Solutions to deconvolution equations using nonperiodic sampling
type: Item
instance of: scholarly article
title: Solutions to deconvolution equations using nonperiodic sampling
zbMATH Open document ID: 0954.42021
DOI: 10.1007/BF02479673
published in: The Journal of Fourier Analysis and Applications
publication date: 2001-02-08T00:00:00Z
full work available at URL: https://eudml.org/doc/59588
review text: Given a collection of \(m (\geq d+1)\) characteristic functions \(\chi_i\) of closed hypercubes \([-4_i, r_i]^d\) in \(\mathbb{R}^d\) such that \(r_i/r_j\) is badly approximated by rationals for all pairs of distinct \(i\), \(j\), then it is known that there are distributions \(\mu_i\) of compact support such that \(\mu_1* \chi_1+\cdots+ \mu_m* \chi_m= \delta\) (Bézout equation). The author presents a practical and explicit approximate construction of such \(\mu_i\) using non-periodic sampling. Namely, for any smooth approximation \(\varphi\) of \(\delta,\varphi\in C^\infty_0(\mathbb{R}^d)\), he constructs explicitly \(\mu_i\), \(\varphi\) such that they yield an approximate solution of the Bézout equation, namely, \(\chi_1* \mu_{1,\varphi}+\cdots+ \chi_m*\mu_{m,\varphi}= \varphi\). The interest of such a result is that it provides concrete construction of systems of detectors for which there is superresolution, i.e., the resolution limit is essentially limited by the sampling size and not by the physical size (or equivalent characteristic of the detectors), thus implementing the scheme proposed in [\textit{C. A. Berenstein}, \textit{B. A. Taylor} and \textit{A. Yger}, J. Optics (Paris) 14, 75-82 (1983)[. This question tends to reappear every so often in the literature (especially in physics). The companion publication by \textit{S. D. Casey} and \textit{D. F. Walnut} [SIAM Rev. 36, No. 4, 537-577 (1994; Zbl 0814.45001)] is a good overview of the same subject.
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 78A55
Mathematics Subject Classification ID: 42A85
zbMATH DE Number: 1269678
zbMATH Keywords: deconvolution
zbMATH Keywords: Bézout equation
zbMATH Keywords: superresolution
zbMATH Keywords: sampling
author: David F. Walnut
reviewed by: Carlos A. Berenstein
MaRDI profile type: Publication
cites work: Analytic Bezout identities
cites work: Le problème de la deconvolution
cites work: Effective Bezout identities in \({\mathbb{Q}}[z_ 1,\dots ,z_ n]\)
cites work: Modulation and sampling techniques for multichannel deconvolution
cites work: Systems of Convolution Equations, Deconvolution, Shannon Sampling, and the Wavelet and Gabor Transforms
cites work: Generators for some rings of analytic functions
cites work: Non-Liouville numbers and a theorem of Hörmander
cites work: Q4307561
cites work: Nonperiodic sampling of bandlimited functions on unions of rectangular lattices.
OpenAlex ID: W2080530013
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description: scientific article; zbMATH DE number 1269678