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 {W_{i}}_{iI} is a fusion frame for ℍ with fusion frame bounds C,D if and
only if for every choice of vectors {f_{ij}}_{j=1}^{Ki} which is a frame for W_{i} with
frame bounds A,B for all i I, the family {f_{ij}}_{j=1, iI}^{Ki} 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 W_{i}. 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 {f_{i}}_{i=1}^{2N} for ℍN, we already
are in trouble. The Rado-Horn Theorem (See [9]) tells us [11] that every
equal norm Parseval frame {f_{i}}_{i=1}^{2N} for ℍN can be divided into
two spanning sets. However, it was recently shown [3] that there does not
exist a constant 0 < c < 1 so that for every equal norm Parseval frame
{f_{i}}_{i=1}^{2N} for ℍN we can divide it into two subsets each of which is a frame
for ℍN with lower frame bound c. It can be shown that dividing a unit norm
frame into Riesz basic sequences (with uniform Riesz basis bounds) will
produce a food fusion frame [2]. But being able to do this for arbitrary unit
norm frames is also equivalent to KS and is known as the Feichtinger
Conjecture in frame theory [10, 6, 11] which asserts: Every bounded frame
can be divided into a finite number of Riesz basic sequences. It is also
known [3] that KS is equivalent to being able to divide every equal norm
Parseval frame {f_{m}}_{m=1}^{2N} for ℍN into a finite number of Riesz basic
sequences.

So in general, it is a very deep and difficult problem to construct fusion frames from arbitrary frames by dividing the frame into subsets.

[1] J. Benedetto and M. Fickus, Finite normalized tight frames, Advances in Computational Mathematics - special issue on Frames, 18 (2003) 357-385.

[2] P.G. Casazza, (Unpublished research).

[3] P.G. Casazza, D. Edidin, D. Kalra and V. Paulsen, Projections and the Kadison-Singer Problem, Operators and Matrices 1 No. 3 (2007) 391-408.

[4] P.G. Casazza and M. Fickus, Fusion frame potential, Preprint.

[5] P.G. Casazza, M. Fickus, J. Kovačević, M. Leon and J.C. Tremain, A physical interpretation of tight frames, Applied Harmonic Analysis and Applications (2006) 51-78.

[6] P.G. Casazza, M. Fickus, J.C. Tremain and E. Weber, The Kadison-Singer Problem in mathematics and engineering: a detailed account, Contemp. Math. 414 (2006) 299-355.

[7] P.G. Casazza and G. Kutyniok, Frames of subspaces, Contemp. Math 345 (2004) 87-113.

[8] P.G. Casazza, G. Kutyniok and S. Li, Fusion frames and distributed processing, Appl. and Comp. Harmonic Anal 25 (2008) 114-132.

[9] P.G. Casazza, G. Kutyniok and D. Speegle, A redundant version of the Rado-Horn theorem, Linear Algebra and Applications 418 (2006) 1-10.

[10] P.G. Casazza, G. Kutyniok and D. Speegle, A decomposition theorem for frames and the Feichtinger Conjecture, Proc. AMS 136 (2008) 2043-2053.

[11] P.G. Casazza and J.C. Tremain, The Kadison-Singer Problem in mathematics and engineering, Proc. National Academy of Sciences 103 No. 7 (2006) 2032-2039.

[12] R. Calderbank, P.G. Casazza, G. Kutyniok and A. Pezeshki, Constructing Parseval fusion frames, Preprint.

[13] R. Kadison and I. Singer, Extensions of pure states, American Jour. Math. 81 (1959), 383–400.

^{*}The author was supported by NSF DMS 0704216.

Department of Mathematics, University of Missouri, Columbia, MO 65211-4100

E-mail address: pete@math.missouri.edu