Solutions to deconvolution equations using nonperiodic sampling
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.
scientific article; zbMATH DE number 1269678
scientific article