Boundary effects on convergence rates for Tikhonov regularization
Let \(H^ k[0,1]\) be the Sobolev space. The main result of the paper deals with the Tikhonov regularization \(f_{\lambda}\) of a smooth function \(f\in H^{2m}[0,1]\) defined as the solution to \[ \min_{u\in H^ m[0,1]}\{\lambda | u|^ 2_ m+| f-u_ 0|^ 2\},\quad \lambda >0. \] It is analysed how the values of f and its derivatives at the boundary points 0 and 1 affect the convergence rates. Some conclusions deduced also help to extend some results of \textit{J. Rice} and \textit{M. Rosenblatt} [Ann. Stat. 11, 141-156 (1983; Zbl 0535.41019)] for cubic smoothing splines.
scientific article; zbMATH DE number 4071621