Excess of Fusion Frames: A Comprehensive Approach

TL;DR

This paper investigates the concept of excess in fusion frames, providing methods to compute it, establishing bounds, and exploring its relationship with $Q$-duals and orthogonal complements, supported by illustrative examples.

Contribution

It introduces explicit methods for computing excess in fusion frames and their $Q$-duals, and characterizes the relationship between excess and orthogonal complements.

Findings

Methods to compute excess of fusion frames and $Q$-duals are developed.

Upper bounds for the excess of $Q$-dual fusion frames are established.

The difference in excess between a fusion frame and its $Q$-dual can be arbitrarily large.

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 fusion frames is studied. 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, unlike ordinary frames, for every $n \in \Bbb{N}$ we can provide a fusion frame with its $Q$-dual whose the difference of their excess is $n$. Furthermore, the connection between the excess of fusion frames and their orthogonal complement is completely characterized. Finally, several examples are exhibited to confirm the obtained results.