Excess of Fusion Frames: A Comprehensive Approach

Ehsan Ameli, Ali Akbar Arefijamaal, Fahimeh Arabyani Neyshaburi
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Abstract

Computing the excess as a method of measuring the redundancy of frames was recently introduced to address certain issues in frame theory. In this paper, the concept of excess for the fusion frame setting is studied. Initially, a local approach is presented to determine exactly which part of each subspace should be considered as redundancy. Then, several explicit methods are provided to compute the excess of fusion frames and their $Q$-duals. In particular, some upper bounds for the excess of $Q$-dual fusion frames are established. It turns out that each fusion frame and its $Q$-dual may not necessarily have the same excess. Along the way, unlike ordinary frames, it follows that for every $n \in \Bbb{N}$, we can provide a fusion frame together an its $Q$-dual such that the difference of their excess is $n$. Furthermore, the connection between the excess of fusion frames and their orthogonal complement fusion frames are completely characterized. Finally, several examples are exhibited to confirm the obtained results.
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融合框架过剩:综合方法
为了解决帧理论中的某些问题,最近引入了计算超差作为衡量帧冗余度的方法。本文研究了融合帧设置中的多余度概念。首先,本文提出了一种局部方法来确定每个子空间的哪一部分应被视为冗余。然后,提供了几种显式方法来计算融合帧及其 $Q$ 对偶的多余量。特别是,建立了一些 Q$ 双融合帧过量的上界。结果发现,每个融合帧及其 Q$二元不一定具有相同的超额。与此同时,与普通框架不同的是,对于每一个 $n \in/Bbb{N}$,我们都可以提供一个融合框架及其 $Q$-dual,使得它们的过量之差为 $n$。此外,我们还完整地描述了融合框架的过量与其正交互补融合框架之间的联系。最后,展示了几个例子来证实所获得的结果。
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