On Partition of Unities Generated by Entire Functions and Gabor Frames in L²(R^d) and ℓ²(Z^d)

We characterize the entire functions P of d variables, d≥2, for which the Z^d-translates of Pχ_[0,N]d satisfy the partition of unity for some N∈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 Pχ_[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 L²(R^d), with small support and high smoothness. By sampling this yields dual pairs of finite Gabor frames in ℓ²(Z^d).