In theory and practice of linear ill-posed problems the approximate solution of linear Fredholm integral equations plays an important role. The paper considers the equation \(Kf(x)=g(x)\), \(x \in \Sigma\), with \(g \in L^ 2 (\Sigma)\) given, possibly subject to error. Variations of the Phillips-Tikhonov regularization \[ \text{minimize } \| Kf-g \|^ 2_{L^ 2(\Sigma)}+\alpha^ 2D(f,\varphi) \text{ subject to } f \geq 0, \] are studied where \(D(f,\varphi)\) is the cross-entropy functional \[ D(f,\varphi)=\int_ \Omega \left\{ f(y) \log {f(y) \over \varphi (y)}+\varphi (y)-f(y) \right\} d \mu(y). \] A series of analytic properties of this approach, which is motivated also from a stochastic point of view, are presented in this paper. A very helpful overview of literature concerning different versions of this so-called maximum entropy regularization method is given. Further paragraphs of the paper deal with finite-dimensional approximations of the method and some results associated with moment problems.
The paper is a valuable contribution to the theory and numerical analysis of linear ill-posed problems.