A Nonlinear Solution of Inverse Problems

We consider here a classical problem of estimating or approximating a density given the information consisting of the values of some of its integrals. The method will have direct applications in many problems of density, spectral density and image analysis which will not be treated in any detail here. One widely used method chooses the density which minimizes a cross entropy functional. We propose a diierent method based on maximizing a functional. However as an estimate we take the reciprocal of the maximizing random variable multiplied by a density. In general, restorations obtained utilizing this new method diier from those obtained by entropy methods. Their properties as well as some examples demonstrate competitiveness of the proposed method.