Error estimates for Hermite interpolation on spheres
type: Item
instance of: scholarly article
title: Error estimates for Hermite interpolation on spheres
zbMATH Open document ID: 1023.41003
DOI: 10.1016/S0022-247X(02)00451-1
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-07-30T00:00:00Z
review text: Let \(S^{d-1}\) denote the unit sphere in the Euclidean space \(\mathbb{R}^d\) and \(C^r(S^{d-1})\) be the set of \(r\)-times differentiable functions on \(S^{d-1}\). Let \(xy\) denote the usual inner product of \(x,y\in\mathbb{R}^d\) and \(d(x, y)=\cos^{-1}xy\) be the geodesic distance between \(x,y\in S^{d-1}\). Let \(\mu\) be the normalised rotation invariant surface measure on \(S^{d-1}\). Furthermore, let \(P_m\) be the subspace of spherical polynomials of degree not greater than \(m\), \(H_m=P_m \cap(P_{m-1})^\perp\) be the set of spherical harmonics of degree \(m\), and \(d_m=\dim (H_m)\). Choose an orthonormal basis \(\{Y_{m,1}, \dots,Y_{m,d_m}\}\) for \(H_m\), \(m=0,1,\dots\). Then, every \(f\in L_1(S^{d-1})\) (the integrable functions on the sphere) has a Fourier series expansion \(f\sim\sum^\infty_{m=0} \sum^{d_m}_{j=1} \widehat f_{m,j}Y_{m,j}\), where \(\widehat f_{m,j}= \int_{S^{d-1}}fY_{m,j} d\mu\). Let \(h:[-\pi,\pi] \to\mathbb{R}\) be a continuous function and \(X=\{x_1, \dots, x_N\}\) be a set of distinct interpolation points. The usual interpolation problem is to find a function \(s(x)=\sum^N_{j=1}c_j h(d(x,x_j))+ p(x)\), \(p\in P_{m-1}\), such that \(s(x_i)=f(x_i)\), \(x_i\in X\), where \(c_j\), \(j=1, \dots,N\), satisfy \(\sum^N_{j=1} c_jq(x_i)=0\), \(\forall q\in P_{m-1}\). The node set \(x\) is assumed to be \(P_{m-1}\)-unisolvent, i.e. if \(p\in P_{m-1}\) and \(p(x)=0\) for all \(x\in X\) then \(p=0\). Furthermore \(N\geq\dim(P_{n-1})\). Let \(r\in \mathbb{N}\) and \(\wedge=\{L_1, \dots,L_N\}\subset (c^r(S^{d-1}))'\) (the dual of \(c^r (S^{d-1}))\). The authors consider the following Hermite interpolation problem: find a function \[ s(x)=\sum^N_{j=1} c_j\cdot L^y_j\biggl(h \bigl(d(x,y)\bigr) \biggr)+p(x),\;p\in P_{m-1}, \] such that \(L_i(s)=L_i(f)\), \(i=1, \dots,N\), where \(c_j\), \(j=1,\dots,N\), satisfy \(\sum^N_{j=1} c_jL_j(q)=0\), \(\forall q\in P_{m-1}\) and \(L^y\) denotes the action on the \(y\) variable. It is assumed that the funct…
Mathematics Subject Classification ID: 41A05
zbMATH DE Number: 1956843
zbMATH Keywords: Hermite interpolation
zbMATH Keywords: pseudodifferential operator
zbMATH Keywords: rotational differential operator
author: Jeremy Levesley
author: Zuhua Luo
MaRDI profile type: Publication
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full work available at URL: s0022-247x(02)00451-1
OpenAlex ID: W2073619140
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description: scientific article; zbMATH DE number 1956843