CHARACTERISATIONS OF PARTITION OF UNITIES GENERATED BY ENTIRE FUNCTIONS IN \mathbb{C}^{d}

Abstract

Collections of functions forming a partition of unity play an important role in analysis. In this paper we characterise for any (Formula presented.) the entire functions (Formula presented.) for which the partition of unity condition (Formula presented.) holds for all (Formula presented.) The general characterisation leads to various easy ways of constructing such entire functions as well. We demonstrate the flexibility of the approach by showing that additional properties like continuity or differentiability of the functions (Formula presented.) can be controlled. In particular, this leads to easy ways of constructing entire functions (Formula presented.) such that the functions in the partition of unity belong to the Feichtinger algebra.