Dynamical Sampling and Systems from Iterative Actions of Operators

Aldroubi, Akram; Petrosyan, Armenak

doi:10.1007/978-3-319-55550-8_2

Cited by 53 publications

(53 citation statements)

References 22 publications

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“…Therefore T can be extended to a bounded operator on H, as claimed. The proof shows that ||T || ≤ √ BA −1 ; the inequality ||T || ≥ 1 is proved in [4]. in Example 1.1.…”

Section: Boundedness Of the Operator Tmentioning

confidence: 84%

Dynamical sampling and frame representations with bounded operators

Christensen

Hasannasab

Rashidi

2018

Journal of Mathematical Analysis and Applications

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The purpose of this paper is to study frames for a Hilbert space H, having the form {T n ϕ} ∞ n=0 for some ϕ ∈ H and an operator T : H → H. We characterize the frames that have such a representation for a bounded operator T, and discuss the properties of this operator. In particular, we prove that the image chain of T has finite length N in the overcomplete case; furthermore {T n ϕ} ∞ n=0 has the very particular property that {T n ϕ} N −1 n=0 ∪{T n ϕ} ∞ n=N +ℓ is a frame for H for all ℓ ∈ N 0 . We also prove that frames of the form {T n ϕ} ∞ n=0 are sensitive to the ordering of the elements and to norm-perturbations of the generator ϕ and the operator T. On the other hand positive stability results are obtained by considering perturbations of the generator ϕ belonging to an invariant subspace on which T is a contraction.

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“…Therefore T can be extended to a bounded operator on H, as claimed. The proof shows that ||T || ≤ √ BA −1 ; the inequality ||T || ≥ 1 is proved in [4]. in Example 1.1.…”

Section: Boundedness Of the Operator Tmentioning

confidence: 84%

Dynamical sampling and frame representations with bounded operators

Christensen

Hasannasab

Rashidi

2018

Journal of Mathematical Analysis and Applications

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“…Hence, the spectrum of a strongly stable operator T is contained in the closed unit disk. It is, moreover, quite easily shown that neither T nor T * can have eigenvalues on the unit circle T. The first statement of the following lemma was proved in [5]. Lemma 3.1.…”

Section: Dynamical Sampling With General Bounded Operatorsmentioning

confidence: 97%

“…Concerning the infinite-dimensional situation, the most interesting result in [3] addresses the one-vector problem (i.e., |I| = 1). This result was further improved in [2] and [5]. Its final version reads as follows: If A is a normal operator, the system (A n f ) n∈N is a frame for H if and only if (a) A = inN λ j · , e j e j with an ONB (e j ) j∈N , (b) the sequence (λ j ) j∈N is uniformly separated in the unit disk (cf.…”

Section: Previous Work On the Topic And Our Contribution The Histormentioning

confidence: 98%

“…Very little is known on the Dynamical Sampling problem for general non-normal bounded operators A. It was only proved in [5] that for (A n f i ) n∈N, i∈I to be a frame for H it is necessary that A * be strongly stable, i.e., (A * ) n f → 0 as n → ∞ for each f ∈ H. Here, we complete this condition to a characterizing set of three conditions (cf. Theorem 3.2).…”

Section: Previous Work On the Topic And Our Contribution The Histormentioning

confidence: 99%

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Dynamical sampling on finite index sets

Cabrelli

Molter

Paternostro

et al. 2020

JAMA

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We consider bounded operators A acting iteratively on a finite set of vectors {fi : i ∈ I} in a Hilbert space H and address the problem of providing necessary and sufficient conditions for the collection of iterates {A n fi : i ∈ I, n = 0, 1, 2, ....} to form a frame for the space H. For normal operators A we completely solve the problem by proving a characterization theorem. Our proof incorporates techniques from different areas of mathematics, such as operator theory, spectral theory, harmonic analysis, and complex analysis in the unit disk. In the second part of the paper we drop the strong condition on A to be normal. Despite this quite general setting, we are able to prove a characterization which allows to infer many strong necessary conditions on the operator A. For example, A needs to be similar to a contraction of a very special kind. We also prove a characterization theorem for the finite-dimensional case.-These results provide a theoretical solution to the so-called Dynamical Sampling problem where a signal f that is evolving in time through iterates of an operator A is spatially sub-sampled at various times and one seeks to reconstruct the signal f from these spatial-temporal samples.2010 Mathematics Subject Classification. 94A20, 42C15, 30J99, 47A53.

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“…We will close the paper with a few operator-theoretical considerations, to which we have referred throughout the paper. 1) Instead of representing a frame on the form {T k f 0 } k∈Z , one could also consider representations on the form {T n f 0 } ∞ n=0 ; this indexing occur, e.g., in dynamical sampling [2,3]. The chosen indexing actually has a serious influence on the properties of the operator T. For example, there exist frames {f k } k∈Z = {T k f 0 } k∈Z where T is a unitary operator (Example 1.1); but if we reindex the frame as {f k } ∞ k=0 it can not be represented on the form {U n f 0 } ∞ n=0 for a unitary operator U, see [3].…”

Section: Appendix: Auxiliary Examplesmentioning

confidence: 99%

Operator Representations of Frames: Boundedness, Duality, and Stability

Christensen¹,

Hasannasab²

2017

Integr. Equ. Oper. Theory

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The purpose of the paper is to analyze frames {f k } k∈Z having the form {T k f 0 } k∈Z for some linear operator T : span{f k } k∈Z → span{f k } k∈Z . A key result characterizes boundedness of the operator T in terms of shift-invariance of a certain sequence space. One of the consequences is a characterization of the case where the representation {f k } k∈Z = {T k f 0 } k∈Z can be achieved for an operator T that has an extension to a bounded bijective operator T : H → H. In this case we also characterize all the dual frames that are representable in terms of iterations of an operator V ; in particular we prove that the only possible operator is V = ( T * ) −1 . Finally, we consider stability of the representation {T k f 0 } k∈Z ; rather surprisingly, it turns out that the possibility to represent a frame on this form is sensitive towards some of the classical perturbation conditions in frame theory. Various ways of avoiding this problem will be discussed. Throughout the paper the results will be connected with the operators and function systems appearing in applied harmonic analysis, as well as with general group representations.

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Dynamical Sampling and Systems from Iterative Actions of Operators

Cited by 53 publications

References 22 publications

Dynamical sampling and frame representations with bounded operators

Dynamical sampling and frame representations with bounded operators

Dynamical sampling on finite index sets

Operator Representations of Frames: Boundedness, Duality, and Stability

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