  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.
