An alternative approach to ill-posed problems
type: Item
instance of: scholarly article
title: An alternative approach to ill-posed problems
zbMATH Open document ID: 0675.65052
DOI: 10.1216/JIE-1988-1-2-287
published in: Journal of Integral Equations and Applications
publication date: 1988-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 65J10
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 47A50
Mathematics Subject Classification ID: 45B05
zbMATH DE Number: 4106153
zbMATH Keywords: Hilbert space
zbMATH Keywords: linear ill-posed problems
zbMATH Keywords: Hilbert-Schmidt integral operator
zbMATH Keywords: Fréchet space
zbMATH Keywords: well-posedness
zbMATH Keywords: duality
zbMATH Keywords: regularizations
author: Gilbert G. Walter
MaRDI profile type: Publication
full work available at URL: jie-1988-1-2-287
OpenAlex ID: W2038905427
description: scientific article; zbMATH DE number 4106153