Weighted Fusion Frame Construction via Spectral Tetris

TL;DR

This paper extends the spectral tetris method to construct arbitrarily weighted fusion frames in Hilbert spaces, providing necessary and sufficient conditions for their existence, thus filling a gap in fusion frame construction techniques.

Contribution

It introduces a novel spectral tetris-based approach for creating weighted fusion frames with prescribed weights and spectra, including the first known conditions for such constructions.

Findings

Successfully constructed weighted fusion frames with prescribed weights.

Derived necessary and sufficient conditions for spectral tetris-based fusion frame construction.

Extended the applicability of spectral tetris to more general fusion frame configurations.

Abstract

Fusion frames consist of a sequence of subspaces from a Hilbert space and corresponding positive weights so that the sum of weighted orthogonal projections onto these subspaces is an invertible operator on the space. Given a spectrum for a desired fusion frame operator and dimensions for subspaces, one existing method for creating unit-weight fusion frames with these properties is the flexible and elementary procedure known as spectral tetris. Despite the extensive literature on fusion frames, until now there has been no construction of fusion frames with prescribed weights. In this paper we use spectral tetris to construct more general, arbitrarily weighted fusion frames. Moreover, we provide necessary and sufficient conditions for when a desired fusion frame can be constructed via spectral tetris.