Frames generated by compact group actions
Frames generated by compact group actions
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DOI：
10.1090/tran/6954
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发表时间：
2015-09
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通讯作者：
Joseph W. Iverson
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作者：
Joseph W. Iverson
设$K$是紧群，$\rho$是Hilbert空间$\mathcal{H}_\rho$上$K$的表示.我们用值域函数对数学{H}_Rho$的不变子空间进行了分类，并研究了形式为$\Rho(\xi)f_i_{\xi\in K，i\in i}$的框架.这首先是在平移不变性的设置中完成的，其中$K$包含在一个更大的组$G$中，而$\rho$在$\mathcal{H}_\rho=L^2(G)$上留下平移。对于这种情况，我们的分析依赖于Zak变换的一个新的、操作员值版本。对于更一般的表示，我们开发了一个称为“括号”的计算系统来分析表示结构和具有单个生成器的框架。文中还探讨了几种应用。然后，我们将注意力转向具有多个生成元的框架，给出了一个对偶定理，它概括了现有关于有限群生成的框架的大部分研究，以及框架和Riesz序列的经典对偶。
Let $K$ be a compact group, and let $\rho$ be a representation of $K$ on a Hilbert space $\mathcal{H}_\rho$. We classify invariant subspaces of $\mathcal{H}_\rho$ in terms of range functions, and investigate frames of the form $\{\rho(\xi) f_i\}_{\xi \in K, i \in I}$. This is done first in the setting of translation invariance, where $K$ is contained in a larger group $G$ and $\rho$ is left translation on $\mathcal{H}_\rho = L^2(G)$. For this case, our analysis relies on a new, operator-valued version of the Zak transform. For more general representations, we develop a calculational system known as a "bracket" to analyze representation structures and frames with a single generator. Several applications are explored. Then we turn our attention to frames with multiple generators, giving a duality theorem that encapsulates much of the existing research on frames generated by finite groups, as well as classical duality of frames and Riesz sequences.