Non-orthogonal fusion frames and the sparsity of fusion frame operators

Fusion frames have become a major tool in the implementation of distributed systems. The effectiveness of fusion frame applications in distributed systems is reflected in the efficiency of the end fusion process. This in turn is reflected in the efficiency of the inversion of the fusion frame operator $S_{\cW}$, which in turn is heavily dependent on the sparsity of $S_{\cW}$. We will show that sparsity of the fusion frame operator naturally exists by introducing a notion of {\it non-orthogonal fusion frames}. We show that for a fusion frame $\{W_i,v_i\}_{i\in I}$, if $\text{dim}(W_i)=k_i$, then the matrix of the non-orthogonal fusion frame operator $\cSw$ has in its corresponding location at most a $k_i\times k_i$ block matrix. We provide necessary and sufficient conditions for which the new fusion frame operator $\cSw$ is diagonal and/or a multiple of an identity. A set of other critical questions are also addressed. A scheme of {\it multiple fusion frames} whose corresponding fusion frame operator becomes an diagonal operator is also examined.