Abstract
Abstract - The paper provides a general framework for studying regularization methods in Sobolev scales. We show that linear regularization methods can be interpreted as smoothing of the generalized inverse or equivalently as the generalized inverse applied to smoothed data. In contrast to [16] we use Sobolev spaces instead of spaces generated by the operator under consideration. Further we give conditions on the degree of the smoothing based on estimates in Sobolev norms. We also show that the regularization methods form a semi group. Conditions for the order optimality of the methods are provided. Some very useful conditions are presented to guarantee that a mollifier of convolution type builds up a regularization method. Finally we verify the results for the Radon transform as the model in computerized tomography.
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