Spectral Tetris Fusion Frame Constructions
This work advances the theoretical foundations of fusion frame construction, benefiting researchers in frame theory and signal processing.
The paper extends the spectral tetris algorithm to construct unit norm frames with any given spectrum, and provides a sufficient condition for constructing fusion frames with prescribed spectrum and subspace dimensions, which is also necessary in the tight case with redundancy >2.
Spectral tetris is a fexible and elementary method to construct unit norm frames with a given frame operator, having all of its eigenvalues greater than or equal to two. One important application of spectral tetris is the construction of fusion frames. We first show how the assumption on the spectrum of the frame operator can be dropped and extend the spectral tetris algorithm to construct unit norm frames with any given spectrum of the frame operator. We then provide a suffcient condition for using this generalization of spectral tetris to construct fusion frames with prescribed spectrum for the fusion frame operator and with prescribed dimensions for the subspaces. This condition is shown to be necessary in the tight case of redundancy greater than two.