Boundary effects on convergence rates for Tikhonov regularization
type: Item
instance of: scholarly article
title: Boundary effects on convergence rates for Tikhonov regularization
zbMATH Open document ID: 0656.42018
DOI: 10.1016/0021-9045(88)90001-9
author: Florencio I. Utreras
published in: Journal of Approximation Theory
publication date: 1988-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 42C15
Mathematics Subject Classification ID: 41A15
Mathematics Subject Classification ID: 42A38
zbMATH DE Number: 4071621
zbMATH Keywords: Birkhoff coefficients
zbMATH Keywords: integrated mean square error
zbMATH Keywords: Sobolev space
zbMATH Keywords: Tikhonov regularization
zbMATH Keywords: cubic smoothing splines
reviewed by: Ganesh Datta Dikshit
MaRDI profile type: Publication
full work available at URL: 0021-9045(88)90001-9
OpenAlex ID: W2008025646
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description: scientific article; zbMATH DE number 4071621