Polyharmonic and related kernels on manifolds: interpolation and approximation

DOI10.1007/s10208-011-9113-5zbMath1259.41005arXiv1012.4852OpenAlexW2056493577MaRDI QIDQ1928234

Francis J. Narcowich, Joseph D. Ward, Thomas Hangelbroek

Publication date: 2 January 2013

Published in: Foundations of Computational Mathematics (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1012.4852

zbMATH Keywords

manifoldSobolev spacespositive definite kernelsleast squares approximation

Mathematics Subject Classification ID

Sobolev spaces and other spaces of ``smooth functions, embedding theorems, trace theorems (46E35) Multidimensional problems (41A63) Interpolation in approximation theory (41A05) Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) (46E22)

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Cites Work

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• Surface spline approximation on \(SO(3)\)
• Polyharmonic approximation on the sphere
• Sobolev error estimates and a Bernstein inequality for scattered data interpolation via radial basis functions
• Approximation power of RBFs and their associated SBFs: a connection
• Bounding homotopy types by geometry
• Boundary layer methods for Lipschitz domains in Riemannian manifolds
• Variational principles and Sobolev-type estimates for generalized interpolation on a Riemannian manifold
• Estimates of \(n\)-widths of Sobolev's classes on compact globally symmetric spaces of rank one.
• A note on the local stability of translates of radial basis functions
• Geometric and transformational properties of Lipschitz domains, Semmes-Kenig-Toro domains, and other classes of finite perimeter domains
• Multivariate cardinal interpolation with radial-basis functions
• Scattered Data Interpolation on Embedded Submanifolds with Restricted Positive Definite Kernels: Sobolev Error Estimates
• Kernel Approximation on Manifolds I: Bounding the Lebesgue Constant
• Kernel Approximation on Manifolds II: The $L_{\infty}$ Norm of the $L_2$ Projector
• $L^p$ Bernstein estimates and approximation by spherical basis functions
• Estimates of n -widths of Besov classes on two-point homogeneous manifolds
• Sur l’erreur d’interpolation des fonctions de plusieurs variables par les $D^m$-splines
• Local stability of translates of polyharmonic splines in even space dimension
• A sampling theorem on homogeneous manifolds
• Sobolev bounds on functions with scattered zeros, with applications to radial basis function surface fitting
• The Mathematical Theory of Finite Element Methods