Convergence of discrete and penalized least squares spherical splines
type: Item
instance of: scholarly article
title: Convergence of discrete and penalized least squares spherical splines
zbMATH Open document ID: 1229.41004
author: Victoria Baramidze
author: Ming-Jun Lai
published in: Journal of Approximation Theory
publication date: 2011-10-10T00:00:00Z
review text: The authors consider the approximation of data \(\mathcal{V}\) on the unit sphere \(\mathbb{S}^2\) in \(\mathbb{R}^3\) by discrete least squares and penalized least squares spherical splines. More precisely, given a set of discrete values \(\{f(v) : v\in \mathcal{V}\}\) of a function \(f:\mathbb{S}^2\to\mathbb{R}\), find a penalized least squares spline \(s_{\lambda,f}\in S^r_d (\Delta)\), where \(d > r \geq 0\) and \(\Delta\) is a triangulation of \(\mathbb{S}^2\). It is assumed that the elements in \(\mathcal{V}\) are evenly distributed over \(\Delta\). Here \(\lambda \geq 0\) denotes the penalty parameter, which when set equal to zero yields the discrete least squares spherical splines. Error bounds for the approximation of sufficiently smooth functions by discrete and penalized least squares spherical splines are derived and it is shown that in the case of discrete least squares spherical splines this error bound depends explicitly on the mesh size of the triangulation \(\Delta\), whereas in the case of penalized least squares spherical splines it also depends on the penalty parameter \(\lambda\).
Mathematics Subject Classification ID: 41A15
Mathematics Subject Classification ID: 43A90
zbMATH DE Number: 5955946
zbMATH Keywords: Spherical splines
zbMATH Keywords: Least squares splines
reviewed by: Peter Massopust
MaRDI profile type: Publication
full work available at URL: j.jat.2011.03.001
OpenAlex ID: W1973036125
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DOI: 10.1016/J.JAT.2011.03.001
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description: scientific article; zbMATH DE number 5955946