The trade-off between regularity and stability in Tikhonov regularization

When deriving rates of convergence for the approximations generated by the application of Tikhonov regularization to ill–posed operator equations, assumptions must be made about the nature of the stabilization (i.e., the choice of the seminorm in the Tikhonov regularization) and the regularity of the least squares solutions which one looks for. In fact, it is clear from works of Hegland, Engl and Neubauer and Natterer that, in terms of the rate of convergence, there is a trade–off between stabilization and regularity. It is this matter which is examined in this paper by means of the best–possible worst–error estimates. The results of this paper provide better estimates than those of Engl and Neubauer, and also include and extend the best possible rate derived by Natterer. The paper concludes with an application of these results to first–kind integral equations with smooth kernels.