Spectral Tetris Fusion Frame Constructions

TL;DR

This paper extends the spectral tetris method to construct unit norm frames and fusion frames with arbitrary spectra, providing conditions for their existence and construction.

Contribution

It generalizes spectral tetris to handle any spectrum for the frame operator and establishes necessary conditions for fusion frame construction.

Findings

Extended spectral tetris to arbitrary spectra

Provided sufficient and necessary conditions for fusion frame construction

Constructed fusion frames with prescribed spectra and dimensions

Abstract

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.