Generalized Sampling in $${L}^{2}({\mathbb{R}}^{d})$$ Shift-Invariant Subspaces with Multiple Stable Generators

Fernández-Morales, H. R.; Garcı́a, Antonio G.; Pérez-Villalón, Gerardo

doi:10.1007/978-1-4614-4145-8_3

Cited by 6 publications

(5 citation statements)

References 43 publications

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“…such that the sequence of reconstruction functions { (⋅ − )} ∈Z; =1,2,..., is a frame for the shift-invariant space 2 . As a consequence, expansions (3) and (5) have the same nature. Recall that a sequence { } ∈Z is a frame for a separable Hilbert space H if there exist constants , > 0 (frame bounds), such that…”

Section: Introductionmentioning

confidence: 96%

See 1 more Smart Citation

On Some Sampling-Related Frames inU-Invariant Spaces

Fernández-Morales

Garcı́a

Hernández-Medina

et al. 2013

Abstract and Applied Analysis

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This paper is concerned with the characterization as frames of some sequences inU-invariant spaces of a separable Hilbert spaceℋwhereUdenotes an unitary operator defined onℋ; besides, the dual frames having the same form are also found. This general setting includes, in particular, shift-invariant or modulation-invariant subspaces inL2ℝ, where these frames are intimately related to the generalized sampling problem. We also deal with some related perturbation problems. In doing so, we need the unitary operatorUto belong to a continuous group of unitary operators.

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Section: Introductionmentioning

confidence: 96%

“…Sampling in shift-invariant spaces of 2 (R) has been profusely treated in the mathematical literature (see, for instance, [2][3][4][5][6][7][8][9][10][11][12][13]).…”

Section: Introductionmentioning

confidence: 99%

On Some Sampling-Related Frames inU-Invariant Spaces

Fernández-Morales

Garcı́a

Hernández-Medina

et al. 2013

Abstract and Applied Analysis

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“…Proof. The uniqueness of the expansion with respect to a Riesz basis gives the stated interpolation property (10).…”

Section: Holdsmentioning

confidence: 97%

“…Indeed, it englobes the most usual sampling settings such as sampling in shift-invariant subspaces of L 2 (R) (see, for instance, Refs. [1,2,5,10,14,15,19,28,29,30,31] and references therein), or sampling periodic extensions of finite signals (see Refs. [13,16]).…”

Section: Introductionmentioning

confidence: 99%

Modeling Sampling in Tensor Products of Unitary Invariant Subspaces

Garcı́a

Ibort

Muñoz-Bouzo

2016

Journal of Function Spaces

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The use of unitary invariant subspaces of a Hilbert space H is nowadays a recognized fact in the treatment of sampling problems. Indeed, shift-invariant subspaces of L 2 (R) and also periodic extensions of finite signals are remarkable examples where this occurs. As a consequence, the availability of an abstract unitary sampling theory becomes a useful tool to handle these problems. In this paper we derive a sampling theory for tensor products of unitary invariant subspaces. This allows to merge the cases of finitely/infinitely generated unitary invariant subspaces formerly studied in the mathematical literature; it also allows to introduce the several variables case. As the involved samples are identified as frame coefficients in suitable tensor product spaces, the relevant mathematical technique is that of frame theory, involving both, finite/infinite dimensional cases.

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“…In order to avoid most of the drawbacks associated with classical Shannon's sampling theory, sampling and reconstruction problems have been investigated in spline spaces, wavelet spaces, or in general shift-invariant spaces (see [16] and references therein for more details and results on sampling in shift-invariant spaces).…”

Section: A Sampling Formula In a Shift-invariant Spacementioning

confidence: 99%

The Kramer Sampling Theorem Revisited

2013

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The classical Kramer sampling theorem provides a method for obtaining orthogonal sampling formulas. Besides, it has been the cornerstone for a significant mathematical literature on the topic of sampling theorems associated with differential and difference problems. In this work we provide, in an unified way, new and old generalizations of this result corresponding to various different settings; all these generalizations are illustrated with examples. All the different situations along the paper share a basic approach: the functions to be sampled are obtaining by duality in a separable Hilbert space H through an H-valued kernel K defined on an appropriate domain.

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Generalized Sampling in $${L}^{2}({\mathbb{R}}^{d})$$ Shift-Invariant Subspaces with Multiple Stable Generators

Cited by 6 publications

References 43 publications

On Some Sampling-Related Frames inU-Invariant Spaces

On Some Sampling-Related Frames inU-Invariant Spaces

Modeling Sampling in Tensor Products of Unitary Invariant Subspaces

The Kramer Sampling Theorem Revisited

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