Translation partitions of unity, symmetry properties, and Gabor frames

Abstract

We consider the general question of constructing a partition of unity formed by translates of a compactly supported function g : DOUBLE-STRUCK CAPITAL Rd -> Double-struck capital 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 DOUBLE-STRUCK CAPITAL Zd. The results are illustrated with constructions of dual pairs of Gabor frames for L2(DOUBLE-STRUCK CAPITAL Rd). 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 L-2(DOUBLE-STRUCK CAPITAL Rd) lead to dual pairs of discrete Gabor frames for l(2)(DOUBLE-STRUCK CAPITAL Zd) and finite Gabor frames for periodic sequences on DOUBLE-STRUCK CAPITAL Zd.