连续和离散动态采样
Continuous and discrete dynamical sampling
摘要：
本文研究复Hilbert空间$\mathcal{H}$中无穷时间连续动态采样问题。本文给出了关于有界线性算子$A\in\mathcal{B}(\mathcal{H})$和向量集$\mathcal{G}\subset \mathcal{H}$的充要条件，从而得到了$\{e^{tA}g\}_{g\in\mathcal{G}, t\in[0,\infty)}$是$\mathcal{H}$的半连续框架.我们研究是否有可能离散的时间变量$t$和仍然有一个框架$\mathcal{H}$。我们还涉及到连续迭代$e^{tA}$在一个集$\mathcal{G}$ $\mathcal{G}^\prime$上的离散迭代$(A^\prime)^n$ $A^\prime$和一个适当的操作集$\mathcal{G}^\prime\subset \mathcal{H}$。
英文摘要：
In this paper we study the continuous dynamical sampling problem at infinite time in a complex Hilbert space $\mathcal{H}$. We find necessary and sufficient conditions on a bounded linear operator $A\in\mathcal{B}(\mathcal{H})$ and a set of vectors $\mathcal{G}\subset \mathcal{H}$, in order to obtain that $\{e^{tA}g\}_{g\in\mathcal{G}, t\in[0,\infty)}$ is a semi-continuous frame for $\mathcal{H}$. We study if it is possible to discretize the time variable $t$ and still have a frame for $\mathcal{H}$. We also relate the continuous iteration $e^{tA}$ on a set $\mathcal{G}$ to the discrete iteration $(A^\prime)^n$ on $\mathcal{G}^\prime$ for an adequate operator $A^\prime$ and set $\mathcal{G}^\prime\subset \mathcal{H}$.
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