Properties of finite dual fusion frames

TL;DR

This paper introduces and explores a new concept of dual fusion frames in finite-dimensional Hilbert spaces, demonstrating its advantages, conditions for uniqueness, and applications in optimal reconstruction under erasures.

Contribution

It extends the notion of dual fusion frames, provides conditions for their uniqueness, and connects them with existing dual systems, enhancing the theoretical framework for fusion frame analysis.

Findings

Existence of multiple duals for overcomplete fusion frames.

Conditions ensuring the uniqueness of dual fusion frames.

Explicit construction of optimal duals for erasure scenarios.

Abstract

A new notion of dual fusion frame has been recently introduced by the authors. In this article that notion is further motivated and it is shown that it is suitable to deal with questions posed in a finite-dimensional real or complex Hilbert space, reinforcing the idea that this concept of duality solves the question about an appropriate definition of dual fusion frames. It is shown that for overcomplete fusion frames there always exist duals different from the canonical one. Conditions that assure the uniqueness of duals are given. The relation of dual fusion frame systems with dual frames and dual projective reconstruction systems is established. Optimal dual fusion frames for the reconstruction in case of erasures of subspaces, and optimal dual fusion frame systems for the reconstruction in case of erasures of local frame vectors are determined. Examples that illustrate the obtained results are exhibited.