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

In this paper, we introduce orthogonal projection between Hilbert space and (the range of the frame transform of traditional tight frame) firstly, and study the relationship between and, then we explore the fusion frame and extend the index set to infinite set through an example. Secondly, we study the dual fusion frames starting with an example which illustrate how the traditional dual frames recover the original signal in the case of data loss. Finally, we obtain some important conclusions mainly including the necessary and sufficient condition, the stability of dual fusion frames and the relationship between canonical dual and alternate dual fusion frames especially the relationship between their respective frame operators.