This paper is concerned with the lower and upper redundancy of infinite frames in a separable Hilbert space. Using unit norm linear operators, a new representation of upper redundancy is given. Also, we show that there is no frame such that its lower redundancy is less than one and its upper redundancy is one. Moreover, for given $(\mu ,\lambda )\in \mathcal{M}:=\left\{\right(\mu ,\lambda );1\le \lceil \mu \rceil \le \lfloor \lambda \rfloor \}\setminus \left\{\right(\mu ,1);0<\mu <1\}$, we construct a frame such that its lower redundancy is μ and its upper redundancy is λ.

中文翻译：

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具有一些给定冗余的无限框架的构建

本文研究的是可分离希尔伯特空间中无限框架的下冗余和上冗余。使用单位范数线性算子，给出了上冗余的新表示。此外，我们证明不存在下冗余度小于 1 而上冗余度为 1 的帧。此外，对于给定$(\mu ,\lambda )\in \mathcal{米}:=\left\{\right(\mu ,\lambda );\mathrm{1个}\le \lceil \mu \rceil \le \lfloor \lambda \rfloor \}\setminus \left\{\right(\mu ,\mathrm{1个});0<\mu <\mathrm{1个}\}$，我们构造一个框架，使其下冗余度为μ，其上冗余度为λ。