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

We investigate the use of a functional analytical version of the Backus-Gilbert Method as a reconstruction strategy to get specific information about the solution of linear and slightly non-linear systems with Frech\'et 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 analyze the involved adjoint equations.