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A generalization of the method of least squares for operator equations in some Fréchet spaces. (English. Russian original) Zbl 0880.46001

Izv. Math. 59, No. 5, 935-948 (1995); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 59, No. 5, 59-72 (1995).
Summary: The classical method of least squares is extended to equations with an operator between Fréchet 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 Fréchet spaces of power series of finite and infinite type, Fréchet spaces of rapidly decreasing sequences and the Fréchet spaces of analytic functions given in Stein’s monograph to themselves.

MSC:

46A04 Locally convex Fréchet spaces and (DF)-spaces
47A50 Equations and inequalities involving linear operators, with vector unknowns
41A35 Approximation by operators (in particular, by integral operators)
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