Stability rates for linear ill-posed problems with compact and non-compact operators
type: Item
instance of: scholarly article
title: Stability rates for linear ill-posed problems with compact and non-compact operators
zbMATH Open document ID: 0941.65055
DOI: 10.4171/ZAA/881
author: G. Fleischer
author: Bernd Hofmann
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 2000-08-10T00:00:00Z
full work available at URL: https://eudml.org/doc/48483
review text: 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 \(A\).
reviewed by: Riho Lepp
Mathematics Subject Classification ID: 65J10
Mathematics Subject Classification ID: 65J20
Mathematics Subject Classification ID: 45B05
Mathematics Subject Classification ID: 45E10
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 65R30
zbMATH DE Number: 1331698
zbMATH Keywords: ill-posed problems
zbMATH Keywords: linear operator equations
zbMATH Keywords: Hilbert spaces
zbMATH Keywords: compact operators
zbMATH Keywords: noncompact operators
zbMATH Keywords: compact convolution operators
zbMATH Keywords: noncompact multiplication operators
MaRDI profile type: Publication
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OpenAlex ID: W2006972588
description: scientific article; zbMATH DE number 1331698