Sampling-related frames in finite U-invariant subspaces

Garcı́a, Antonio G.; Muñoz-Bouzo, María J.

doi:10.1016/j.acha.2014.09.008

Cited by 14 publications

(29 citation statements)

References 17 publications

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“…[2,4,10,11,17,20,21], and sampling in U -invariant subspaces Refs. [8,12,13,19]. Besides, as it was showed in Section 4.2, the present approach opens new sampling settings: for instance, those related with crystallographic groups involving examples of practical interest.…”

Section: Some Final Commentsmentioning

confidence: 72%

“…[2]. Besides, formula (12) is the expected reconstruction formula in the U -invariant subspace V Φ . The differences can be found in conditions (a)-(c)-(d) used here since these conditions are directly related to the filtering process defining the samples (8).…”

Section: Some Final Commentsmentioning

confidence: 99%

“…There exists a frame S g,m g∈G; m=1,2,...,M for V Φ such that for each f ∈ V Φ the expansionf = M m=1 g∈G L m f (g) S g,m in H holds.In this case, the reconstruction elements {S m } m=1,2,...,M in V Φ in formula(12) are necessarily obtained from the columns b 1 , b 2 , . .…”

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confidence: 99%

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Convolution Systems on Discrete Abelian Groups as a Unifying Strategy in Sampling Theory

2020

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A regular sampling theory in a multiply generated unitary invariant subspace of a separable Hilbert space H is proposed. This subspace is associated to a unitary representation of a countable discrete abelian group G on H. The samples are defined by means of a filtering process which generalizes the usual sampling settings. The multiply generated setting allows to consider some examples where the group G is non-abelian as, for instance, crystallographic groups. Finally, it is worth to mention that classical average or pointwise sampling in shift-invariant subspaces are particular examples included in the followed approach.

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Section: Some Final Commentsmentioning

confidence: 72%

Section: Some Final Commentsmentioning

confidence: 99%

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Convolution Systems on Discrete Abelian Groups as a Unifying Strategy in Sampling Theory

2020

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“…See, among others, Refs. [6,7,8,9,10,11,12,21]. For non Abelian groups a classical Fourier transform is not available and consequently other techniques should be considered as in [1]; for the finite case, see, for instance, Part II of Ref.…”

Section: Statement Of the Problemmentioning

confidence: 99%

Knit Product of Finite Groups and Sampling

2019

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A finite sampling theory associated with a unitary representation of a finite non Abelian group G on a Hilbert space is stablished. The non Abelian group G is a knit product N ⊲⊳ H of two finite subgroups N and H. Sampling formulas where the samples are indexed by either N or H are obtained. Using suitable expressions for the involved samples, the problem is reduced to obtain dual frames in the Hilbert space ℓ 2 (G) having a unitary invariance property; this is done by using matrix analysis techniques. An example involving dihedral groups illustrates the obtained sampling results.

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“…Other similar works are about the generalized sampling problems [4,17,18,19,25] and the distributed sampling problems [11,13,14,20,29]. For example, in [17], the authors work in a U -invariant space, and study linear systems {L j : j = 1, · · · , s} such that one can recover any f in the U -invariant space by uniformly downsampling the functions {(L j f ) : j = 1, · · · , s}, i.e.…”

Section: Problem 12mentioning

confidence: 99%

Universal Spatiotemporal Sampling Sets for Discrete Spatially Invariant Evolutionary Systems

Tang

2017

IEEE Trans. Inform. Theory

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Abstract. Let (I, +) be a finite abelian group and A be a circular convolution operator on ℓ 2 (I). The problem under consideration is how to construct minimal Ω ⊂ I and li such that Y = {ei, Aei, · · · , A l i ei : i ∈ Ω} is a frame for ℓ 2 (I), where {ei : i ∈ I} is the canonical basis of ℓ 2 (I). This problem is motivated by the spatiotemporal sampling problem in discrete spatially invariant evolution systems. We will show that the cardinality of Ω should be at least equal to the largest geometric multiplicity of eigenvalues of A, and we consider the universal spatiotemporal sampling sets (Ω, li) for convolution operators A with eigenvalues subject to the same largest geometric multiplicity. We will give an algebraic characterization for such sampling sets and show how this problem is linked with sparse signal processing theory and polynomial interpolation theory. IntroductionLet (I, +) be a finite abelian group, we denote by ℓ 2 (I) the space of square summable complex-valued functions defined on I with the inner product given byWe denote by {e i : i ∈ I} the canonical basis of ℓ 2 (I), where e i is the characteristic function of {i} ⊂ I.Definition 1.1. The operator A on ℓ 2 (I) is called a circular convolution operator if there exists a complex-valued function a defined on I such thatThe function a is said to be the convolution kernel of A.2010 Mathematics Subject Classification. Primary 94A20, 94A12, 42C15, 15A29.

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Sampling-related frames in finite U-invariant subspaces

Cited by 14 publications

References 17 publications

Convolution Systems on Discrete Abelian Groups as a Unifying Strategy in Sampling Theory

Convolution Systems on Discrete Abelian Groups as a Unifying Strategy in Sampling Theory

Knit Product of Finite Groups and Sampling

Universal Spatiotemporal Sampling Sets for Discrete Spatially Invariant Evolutionary Systems

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