Maximum Entropy Regularization for Fredholm Integral Equations of the First Kind

The regularization of Fredholm integral equations of the first kind is considered with positive solutions by means of maximum entropy. The regularized solution is the minimizes of a functional analogous to the case of Phillips–Tikhonov regularization. The regularized solution is shown to converge to the solution of the maximum entropy least squares problem, assuming it exists. Under additional regularity conditions akin to those for Phillips–Tikhonov regularization error estimates are obtained as well. In addition it is shown that the regularity conditions are necessary for these estimates to hold. Approximation from finite-dimensional subspaces are also considered, as well as exact and approximate moment problems for the integral equations. The basic tools in the analysis are the weak compactness of subsets of $L_1$ consisting of functions of bounded entropy, and an inequality for convex optimization problems with Bregman functionals.