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 to dual pairs of discrete Gabor frames for ℓ²(ℤd) and finite Gabor frames for periodic sequences on ℤd.

我们考虑构建由紧支持函数g的平移形成的统一分区的一般问题：ℝ d → ℂ。特别是，我们证明了这样的函数具有特殊的结构，可以简化具有特定属性的统一分区的构造。我们还证明，可以修改函数g使其相对于 ℤ d上的给定对称群对称。结果用L 2(ℝ d）。此外，我们获得了构造双 Gabor 框架的一般方法，其窗函数关于任意对称群对称。通过采样和周期化，这些L ² (ℝ d ) 的双 Gabor 帧导致ℓ ² (ℤ d ) 的双离散 Gabor 帧对和 ℤ d上的周期序列的有限 Gabor 帧。