Perturbations of frames

Chen, Dong Yang; Li, Lei; Zheng, Bentuo

doi:10.1007/s10114-014-2804-5

Cited by 12 publications

(13 citation statements)

References 19 publications

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“…Moreover, he showed that in the case when we lose the assumption q < m Φ , the sequence Ψ is not a frame for the whole space . Hence, in the next two results, inspired by Chen et al, 12 we formulate similar results as above for quadratically close sequences Φ and Ψ whenever Φ is a frame for  and we have m Φ ≤q. Particularly, Example 4.7 below shows that these investigations expand our results to the sequences which do not satisfy the Paley and Wiener perturbation condition.…”

Section: The Perturbation Effect On the Dualsmentioning

confidence: 60%

“…We first study some properties of (canonical) approximately dual frames which have not been touched so far. Second, we show that if we do a sufficiently small perturbation of a frame in the sense of Chen et al and Christensen, 12,13 then their specific approximately dual frames are close to each other. In contrast to the work, 7 we consider the problem in a much more general setting.…”

Section: Introductionmentioning

confidence: 61%

“…12, Example 2.5 Let for each n ∈ N, 𝛼 n = 2 n and k n = 4 n and let t 1 = 3 and t n = 2 for all n ≥ 2. Suppose also that Φ and Ψ are the following sequences:…”

mentioning

confidence: 99%

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The effect of perturbations of frames on their alternate and approximately dual frames

Javanshiri

Hajiabootorabi

Mardanbeigi

2021

Math Methods in App Sciences

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Approximately dual frames as a generalization of duality notion in Hilbert spaces have applications in Gabor systems, wavelets, coorbit theory, and sensor modeling. In recent years, the computing of the associated deviations of the canonical and alternate dual frames from the original ones has been considered by functional analysts. In this paper, we consider the quantitative measurement of the associated deviations of the alternate and approximately dual frames from the original ones. More precisely, it is proved that if the sequence Ψ is sufficiently close to the frame Φ, then for given approximately dual frame Φ ad of Φ, the approximately dual frame Ψ ad of Ψ can be found which is close to Φ ad , and particularly, we estimate their deviation in terms of the distance of the operators  1 ∶= T Φ U Φ ad and  2 ∶= T Ψ U Ψ ad and their approximation rates, where T X and U X denote the synthesis and analysis operators of the frame X, respectively. It is worth mentioning that some of our perturbation conditions are quite different from those used in the previous literatures on this topic.

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Section: The Perturbation Effect On the Dualsmentioning

confidence: 60%

Section: Introductionmentioning

confidence: 61%

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The effect of perturbations of frames on their alternate and approximately dual frames

Javanshiri

Hajiabootorabi

Mardanbeigi

2021

Math Methods in App Sciences

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“…By assumptions, we have W g j ≤ λ g j , T g j ≤ λ g j , j ∈ I. Therefore, [ [11], Theorem 2.14] leads to the desired result.…”

Section: Frame Representation Of the Form {Amentioning

confidence: 87%

Dynamical sampling: mixed frame operators, representations and perturbations

Rashidi,

Najati,

Osgooei

2019

Preprint

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Motivated by recent progress in operator representation of frames, we investigate the frames of the form {T n ϕ} n∈I for I = N, Z, and answer questions about representations, perturbations and frames induced by the action of powers of bounded linear operators. As a particular case, we discuss problems concerning representation of frames in terms of iterations of the mixed frame operators. As our another contribution, we consider frames of the form {a n T n ϕ} ∞ n=0 for some non-zero scalars {a n } ∞ n=0 , and we obtain some new results in dynamical sampling. Finally, we will present some auxiliary results related to the perturbation of sequences of the form {T n ϕ} ∞ n=0 .2010 Mathematics Subject Classification. 42C15, 47B40.

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“…Alternative norm-perturbation conditions are formulated in [6]; however, they need that we have access to information about a dual frame, which is not the case in the current paper. Note also that a number of classical results about norm-perturbation (typically for orthonormal sequences) are collected in [17].…”

Section: Introductionmentioning

confidence: 98%

Completion versus removal of redundancy by perturbation

Christensen¹,

Hasannasab²

2021

Preprint

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A sequence {g k } ∞ k=1 in a Hilbert space H has the expansion prop-In this paper we analyze the question whether there exist small norm-perturbations of {g k } ∞ k=1 which allow to represent all f ∈ H; the answer turns out to be yes for frame sequences and Riesz sequences, but no for general basic sequences. The insight gained from the analysis is used to address a somewhat dual question, namely, whether it is possible to remove redundancy from a sequence with the expansion property via small norm-perturbations; we prove that the answer is yes for frames {g k } ∞ k=1 such that g k → 0 as k → ∞, as well as for frames with finite excess. This particular question is motivated by recent progress in dynamical sampling.

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Perturbations of frames

Cited by 12 publications

References 19 publications

The effect of perturbations of frames on their alternate and approximately dual frames

The effect of perturbations of frames on their alternate and approximately dual frames

Dynamical sampling: mixed frame operators, representations and perturbations

Completion versus removal of redundancy by perturbation

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