Given discrete groups \(\Gamma \subset \Delta \) we characterize \((\Gamma ,\sigma )\)-invariant spaces that are also invariant under \(\Delta \). This will be done in terms of subspaces that we define using an appropriate Zak transform and a particular partition of the underlying group. On the way, we obtain a new characterization of principal \((\Gamma ,\sigma )\)-invariant spaces in terms of the Zak transform of its generator. This result is in the spirit of the well-known characterization of shift-invariant spaces in terms of the Fourier transform. As a consequence of our results, we give a solution for the problem of finding the \((\Gamma ,\sigma )\)-invariant space nearest—in the sense of least squares—to a given set of data.

给定离散群\(\Gamma \subset \Delta \)我们刻画\((\Gamma ,\sigma )\) -不变空间，在\(\Delta \)下也是不变的。这将根据我们使用适当的 Zak 变换和底层组的特定分区定义的子空间来完成。在此过程中，我们根据其生成器的 Zak 变换获得了主\((\Gamma ,\sigma )\)不变空间的新特征。这个结果是在傅立叶变换方面的众所周知的移不变空间表征的精神。根据我们的结果，我们给出了寻找\((\Gamma ,\sigma )\)- 在最小二乘意义上，最接近给定数据集的不变空间。