Fusion Frames and Theoretical Applications: for the Fusion Frame Web Page

The deepest and most difficult question in Fusion Frame Theory is the construction of fusion frames with added properties for specific applications. In frame theory we have a powerful tool introduced by Benedetto and Fickus [1] called frame potentials (See [5] for a deep analysis of frame potentials). Frame potentials are a valuable tool for showing the existence of frames with certain specified properties but have the drawback that they do not show how to construct the frames. Recently, Casazza and Fickus [4] have developed a Fusion Frame Potential theory. This theory is much more complicated than regular frame potentials and therefore is not as exact at this time. Part of the problem stems from the fact that certain simple examples of fusion frames just do not exist (See [12]). The fusion frame potential theory thus works best if the size of the fusion frame (I.e. The sums of the dimensions of the subspaces in the fusion frame) is quite large comparted to the dimension of the space. This topic needs more development at this time. The results of Casazza and Kutyniok [7] (See also [8]) show: A family of subspaces {Wi}i∈I is a fusion frame for H with fusion frame bounds C, D if and only if for every choice of vectors {fij} j=1 which is a frame for Wi with frame bounds A, B for all i ∈ I, the family {fij} j=1, i∈I is a frame for H with frame bounds AC, BD. So fusion frames come from dividing a frame into subsets which are ”good” local frames for the subspaces Wi. For this process to work, we need the local frames to have (uniformly) good lower frame bounds, since these bounds control the computational complexity of reconstruction. So a natural way to find fusion frames is to take a frame and divide it into subsets which are good frame sequences. This seems like a reasonable approach to fusion frame constructions, but it has been recently shown that this approach is extremely difficult in general. In particular, it is now known that the problem of dividing an arbitrary (even equal norm) Parseval frame into a finite number of subsets each of which has good lower frame bounds is equivalent to one of the deepest and most intractable problems in mathematics: The 1959 Kadison-Singer Problem in C∗-Algebras [6, 11, 13]. Trying to simplify the problem does not help. If we consider such a simple case as taking an equal norm Parseval frame {fi} i=1 for HN , we already are in trouble.