On ill-posedness measures and space change in Sobolev scales
type: Item
instance of: scholarly article
title: On ill-posedness measures and space change in Sobolev scales
zbMATH Open document ID: 0896.65043
DOI: 10.4171/ZAA/800
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 1998-09-22T00:00:00Z
review text: Let \(A:X \rightarrow Y\) be a non-degenerate, compact, linear operator between infinite-dimensional, separable Hilbert spaces \(X\), \(Y\). Hence the range is non-closed and the equation (1) \(Ax = y\) is ill-posed. Let \(\{ s_i(A)\}\) be the (ordered) sequence of positive singular values of \(A\). Then the (finite or infinite) interval \[ [\underline{\mu}(A), \bar{\mu}(A)] = \Big[ \liminf_{i \to \infty} {{\log s_i(A)} \over {\log i}}, \limsup_{i \to \infty} {{\log s_i(A)} \over {\log i}} \Big] \] is defined as the interval of ill-posedness of (1) and \(\underline{\mu} (A)\) as the degree of ill-posedness. This interval is compared with alternative interval definitions obtained from compactness measures as \(\varepsilon\)-entropy or \(\varepsilon\)-capacity. For the specific case \(X = L^2(0,1)\) the space-change problem of shifting the space \(X\) along a Sobolev scale is treated. In particular, the change of the ill-posedness interval is analyzed in detail when the solutions are restricted to \(W^1_2[0,1]\). In addition, some inter-dependencies between ill-posedness measures, embedding operators, Hilbert and Sobolev scales are discussed.
Mathematics Subject Classification ID: 65J10
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 65R30
Mathematics Subject Classification ID: 65J20
Mathematics Subject Classification ID: 47B07
Mathematics Subject Classification ID: 45B05
zbMATH DE Number: 1129180
zbMATH Keywords: ill-posed problems
zbMATH Keywords: compact linear operators
zbMATH Keywords: ill-posedness measures
zbMATH Keywords: interval of ill-posedness
zbMATH Keywords: Hilbert scale
zbMATH Keywords: Sobolev scale
zbMATH Keywords: embedding operators
author: Bernd Hofmann
author: Ulrich Tautenhahn
reviewed by: Werner C. Rheinboldt
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.4171/zaa/800
OpenAlex ID: W2068678104
description: scientific article; zbMATH DE number 1129180