Fusion frames and distributed processing

Casazza, Peter G.; Kutyniok, Gitta; Li, Shidong

doi:10.1016/j.acha.2007.10.001

Cited by 286 publications

(229 citation statements)

References 30 publications

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“…We wish to remark that the orthogonal family of finite dimensional subspaces forms an orthonormal basis of subspaces and in this sense is a special case of a Parseval fusion frame [9,10].…”

Section: Definitions and Basic Resultsmentioning

confidence: 99%

“…Our idea of an ideal sequence is a sequence for which there exists a partition of its elements into finite sets such that the spans of the elements of those sets are mutually orthogonal; thus, properties of the sequence are completely determined by properties of its local components. This definition is inspired by a more general notion called fusion frames [9,10], which were designed to model distributed processing applications. This paper is organized as follows.…”

Section: Theorem 14 Every Unit Norm Bessel Sequence Which Is Finitementioning

confidence: 99%

See 1 more Smart Citation

A decomposition theorem for frames and the Feichtinger Conjecture

Casazza¹,

Kutyniok²,

Speegle³

et al. 2008

Proc. Amer. Math. Soc.

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Abstract. In this paper we study the Feichtinger Conjecture in frame theory, which was recently shown to be equivalent to the 1959 Kadison-Singer Problem in C * -Algebras. We will show that every bounded Bessel sequence can be decomposed into two subsets each of which is an arbitrarily small perturbation of a sequence with a finite orthogonal decomposition. This construction is then used to answer two open problems concerning the Feichtinger Conjecture: 1. The Feichtinger Conjecture is equivalent to the conjecture that every unit norm Bessel sequence is a finite union of frame sequences. 2. Every unit norm Bessel sequence is a finite union of sets each of which is ω-independent for 2 -sequences.

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“…We wish to remark that the orthogonal family of finite dimensional subspaces forms an orthonormal basis of subspaces and in this sense is a special case of a Parseval fusion frame [9,10].…”

Section: Definitions and Basic Resultsmentioning

confidence: 99%

Section: Theorem 14 Every Unit Norm Bessel Sequence Which Is Finitementioning

confidence: 99%

A decomposition theorem for frames and the Feichtinger Conjecture

Casazza¹,

Kutyniok²,

Speegle³

et al. 2008

Proc. Amer. Math. Soc.

Self Cite

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show abstract

“…We developed the theory of operator-valued frames to provide a framework for such problems and we tested this model by solving a problem concerning norm path-connectedness. It has been brought to our attention that a few other recent papers in the literature overlap to some extent with our approach, notably works of Casazza, Kutyniok and Li [6] on "fusion frames", and also recent work of Bodmann [2] on quantum computing and work of W. Sun [25] on G-frames. These do not deal however with the path-connectedness that we address.…”

Section: Introductionmentioning

confidence: 99%

Operator-valued frames

Kaftal

Larson

Zhang

2009

Trans. Amer. Math. Soc.

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Abstract. We develop a natural generalization of vector-valued frame theory, which we term operator-valued frame theory, using operator-algebraic methods. This extends work of the second author and D. Han which can be viewed as the multiplicity one case and extends to higher multiplicity their dilation approach. We prove several results for operator-valued frames concerning duality, disjointedness, complementarity, and composition of operator-valued frames and the relationship between the two types of similarity (left and right) of such frames. A key technical tool is the parametrization of Parseval operatorvalued frames in terms of a class of partial isometries in the Hilbert space of the analysis operator. We apply these notions to an analysis of multiframe generators for the action of a discrete group G on a Hilbert space. One of the main results of the Han-Larson work was the parametrization of the Parseval frame generators in terms of the unitary operators in the von Neumann algebra generated by the group representation, and the resulting norm pathconnectedness of the set of frame generators due to the connectedness of the group of unitary operators of an arbitrary von Neumann algebra. In this paper we generalize this multiplicity one result to operator-valued frames. However, both the parametrization and the proof of norm path-connectedness turn out to be necessarily more complicated, and this is at least in part the rationale for this paper. Our parametrization involves a class of partial isometries of a different von Neumann algebra. These partial isometries are not path-connected in the norm topology, but only in the strong operator topology. We prove that the set of operator frame generators is norm pathwise-connected precisely when the von Neumann algebra generated by the right representation of the group has no minimal projections. As in the multiplicity one theory there are analogous results for general (non-Parseval) frames.

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“…The entire family (P i ) is to be treated as a fusion frame [5,6]. Fusion frames are defined in Definition 4.12 below.…”

Section: General Backgroundmentioning

confidence: 99%