Applications of the Backus-Gilbert method to linear and some nonlinear equations

We investigate the use of a functional analytical version of the Backus-Gilbert method as a reconstruction strategy to obtain specific information on the solution of linear and slightly nonlinear systems with Frechet derivable operators. Some a priori error estimates are shown and tested for two classes of problems: a nonlinear moment problem and a linear elliptic Cauchy problem. For this second class of problems a special version of the Green formula is developed in order to analyse the involved adjoint equations.