Constructing fusion frames with desired parameters

Abstract

A fusion frame is a frame-like collection of subspaces in a Hilbert space. It generalizes the concept of a frame system for signal representation. In this paper, we study the existence and construction of fusion frames. We first introduce two general methods, namely the spatial complement and the Naimark complement, for constructing a new fusion frame from a given fusion frame. We then establish existence conditions for fusion frames with desired properties. In particular, we address the following question: Given M, N, m ∈ N and {λj}Mj =1, does there exist a fusion frame in RM with N subspaces of dimension m for which {λj} Mj =1are the eigenvalues of the associated fusion frame operator? We address this problem by providing an algorithm which computes such a fusion frame for almost any collection of parameters M, N, m ∈ N and {λj}Mj =1. Moreover, we show how this procedure can be applied, if subspaces are to be added to a given fusion frame to force it to become tight.© 2009 SPIE.