On Partition of Unities Generated by Entire Functions and Gabor Frames in L2(Rd) and l2(Zd)

We characterize the entire functions P of d variables, for which the -translates of satisfy the partition of unity for some 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 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 with small support and high smoothness. By sampling this yields dual pairs of finite Gabor frames in l(2)(Z(d)).