Translation partitions of unity, symmetry properties, and Gabor frames

We consider the general question of constructing a partition of unity formed by translates of a compactly supported function g : R^d → C. In particular, we prove that such functions have a special structure that simplifies the construction of partitions of unity with specific properties. We also prove that it is possible to modify the function g in such a way that it becomes symmetric with respect to a given symmetry group on Z^d. The results are illustrated with constructions of dual pairs of Gabor frames for L²(R^d). In addition, we obtain general approaches to construct dual Gabor frames whose window functions are symmetric with respect to an arbitrary symmetry group. Through sampling and periodization, these dual Gabor frames for L²(R^d) lead.