We consider the general question of constructing a partition of unity formed by translates of a compactly supported function g : Rd → 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 Zd. The results are illustrated with constructions of dual pairs of Gabor frames for L2(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 L2(Rd) lead.
- Translation partitions of unity
- Gabor systems
- Dual frame pairs