoothness conditions for trigonometric polynomial generators.

Explicit constructions of Gabor frames with small support and high smoothness.

Abstract

We characterize the entire functions $P$ of $d$ variables, $d\ge 2,$ for which the $\mzd$-translates of $P\chi_{[0,N]^d}$ satisfy the partition of unity for some $N\in \mn.$ 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 $\ltrd,$ with small support and high smoothness. By sampling this yields dual pairs of finite Gabor frames in $\ell^2(\mzd).$