Using a natural deconvolution for analysis of perturbed integer sampling in shift-invariant spaces
type: Item
instance of: scholarly article
title: Using a natural deconvolution for analysis of perturbed integer sampling in shift-invariant spaces
zbMATH Open document ID: 1198.94067
author: Stefan Ericsson
author: Niklas Grip
published in: Journal of Mathematical Analysis and Applications
publication date: 2010-10-22T00:00:00Z
full work available at URL: http://urn.kb.se/resolve
review text: Let \(V \subset L^2(\mathbb R)\) be a shift-invariant space spanned by a Riesz basis of integer-translates of a single function. The authors discuss the existence of a numerically stable reconstruction formula that recovers any \(f\in V\) from the samples \(f(k + \varepsilon_k)\) \((k \in \mathbb Z)\) with perturbed integer sampling points \(k + \varepsilon_k\), where \(\sup_k |\varepsilon_k| \leq \varepsilon\) for some jitter error bound \(\varepsilon \in (0,\, \frac{1}{2})\). It is shown that under mild assumptions, \(V\) can also be generated by integer-translates of a function \(\varphi\), where its Fourier transform reads \[ {\hat \varphi}(v) = \int_{v-\pi}^{v+\pi} g(x) \, dx \] for some \(g\) with \(\int_{-\infty}^{\infty} g(x)\, dx = 1\). The authors adapt the analysis techniques from their previous paper [Int. J. Wavelets Multiresolut. Inf. Process. 3, No. 3, 301--319 (2005; Zbl 1075.42013)] into new sampling theorems with \(g\) as a design parameter. The method is demonstrated on different shift-invariant spaces for which \(g\) has compact support and bounded variation (including all spaces generated by Meyer wavelet scaling functions). Further, jitter error bounds \(\varepsilon\) are computed for particular choices of \(\varphi\).
reviewed by: Manfred Tasche
Mathematics Subject Classification ID: 94A20
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 65T60
zbMATH DE Number: 5804460
zbMATH Keywords: shift-invariant space
zbMATH Keywords: irregular sampling
zbMATH Keywords: perturbed integer sampling
zbMATH Keywords: deconvolution
zbMATH Keywords: numerically stable reconstruction
zbMATH Keywords: reproducing kernel
zbMATH Keywords: scaling function
zbMATH Keywords: jitter error bound
MaRDI profile type: Publication
OpenAlex ID: W2052174719
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DOI: 10.1016/J.JMAA.2010.07.021
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description: scientific article; zbMATH DE number 5804460