Solving Fredholm equations by maximum entropy on the mean. Application to superresolution
type: Item
instance of: scholarly article
title: Solving Fredholm equations by maximum entropy on the mean. Application to superresolution
zbMATH Open document ID: 0884.45001
DOI: 10.1016/S0895-7177(97)00046-0
author: Henryk Gzyl
author: Fabrice Gamboa
published in: Mathematical and Computer Modelling
publication date: 1998-03-17T00:00:00Z
review text: The authors look for a solution to a first kind Fredholm integral equation \[ \int_U K(x,y)f(x)dP(x)=c(y), \quad y\in V,\tag{1} \] where \(U\) and \(V\) are compact sets, \(P\) is a probability measure on \(U\); the solution is subject to constraint \[ a<f(x)<b, \qquad -\infty\leq a<b\leq \infty.\tag{2} \] Solving (1) we need to apply some regularization. Instead of the exact one the authors suggest to look for a solution in a form \[ \bigg|\int\limits_U K(x,y) f(x) dP(x)-c(y)\bigg|\leq \sigma, \quad \sigma>0.\tag{3} \] Discretizing condition (2) it is replaced by the following generalized moment problem \[ \bigg|\int\limits_U \Phi(x)f(x) dP(x)-\widetilde c\bigg|_\infty\leq \sigma,\tag{4} \] where \(\Phi : U\rightarrow \mathbb{R}^d\) is a vector-function and \(\widetilde c\in \mathbb{R}^d\) is a vector. The maximum entropy method on the mean (M.E.M.) provides a general strategy to pick up a solution to (4). Superresolution is invoked to specify cases when (2), (3) has only one solution, which is found with the help of M.E.M.
Mathematics Subject Classification ID: 45B05
zbMATH DE Number: 1067222
zbMATH Keywords: maximum entropy
zbMATH Keywords: first kind Fredholm integral equation
zbMATH Keywords: regularization
zbMATH Keywords: moment problem
zbMATH Keywords: maximum entropy method
zbMATH Keywords: superresolution
reviewed by: Roland Duduchava
MaRDI profile type: Publication
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full work available at URL: s0895-7177(97)00046-0
OpenAlex ID: W2031130978
description: scientific article; zbMATH DE number 1067222