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Abstract
The article is concerned with ill-posed operator equations Ax = y where A:X →Y is an injective bounded linear operator with non-closed range and X and Y are Hilbert spaces. The solution x=x
† is assumed to be in the range
of some selfadjoint strictly positive bounded linear operator G:X →X. Under several assumptions on G, such as
or more generally
, inequalities of the form
, or range inclusions
, convergence rates for the regularization error
of Tikhonov regularization are established. We also show that part of our assumptions automatically imply so-called source conditions. The article contains a series of new results but also intends to uncover cross-connections between the different kinds of smoothness conditions that have been discussed in the literature on convergence rates for Tikhonov regularization.
Acknowledgments
The authors express their sincere thanks to Dr Dana Düvelmeyer (TU Chemnitz) for a couple of helpful comments on the paper, which were made during her stay in Tokyo in July 2005. The fourth named author is partly supported by Grant 15340027 from the Japan Society for the Promotion of Science and Grant 17654019 from the Ministry of Education, Culture, Sports and Technology.