A Survey on Constructing Parseval Fusion Frames via Scaling Weights

TL;DR

This paper surveys methods for constructing Parseval fusion frames by adjusting weights, extending scalability concepts, and providing conditions for various fusion frame types, with examples confirming theoretical results.

Contribution

It introduces the extension of scalability notions to fusion frames and characterizes conditions for creating Parseval fusion frames through weight adjustments.

Findings

Characterization of scalable fusion Riesz bases

Conditions for the scalability of k-excess fusion frames

Examples illustrating the theoretical results

Abstract

The construction of Parseval fusion frames is highly desirable in a wide range of signal processing applications. In this paper, we study the problem of modifying the weights of a fusion frame in order to generate a Parseval fusion frame. To this end, we extend the notion of the scalability to the fusion frame setting. We then proceed to characterize scalable fusion Riesz bases and $1$-excess fusion frames. Furthermore, we provide the necessary and sufficient conditions for the scalability of certain $k$-excess fusion frames, $k\geq 2$. Finally, we present several pertinent examples to confirm the obtained results.