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Stability rates for linear ill-posed problems with compact and non-compact operators. (English) Zbl 0941.65055
Two types of ill-posedness of linear operator equations $$Ax= y$$ in Hilbert spaces are considered: type I, if $$A$$ is not compact, but for the range $$R(A)$$ we have $$R(A)\neq \text{cl }R(A)$$, and type II for compact operators $$A$$. Ill-posedness of type I means that range $$R(A)$$ contains a closed infinite-dimensional subspace. Conditions are presented that can be ensured for ill-posed equations of type II by appropriate choice of an orthonormal system, but for ill-posed equations of type I are never satisfied. Since noncompact operators $$A$$ have no singular values, the authors introduce stability rates in order to have a common measure for both, compact and noncompact ones. Properties of these rates are analyzed for compact convolution operators and for noncompact multiplication operators $$A$$.
Reviewer: R.Lepp (Tallinn)

MSC:
 65J10 Numerical solutions to equations with linear operators 65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization 45B05 Fredholm integral equations 45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) 65R20 Numerical methods for integral equations 65R30 Numerical methods for ill-posed problems for integral equations
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