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Zeitschrift für Analysis und ihre Anwendungen


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Volume 32, Issue 3, 2013, pp. 299–312
DOI: 10.4171/ZAA/1486

Published online: 2013-07-03

Regularizability of Ill-Posed Problems and the Modulus of Continuity

Radu Ioan Boţ[1], Bernd Hofmann[2] and Peter Mathé[3]

(1) Technische Universität Chemnitz, Germany
(2) Technische Universität Chemnitz, Germany
(3) Weierstrass Institut für Angewandte Analysis und Stochastik, Berlin, Germany

The regularization of linear ill-posed problems is based on their conditional well-posedness when restricting the problem to certain classes of solutions. Given such class one may consider several related real-valued functions, which measure the well-posedness of the problem on such class. Among those functions the modulus of continuity is best studied. For solution classes which enjoy the additional feature of being star-shaped at zero, the authors develop a series of results with focus on continuity properties of the modulus of continuity. In particular it is highlighted that the problem is conditionally well-posed if and only if the modulus of continuity is right-continuous at zero. Those results are then applied to smoothness classes in Hilbert space. This study concludes with a new perspective on a concavity problem for the modulus of continuity, recently addressed by two of the authors in [Trudy Inst. Mat. i Mekh. UrO RAN 18 (2012)(1), 34{41].

Keywords: Linear ill-posed problems, modulus of continuity, conditional stability

Boţ Radu Ioan, Hofmann Bernd, Mathé Peter: Regularizability of Ill-Posed Problems and the Modulus of Continuity. Z. Anal. Anwend. 32 (2013), 299-312. doi: 10.4171/ZAA/1486