Regularizability of Ill-Posed Problems and the Modulus of Continuity

  • Radu Ioan Boţ

    Technische Universität Chemnitz, Germany
  • Bernd Hofmann

    Technische Universität Chemnitz, Germany
  • Peter Mathé

    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].

Cite this article

Radu Ioan Boţ, Bernd Hofmann, Peter Mathé, Regularizability of Ill-Posed Problems and the Modulus of Continuity. Z. Anal. Anwend. 32 (2013), no. 3, pp. 299–312

DOI 10.4171/ZAA/1486