Oversmoothing Tikhonov regularization in Banach spaces * *The work of DC was financially supported by National Natural Science Foundation of China (Nos. 11701205 and 11871240). The work of BH was financially supported by German Research Foundation (DFG Grant HO 1454/12-1). The work of IY was financially supported by German Research Foundation (DFG Grants YO159/2-2 and YO159/4-1).

Abstract

This paper develops a Tikhonov regularization theory for nonlinear ill-posed operator equations in Banach spaces. As the main challenge, we consider the so-called oversmoothing state in the sense that the Tikhonov penalization is not able to capture the true solution regularity and leads to the infinite penalty value in the solution. We establish a vast extension of the Hilbertian convergence theory through the use of invertible sectorial operators from the holomorphic functional calculus and the prominent theory of interpolation scales in Banach spaces. Applications of the proposed theory involving ℓ 1, Bessel potential spaces, and Besov spaces are discussed.