Convergence rates for oversmoothing Banach space regularization

This paper studies Tikhonov regularization for finitely smoothing operators in Banach spaces when the penalizationenforces too much smoothness sense that penalty term is not finite at truesolution. In a Hilbert space setting, Natterer [Applicable Anal., 18 (1984), pp. 29-37] showed with help of spectral theory optimal rates can be achieved this situation. ("Oversmoothing does harm.") For oversmoothing variational spaces, only very recently has progressbeen several papers different settings, all which construct families smooth approximations to true solution. we propose such familyof based on K-interpolation theory. We demonstrate thisleads simple, self-contained proofs and rather general results.In particular, obtain convergencerates bounded variation regularization, Besov terms, $\ell^p$ wavelet penalization $p<1$, cannotbe treated by previous approaches. also derive minimax white noisemodels. Our theoretical results are confirmed numerical experiments.