Convergence rates for multivariate smoothing spline functions
type: Item
instance of: scholarly article
title: Convergence rates for multivariate smoothing spline functions
zbMATH Open document ID: 0646.41006
DOI: 10.1016/0021-9045(88)90034-2
author: Florencio I. Utreras
published in: Journal of Approximation Theory
publication date: 1988-01-01T00:00:00Z
review text: Given the set \(\{t_ i\}\) \(n_ 1\) of scattered data points in an open bounded subset \(\Omega\) of the d-dimensional space \({\mathbb{R}}^ d \)and the values \(\{z_ i\}\) \(n_ 1\), the thin plate smoothing spline \(\sigma_{\lambda}\) is defined as the unique minimizer of the expression \(\lambda | u|^ 2_{m,\Omega}+(1/n)\sum^{n}_{i-1}(u(t_ i)-z_ i)^ 2,\) Here \(| \cdot |\) \(2_{m,\Omega}\) is the Sobolev semi-norm in H m(\(\Omega)\). The author studies the approximation of functions \(f\in H\) m(\(\Omega)\) by a thin plate smoothing spline \(\sigma_{\lambda}\) based on the values \(z_ i=f(t_ i)+\epsilon_ i\), \(i=1,...,n\) of f dt n scattered points in \(\Omega\) known with random errors \(\epsilon_ i\). He obtains the following estimate \[ E[| f- \sigma_{\lambda}|^ 2_{k,\Omega}]\leq C[\lambda^{(m-k)/m}| f|^ 2_{m,\Omega}+D/(n\lambda^{(2k+d)/2m})],\quad k=0,...,m-1 \] provided the boundary of \(\Omega\) is smooth and the points satisfy a ``quasi-uniform'' condition.
Mathematics Subject Classification ID: 41A15
Mathematics Subject Classification ID: 41A25
Mathematics Subject Classification ID: 65D10
zbMATH DE Number: 4054226
zbMATH Keywords: random error
zbMATH Keywords: scattered data points
zbMATH Keywords: thin plate smoothing spline
zbMATH Keywords: Sobolev semi-norm
reviewed by: Borislav D. Bojanov
MaRDI profile type: Publication
full work available at URL: 0021-9045(88)90034-2
OpenAlex ID: W2033663831
cites work: Equivalent Norms for Sobolev Spaces
cites work: On kernels, eigenvalues, and eigenfunctions of operators related to elliptic problems
cites work: Q5555231
cites work: Q3890207
cites work: Q4001895
cites work: Multivariate Smoothing Spline Functions
cites work: Smoothing noisy data with spline functions: Estimating the correct degree of smoothing by the method of generalized cross-validation
cites work: Q4110059
cites work: Sur l’erreur d’interpolation des fonctions de plusieurs variables par les $D^m$-splines
cites work: Error bounds for derivative estimates based on spline smoothing of exact or noisy data
cites work: Q5528072
cites work: Q3893210
cites work: Natural spline functions, their associated eigenvalue problem
cites work: Q3896398
cites work: Q3262445
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description: scientific article; zbMATH DE number 4054226