Tight p -fusion frames

Abstract

Fusion frames enable signal decompositions into weighted linear subspace components. For positive integers p, we introduce p-fusion frames, a sharpening of the notion of fusion frames. Tight p-fusion frames are closely related to the classical notions of designs and cubature formulas in Grassmann spaces and are analyzed with methods from harmonic analysis in the Grassmannians. We define the p-fusion frame potential, derive bounds for its value, and discuss the connections to tight p-fusion frames.

On symmetric decompositions of positive operators

“…Since the matrices R j have unit Schatten 2-norm and are also traceless, they have eigenvalues ±1/ √ 2, so the optimal constants x and a from Proposition 3.8 read, respectively, x = √ 2/4 and a = 1/4, which is indeed the upper bound (18). We show now (1) =⇒ (2). Les us consider the inequality (19) which leads to the upper bound (18).…”

“…The idea, originating in [9, Corollary 5.2], is that among N subspaces of fixed dimension in C d , there must be at least a pair with "small" principal angles. This result has been generalized to subspaces with weights in [2,Theorem 3.4], and then to arbitrary positive semidefinite operators with fixed trace in [4,Proposition 4.1]. In the result below, we slightly generalize this last result, by removing the fixed trace condition.…”

“…In the following we give some extensions and generalizations of a result from [21]. We use an idea from [2], which requires to introduce scalar weights v i .…”

Welch bounds for cross correlation of subspaces and generalizations

“…The main objective here is to put on common ground lower bounds on the maximal cross correlation among vectors indexed by a discrete or continuous set, subspaces, and random vectors in a (discrete) set with the underlying space being some finite dimensional Hilbert space. The author has recently become aware of work on p-fusion frames [10] where such lower bounds also arise. Even though the fusion frame potential is used here to derive the required lower bounds, p-fusion frames are not studied in this work.…”

Points on manifolds with asymptotically optimal covering radius

“…Our quasi-Monte Carlo integration points are cubatures (in fact designs) in Grassmannians that have been studied in [2,3,4,5] from a theoretical point of view, see [18] for the construction through numerical minimization. For related results on cubatures in more classical settings, see [17,23,24,26,28,31,32] and, for further related results, we refer to [10,13,14,19,20,22,27] and references therein.…”