A note on finite dual frame pairs

Christensen, Ole; Powell, Alexander M.; Xiao, Xifeng

doi:10.1090/s0002-9939-2012-11256-0

Cited by 10 publications

(4 citation statements)

References 16 publications

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“…Remark 3. If we assume N > d, the statements in the previous corollary are also equivalent to say that N m=1 α m = d. This is a consequence of Proposition 2 and of Corollary 3.7 in [8].…”

Section: Critical Pairs Of Sequences Of the Mixed Frame Potentialmentioning

confidence: 80%

Critical Pairs of Sequences of a Mixed Frame Potential

Carrizo

Heineken

2014

Numerical Functional Analysis and Optimization

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The classical frame potential in a finite dimensional Hilbert space has been introduced by Benedetto and Fickus, who showed that all finite unit-norm tight frames can be characterized as the minimizers of this energy functional. This was the start point of a series of new results in frame theory, related to finding tight frames with determined length. The frame potential has been studied in the traditional setting as well as in the finite-dimensional fusion frame context. In this work we introduce the concept of mixed frame potential, which generalizes the notion of the Benedetto-Fickus frame potential. We study properties of this new potential, and give the structure of its critical pairs of sequences on a suitable restricted domain. For a given sequence {αm} m=1,...,N in K, where K is R or C, we obtain necessary and sufficient conditions in order to have a dual pair of frames {fm} m=1,...,N , {gm} m=1,...,N such that fm, gm = αm for all m = 1, ..., N.

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“…Remark 3. If we assume N > d, the statements in the previous corollary are also equivalent to say that N m=1 α m = d. This is a consequence of Proposition 2 and of Corollary 3.7 in [8].…”

Section: Critical Pairs Of Sequences Of the Mixed Frame Potentialmentioning

confidence: 80%

Critical Pairs of Sequences of a Mixed Frame Potential

Carrizo

Heineken

2014

Numerical Functional Analysis and Optimization

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“…In this section we derive generalizations of the Welch bounds for tight frames to dual pairs of frames. We will repeatedly need the following result, which is Theorem 3.1 in [3].…”

Section: Equiangular Dual Framesmentioning

confidence: 99%

Equiangular frames and generalizations of the Welch bound to dual pairs of frames

Christensen

Datta

Kim

2019

Linear and Multilinear Algebra

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“…Characterizing dual frames and general frame decompositions is an important problem in pure and applied fields. In [17][18][19][20][21][22], the authors study the dual frames and frame-like decompositions in Hilbert spaces. In particular, Li derived a general parametric and algebraic formula for all duals of a frame in [17] and introduced the pseudoframe decompositions in [18].…”

Section: Journal Of Function Spacesmentioning

confidence: 99%

Decompositions ofg-Frames and Duals and Pseudoduals ofg-Frames in Hilbert Spaces

Guo

2015

Journal of Function Spaces

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Firstly, we study the representation ofg-frames in terms of linear combinations of simpler ones such asg-orthonormal bases,g-Riesz bases, and normalized tightg-frames. Then, we study the dual and pseudodual ofg-frames, which are critical components in reconstructions. In particular, we characterize the dualg-frames in a constructive way; that is, the formulae for dualg-frames are given. We also give someg-frame like representations for pseudodualg-frame pairs. The operator parameterizations ofg-frames and decompositions of bounded operators are the key tools to prove our main results.

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A note on finite dual frame pairs

Cited by 10 publications

References 16 publications

Critical Pairs of Sequences of a Mixed Frame Potential

Critical Pairs of Sequences of a Mixed Frame Potential

Equiangular frames and generalizations of the Welch bound to dual pairs of frames

Decompositions ofg-Frames and Duals and Pseudoduals ofg-Frames in Hilbert Spaces

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