A generalization of the method of least squares for operator equations in some Frechet spaces

Abstract

The classical method of least squares is extended to equations with an operator between Frechet spaces. Approximate solutions are obtained by minimizing the discrepancy relative to a metric, which in the Hilbert space case coincides with the metric induced by the scalar product. The convergence of a sequence of approximate solutions to the exact solution is proved. A concrete realization of the results obtained is given in the case of continuously invertible and so-called tamely invertible operators that map Frechet spaces of power series of finite and infinite type, Frechet spaces of rapidly decreasing sequences and the Frechet spaces of analytic functions given in Stein's monograph to themselves.