有限指标集上的动态抽样
Dynamical Sampling on Finite Index Sets
摘要：
我们考虑有界算子$A$迭代地作用在Hilbert空间$\mathcal H$中的有限向量集$\{f_i : i\in I\}$上，并解决了为迭代集$\{A^nf_i : i\in I,\,n=0,1,2,\ldots\}$形成空间$\mathcal H$的框架提供充要条件的问题。对于正规算子$A$，我们通过证明一个特征定理完全解决了这个问题。我们的证明结合了来自不同数学领域的技术，如算子理论，谱理论，调和分析和单位圆盘的复分析。在本文的第二部分，我们放弃了$A$正规的强条件。尽管这个相当一般的设置，我们能够证明一个特征，它允许推断出许多强大的必要条件的运营商$A$。例如，$A$需要类似于一种非常特殊的收缩。我们还证明了有限维情况下的特征定理。这些结果提供了对所谓的动态采样问题的理论解决方案，其中通过算子$A$的迭代在时间上演变的信号$f$在不同时间被空间子采样，并且人们试图从这些空间-时间采样重构信号$f$。
英文摘要：
We consider bounded operators $A$ acting iteratively on a finite set of vectors $\{f_i : i\in I\}$ in a Hilbert space $\mathcal H$ and address the problem of providing necessary and sufficient conditions for the collection of iterates $\{A^nf_i : i\in I,\,n=0,1,2,\ldots\}$ to form a frame for the space $\mathcal H$. For normal operators $A$ we completely solve the problem by proving a characterization theorem. Our proof incorporates techniques from different areas of mathematics, such as operator theory, spectral theory, harmonic analysis, and complex analysis in the unit disk. In the second part of the paper we drop the strong condition on $A$ to be normal. Despite this quite general setting, we are able to prove a characterization which allows to infer many strong necessary conditions on the operator $A$. For example, $A$ needs to be similar to a contraction of a very special kind. We also prove a characterization theorem for the finite-dimensional case. --- These results provide a theoretical solution to the so-called Dynamical Sampling problem where a signal $f$ that is evolving in time through iterates of an operator $A$ is spatially sub-sampled at various times and one seeks to reconstruct the signal $f$ from these spatial-temporal samples.
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