On the Stability of Multivariate Trigonometric Systems

Sun, Wei; Zhou, Xingwei

doi:10.1006/jmaa.1999.6386

Cited by 33 publications

(25 citation statements)

References 5 publications

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“…(ii) The relation between Hilbert frames and sampling of bandlimited functions is well known [14,48]. Sampling in shift-invariant spaces is more recent, and the relation between frames and sampling in shift-invariant spaces (with p = 2 and ν = 1) can be found in [5,30,75,77,102]. (iii) The relation between Hilbert frames and the weighted average sampling mentioned in Remark 3.1 can be found in [1].…”

Section: Framesmentioning

confidence: 99%

“…Nonuniform Sampling in Shift-Invariant Spaces. The problem of nonuniform sampling in general shift-invariant spaces is more recent [4,5,30,66,75,76,77,102,119]. The earliest results [31,77] concentrate on perturbation of regular sampling in shift-invariant spaces and are therefore similar in spirit to Kadec's result for bandlimited functions.…”

mentioning

confidence: 93%

“…The earliest results [31,77] concentrate on perturbation of regular sampling in shift-invariant spaces and are therefore similar in spirit to Kadec's result for bandlimited functions. For the L 2 case in dimension d = 1, and under some restrictions on the shift-invariant spaces, several theorems on nonuniform sampling can be found in [76,102]. Moreover, a lower bound on the maximal distance between two sampling points needed for reconstructing a function from its samples was given for the case of polynomial splines and other special cases of shift-invariant spaces in [76].…”

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confidence: 99%

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Nonuniform Sampling and Reconstruction in Shift-Invariant Spaces

Aldroubi¹,

Gröchenig²

2001

SIAM Rev.

624

541

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Abstract. This article discusses modern techniques for nonuniform sampling and reconstruction of functions in shift-invariant spaces. It is a survey as well as a research paper and provides a unified framework for uniform and nonuniform sampling and reconstruction in shiftinvariant spaces by bringing together wavelet theory, frame theory, reproducing kernel Hilbert spaces, approximation theory, amalgam spaces, and sampling. Inspired by applications taken from communication, astronomy, and medicine, the following aspects will be emphasized: (a) The sampling problem is well defined within the setting of shift-invariant spaces. (b) The general theory works in arbitrary dimension and for a broad class of generators. (c) The reconstruction of a function from any sufficiently dense nonuniform sampling set is obtained by efficient iterative algorithms. These algorithms converge geometrically and are robust in the presence of noise. (d) To model the natural decay conditions of real signals and images, the sampling theory is developed in weighted L p -spaces.

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Section: Framesmentioning

confidence: 99%

mentioning

confidence: 93%

mentioning

confidence: 99%

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Nonuniform Sampling and Reconstruction in Shift-Invariant Spaces

Aldroubi¹,

Gröchenig²

2001

SIAM Rev.

624

541

View full text Add to dashboard Cite

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“…Kadec's method of proof is to expand e iδx with respect to the orthogonal basis 1, cos(nx), sin n − Sun and Zhou (see [13] second half of Theorem 1.3) refined Kadec's argument to obtain a partial generalization of his result in higher dimensions:…”

Section: Generalizations Of Kadec's 1/4 Theoremmentioning

confidence: 99%

Sampling and recovery of multidimensional bandlimited functions via frames

Bailey

2010

Journal of Mathematical Analysis and Applications

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In this paper, we investigate frames for L 2 [−π , π ] d consisting of exponential functions in connection to oversampling and nonuniform sampling of bandlimited functions. We derive a multidimensional nonuniform oversampling formula for bandlimited functions with a fairly general frequency domain. The stability of said formula under various perturbations in the sampled data is investigated, and a computationally manageable simplification of the main oversampling theorem is given. Also, a generalization of Kadec's 1/4 theorem to higher dimensions is considered. Finally, the developed techniques are used to approximate biorthogonal functions of particular exponential Riesz bases for L 2 [−π , π ], and a wellknown theorem of Levinson is recovered as a corollary.

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“…We assume throughout the paper that the functions in the shift-invariant space V Φ are continuous on R. This is equivalent to the generators Φ being continuous on R with n∈Z |Φ(t − n)| 2 uniformly bounded on R. A proof of this equivalence can be found in [20] for the case of a unique generator ( = 1). The general case can be proved similarly.…”

Section: Preliminariesmentioning

confidence: 99%

Generalized sampling in shift-invariant spaces with multiple stable generators

Garcı́a

Hernández-Medina

Pérez-Villalón³

2008

Journal of Mathematical Analysis and Applications

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The aim of this paper is to derive stable generalized sampling in a shift-invariant space with stable generators. This is done in the light of the theory of frames in the product Hilbert space L 2 (0, 1) := L 2 (0, 1) × · · · × L 2 (0, 1) ( times). The generalized samples are expressed as the frame coefficients of an appropriate function in L 2 (0, 1) with respect to some particular frame in L 2 (0, 1). Since any multiply stable generated shift-invariant space is the image of L 2 (0, 1) by means of a bounded invertible operator, the generalized sampling is obtained from some dual frame expansions in L 2 (0, 1). An example in the setting of the Hermite cubic splines is exhibited.

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On the Stability of Multivariate Trigonometric Systems

Cited by 33 publications

References 5 publications

Nonuniform Sampling and Reconstruction in Shift-Invariant Spaces

Nonuniform Sampling and Reconstruction in Shift-Invariant Spaces

Sampling and recovery of multidimensional bandlimited functions via frames

Generalized sampling in shift-invariant spaces with multiple stable generators

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