Convolution, Average Sampling, and a Calderon Resolution of the Identity for Shift-Invariant Spaces

Aldroubi, Akram; Sun, Qiyu; Tang, Wai-Shing

doi:10.1007/s00041-005-4003-3

Cited by 105 publications

(119 citation statements)

References 24 publications

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“…The modulus of continuity is a delicate tool in mathematical analysis [18,46], and it has also been used to estimate the density of a stable sampling set (see [1,3,6] and references therein). The modulus of continuity can also be used to estimate the density of a local determining sampling set.…”

Section: Theorem 21 Let V Be a Linear Space Of Functions On The Linementioning

confidence: 99%

“…is called to be a relatively-separated set, which has been widely used in the nonuniform sampling [3,6,43]. In Section 2, we also show that for any locally finite-dimensional shift-invariant space V of continuous signals (functions), any element in V can be locally reconstructed from its samples taken on a uniform sampling set with sufficiently large density (Theorem 2.2).…”

Section: Introductionmentioning

confidence: 98%

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Local reconstruction for sampling in shift-invariant spaces

Sun

2008

Adv Comput Math

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The local reconstruction from samples is one of most desirable properties for many applications in signal processing, but it has not been given as much attention. In this paper, we will consider the local reconstruction problem for signals in a shift-invariant space. In particular, we consider finding sampling sets X such that signals in a shift-invariant space can be locally reconstructed from their samples on X. For a locally finite-dimensional shiftinvariant space V we show that signals in V can be locally reconstructed from its samples on any sampling set with sufficiently large density. For a shiftinvariant space V(φ 1 , . . . , φ N ) generated by finitely many compactly supported functions φ 1 , . . . , φ N , we characterize all periodic nonuniform sampling sets X such that signals in that shift-invariant space V(φ 1 , . . . , φ N ) can be locally reconstructed from the samples taken from X. For a refinable shift-invariant space V(φ) generated by a compactly supported refinable function φ, we prove that for almost all (x 0 , x 1 ) ∈ [0, 1] 2 , any signal in V(φ) can be locally reconstructed from its samples from {x 0 , x 1 } + Z with oversampling rate 2. The proofs of our results on the local sampling and reconstruction in the refinable shift-invariant space V(φ) depend heavily on the linear independent shifts of a refinable function on measurable sets with positive Lebesgue measure and the almost ripplet property for a refinable function, which are new and interesting by themselves.

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Section: Theorem 21 Let V Be a Linear Space Of Functions On The Linementioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

Local reconstruction for sampling in shift-invariant spaces

Sun

2008

Adv Comput Math

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“…Over the past decade, the paradigm for representing band-limited signals in the Shannon's sampling theory has been extended to signals in shift-invariant spaces. It is well known [1,2,32,33] that any signal x in the shift-invariant space…”

Section: Introductionmentioning

confidence: 99%

Reconstructing Signals with Finite Rate of Innovation from Noisy Samples

Nashed

Sun

2009

Acta Appl Math

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A signal is said to have finite rate of innovation if it has a finite number of degrees of freedom per unit of time. Reconstructing signals with finite rate of innovation from their exact average samples has been studied in Sun (SIAM J. Math. Anal. 38, 1389-1422. In this paper, we consider the problem of reconstructing signals with finite rate of innovation from their average samples in the presence of deterministic and random noise. We develop an adaptive Tikhonov regularization approach to this reconstruction problem. Our simulation results demonstrate that our adaptive approach is robust against noise, is almost consistent in various sampling processes, and is also locally implementable.

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“…Sampling theory has been extended for the general shift-invariant spaces generated by splines or wavelets, which exhibit better localization properties than the ideal sinc kernel [4][5][6]. Although a large body of work has been dedicated to sampling in shift-invariant spaces, [7][8][9][10][11][12][13][14][15][16][17][18][19][20] just to name a few, a general theory for sampling non-decaying signals seems to be still missing. There were some attempts to generalize the sampling theorem for bandlimited signals of polynomial growth [21][22][23][24], but the bandlimitedness requirement is too restrictive.…”

Section: Introductionmentioning

confidence: 99%

A sampling theory for non-decaying signals

Nguyen

Unser

2017

Applied and Computational Harmonic Analysis

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The classical assumption in sampling and spline theories is that the input signal is square-integrable, which prevents us from applying such techniques to signals that do not decay or even grow at infinity. In this paper, we develop a sampling theory for multidimensional non-decaying signals living in weighted L p spaces. The sampling and reconstruction of an analog signal can be done by a projection onto a shift-invariant subspace generated by an interpolating kernel. We show that, if this kernel and its biorthogonal counterpart are elements of appropriate hybridnorm spaces, then both the sampling and the reconstruction are stable. This is an extension of earlier results by Aldroubi and Gröchenig. The extension is required because it allows us to develop the theory for the ideal sampling of non-decaying signals in weighted Sobolev spaces. When the d-dimensional signal and its d/p + ε derivatives, for arbitrarily small ε > 0, grow no faster than a polynomial in the L p sense, the sampling operator is shown to be bounded even without a sampling kernel. As a consequence, the signal can also be interpolated from its samples with a nicely behaved interpolating kernel.

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Convolution, Average Sampling, and a Calderon Resolution of the Identity for Shift-Invariant Spaces

Cited by 105 publications

References 24 publications

Local reconstruction for sampling in shift-invariant spaces

Local reconstruction for sampling in shift-invariant spaces

Reconstructing Signals with Finite Rate of Innovation from Noisy Samples

A sampling theory for non-decaying signals

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