A sequence $\left \{g_k\right \}_{k=1}^{\infty }$ in a Hilbert space ${\cal H}$ has the expansion property if each $f\in \overline {\text {span}} \left \{g_k\right \}_{k=1}^{\infty }$ has a representation $f=\sum _{k=1}^{\infty } c_k g_k$ for some scalar coefficients $c_k.$ In this paper, we analyze the question whether there exist small norm-perturbations of $\left \{g_k\right \}_{k=1}^{\infty }$ which allow to represent all $f\in {\cal H};$ the answer turns out to be yes for frame sequences and Riesz sequences, but no for general basic sequences. The insight gained from the analysis is used to address a somewhat dual question, namely, whether it is possible to remove redundancy from a sequence with the expansion property via small norm-perturbations; we prove that the answer is yes for frames $\left \{g_k\right \}_{k=1}^{\infty }$ such that $g_k\to 0$ as $k\to \infty ,$ as well as for frames with finite excess. This particular question is motivated by recent progress in dynamical sampling.

希尔伯特空间 ${\cal H}$ 中的序列 $\left \{g_k\right \}_{k=1}^{\infty }$ 如果每个 $f\in \overline {\text {span}} \left \{g_k\right \}_{k=1}^{\infty }$ 有一个表示 $f=\sum _{k=1}^{\infty } c_k g_k$ 对于一些标量系数 $c_k.$ 在本文中，我们分析了是否存在 $\left \{g_k\right \}_{k=1}^{\infty }$ 的小范数扰动可以表示所有 $f \in {\cal H};$ 对于帧序列和 Riesz 序列，答案是肯定的，但对于一般基本序列来说不是。从分析中获得的见解用于解决一个有点双重的问题，即是否可以通过小的范数扰动从具有扩展属性的序列中去除冗余？我们证明答案是肯定的帧 $\left \{g_k\right \}_{k=1}^{\infty }$ 使得 $g_k\to 0$ 为 $k\to \infty ,$ 以及至于有限过剩的框架。这个特定问题的动机是动态采样的最新进展。