The Studies of (Dual) Fusion Frames on Hilbert Space and Generalization of the Index Set

In this paper, we begin with the classical concept of tight frames in Hilbert spaces. First, we introduce the orthogonal projection P between H and θ(H) (the range of the frame transform θ associated with a traditional tight frame) and investigate the relationship between P and θ. We then explore fusion frames and extend the index set to an infinite set through a concrete example. Second, we examine the role of orthogonal projections in fusion frames with particular emphasis on robustness and redundancy illustrated by examples. Finally, we study dual fusion frames and establish several important results, especially concerning the relationship between the frame operators of two types of dual fusion frames.