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 equa tions, assumptions must be made about the nature of tile 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 Natter er that, in terms of the rate of convergence, there is a trade-off bet ween stabilization and regularity. It. is this matter which is examine d in this paper by means of the best-possible worst-error estimates. T he results of this paper provide better estimates than those of Engl a nd Neubauer, and also include and extend the best possible rate derive d by Natterer. The paper concludes with an application of these result s to first-kind integral equations with smooth kernels.