使用Nagy型和Pollard-Hilding型框架扰动从动态样本中稳定恢复初始状态
Stable Initial-State Recovery from Dynamical Samples using Nagy-Type and Pollard-Hilding-Type Frame Perturbations
AI总结：
针对离散动力系统，研究在Nagy型和Pollard-Hilding型框架扰动下，从扰动动态样本中稳定恢复初始状态的充分条件。
摘要：
受Aldroubi及其合作者最近关于动态采样工作的启发，我们考虑形式为$f_n = A f_{n-1}= A^{n}f, \quad f_0 = f$的齐次离散动力系统，其中$A$是可分希尔伯特空间$\mathcal{H}$上的有界线性算子（称为演化算子），$f_0 \in \mathcal{H}$是未知初始状态。相关的动态样本由集合$\{\langle A^n f, g\rangle: g \in \mathcal{G, 0 \leq n < n < L(g)}\}$给出，其中$\mathcal{G} \subset H$是有限或可数采样集，$L$是函数$L: G \rightarrow \mathbb{N} \bigcup \{\infty\}$。我们在Nagy和Pollard-Hilding意义下分析扰动动态采样系统的稳定性。更准确地说，我们在Nagy型和Pollard-Hilding型框架扰动框架下，通过改变采样向量、演化算子或同时改变两者，建立了从扰动动态样本中稳定恢复初始状态的充分条件。
英文摘要：
Stimulated by Aldroubi and his collaborator's recent work on dynamical sampling, we consider a homogeneous discrete dynamical system of the form $f_n = A f_{n-1}= A^{n}f, \quad f_0 = f$, where A is a bounded linear operator on a separable Hilbert space H, which is known as the evolution operator, and $f_0 \in \mathcal{H}$ is the unknown initial state. The associated dynamical samples are given by the collection $\{\langle A^n f, g\rangle: g \in \mathcal{G, 0 \leq n < n < L(g)}\}$, where $\mathcal{G} \subset H$, is a finite or countable sampling set and $L$ is a function $L: G \rightarrow \mathbb{N} \bigcup \{\infty\}$. We analyze the stability of perturbed dynamical sampling systems in the sense of Nagy and Pollard-Hilding. More precisely, we establish sufficient conditions for the stable recovery of an initial state from perturbed dynamical samples obtained by changing the sampling vector, the evolution operator, ro simultaneously both, within the framework of Nagy-type and Pollard-Hilding-type perturbation of frames.
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