Convolutional frames and the frame potential

Fickus, Matthew; Johnson, Brody Dylan; Kornelson, Keri; Okoudjou, Kasso A.

doi:10.1016/j.acha.2005.02.005

Cited by 28 publications

(43 citation statements)

References 6 publications

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“…As such, in the third section, we study the minimization of (2) in greater detail, strengthening and simplifying several of the main results of [1,5,10], as summarized in Theorem 3. In the final section, we then use these results to prove Theorem 4, which places a strong necessary structure on any local minimizer of the fusion frame potential.…”

Section: Introductionmentioning

confidence: 90%

“…Though Proposition 1 characterizes tight fusion frames, should they exist, as the global minimizers of FFP : P({L k } K k=1 ) → R, our approach, paralleling that of [1,5,10], is to study the local minimizers of this functional. Here, we take the distance between any…”

Section: Proposition 1 For Any Sequence Of Positive Integersmentioning

confidence: 99%

“…The frame potential was introduced in [1], with its domain of optimization being later generalized in [5]. It has been used to characterize tight filter bank frames [10,11]. Recently, generalized frame potentials have been a subject of interest [2,14].…”

Section: Introductionmentioning

confidence: 99%

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Minimizing Fusion Frame Potential

Casazza

Fickus

2008

Acta Appl Math

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Fusion frames are an emerging topic of frame theory, with applications to encoding and distributed sensing. However, little is known about the existence of tight fusion frames. In traditional frame theory, one method for showing that unit norm tight frames exist is to characterize them as the minimizers of an energy functional, known as the frame potential. We generalize the frame potential to the fusion frame setting. In particular, we introduce the fusion frame potential, and show how its minimization is equivalent to the minimization of the traditional frame potential over a particular domain. We then study this minimization problem in detail. Specifically, we show that if the desired number of fusion frame subspaces is large, and if the desired dimension of these subspaces is small compared to the dimension of the underlying space, then a tight fusion frame of those dimensions will necessarily exist, being a minimizer of the fusion frame potential.

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Section: Introductionmentioning

confidence: 90%

Section: Proposition 1 For Any Sequence Of Positive Integersmentioning

confidence: 99%

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Minimizing Fusion Frame Potential

Casazza

Fickus

2008

Acta Appl Math

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“…A special case of this situation is given on convolutional frames studied in [9] (see also [11] where the more general case of a convolutional frame based on a finite -not necessarily cyclic -abelian group is considered). In particular, the previous theorem can be seen as a partial generalization to [9,Thm.…”

Section: Remark 415mentioning

confidence: 99%

Minimization of convex functionals over frame operators

Massey

Ruiz

2008

Adv Comput Math

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We present results about minimization of convex functionals defined over a finite set of vectors in a finite-dimensional Hilbert space, that extend several known results for the Benedetto-Fickus frame potential. Our approach depends on majorization techniques. We also consider some perturbation problems, where a positive perturbation of the frame operator of a set of vectors is realized as the frame operator of a set of vectors which is close to the original one.

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“…For example, if μ is the middlethird Cantor measure, the question of whether of not L 2 (μ) admits Fourier frames has implications for the Kadison-Singer conjecture. For some of these results regarding frames, the reader is referred to [15,17,21,23,32].…”

Section: Overview Of Prior Literaturementioning

confidence: 99%

Families of Spectral Sets for Bernoulli Convolutions

Jørgensen

Kornelson

Shuman

2010

J Fourier Anal Appl

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We study the harmonic analysis of Bernoulli measures μ λ , a one-parameter family of compactly supported Borel probability measures on the real line. The parameter λ is a fixed number in the open interval (0, 1). The measures μ λ may be understood in any one of the following three equivalent ways: as infinite convolution measures of a two-point probability distribution; as the distribution of a random power series; or as an iterated function system (IFS) equilibrium measure determined by the two transformations λ(x ± 1). For a given λ, we consider the harmonic analysis in the sense of Fourier series in the Hilbert space L 2 (μ λ ). For L 2 (μ λ ) to have infinite families of orthogonal complex exponential functions e 2πis(·) , it is known that λ must be a rational number of the form m 2n , where m is odd. We show that L 2 (μ 1 2n ) has a variety of Fourier bases; i.e. orthonormal bases of exponential functions. For some other rational values of λ, we exhibit maximal Fourier families that are not orthonormal bases.

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Convolutional frames and the frame potential

Cited by 28 publications

References 6 publications

Minimizing Fusion Frame Potential

Minimizing Fusion Frame Potential

Minimization of convex functionals over frame operators

Families of Spectral Sets for Bernoulli Convolutions

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