Solving Fredholm equations by maximum entropy on the mean. Application to superresolution (Q1368526)

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.