An alternative approach to ill-posed problems
The author suggests the following approach to linear ill-posed problems of the form \(Tx=y\), where T is a Hilbert-Schmidt integral operator on \(L^ 2:\) Let \(A:=\cap_{n\in {\mathbb{N}}}R(T^ n).\) A Fréchet space structure is introduced on A in a natural way. It is shown that T, considered as a selfmap on A, has a continuous inverse. Thus, the problem is well-posed on A. However, it has to be remarked that this concept of ``well-posedness'' allows only for perturbations in the data which are ``very smooth'', which seems to restrict the applicability of this approach. By duality, the equation is then interpreted as an equation on \(A^*\supset L^ 2\), which is again well-posed. However, there is no discussion on the meaning of ``well-posedness'' in this setting. In a relatively separate part of the paper, methods from analytic function theory are used to construct regularizations in this setting.
scientific article; zbMATH DE number 4106153