A unified theory of radial basis functions. Native Hilbert spaces for radial basis functions. II
type: Item
instance of: scholarly article
title: A unified theory of radial basis functions. Native Hilbert spaces for radial basis functions. II
zbMATH Open document ID: 0984.46014
DOI: 10.1016/S0377-0427(00)00345-9
author: Robert Schaback
published in: Journal of Computational and Applied Mathematics
publication date: 2001-03-05T00:00:00Z
review text: [For part I see ISNM, Int. Ser. Numer. Math. 132, 255-282 (1999; Zbl 0944.46017).] Given some symmetric positive definite function \(\Phi\) on some domain of Euclidean space, one can define the native Hilbert space as the completion of the linear hull of the functions \(\Phi(x,\cdot)\) with respect to the inner product \((\Phi(x,\cdot),\Phi(y,\cdot))_\Phi= \Phi(x,y)\) (as a model serves the case of radial basis functions \(\Phi(x,y)=\phi(\|x-y\|)\)). Under a mild boundedness condition, the native Hilbert space can be embedded into \(L_2\), and the adjoint of this embedding is the integral operator with kernel \(\Phi\). Using eigenfunction expansion for this operator in \(L_2\), a different characterization of the native space is obtained. Also some convergence results are given.
Mathematics Subject Classification ID: 46E22
Mathematics Subject Classification ID: 41A65
Mathematics Subject Classification ID: 46B15
zbMATH DE Number: 1533133
zbMATH Keywords: native Hilbert space
zbMATH Keywords: radial basis function
zbMATH Keywords: reproducing kernel
zbMATH Keywords: positive definite function
zbMATH Keywords: eigenfunction expansion
MaRDI profile type: Publication
cites work: Variational principles and Sobolev-type estimates for generalized interpolation on a Riemannian manifold
cites work: Q3482292
cites work: Generalized Hermite interpolation and positive definite kernels on a Riemannian manifold
cites work: Q3995224
cites work: Error estimates and condition numbers for radial basis function interpolation
cites work: Approximation by radial basis functions with finitely many centers
cites work: Q4699405
cites work: Improved error bounds for scattered data interpolation by radial basis functions
cites work: Operators on radial functions
Recommended article: Q4699405
Recommended article: Q3201634
Recommended article: Q3611743
Recommended article: Radial Basis Functions
Recommended article: Q4719288
Recommended article: Q3001713
Recommended article: Q4783882
Recommended article: Radial basis functions as approximate particular solutions: Review of recent progress
Recommended article: Characterization and construction of radial basis functions.
description: scientific article; zbMATH DE number 1533133