Polyharmonic and Related Kernels on Manifolds: Interpolation and Approximation

This article is devoted to developing a theory for effective kernel interpolation and approximation in a general setting. For a wide class of compact, connected C ∞ Riemannian manifolds, including the important cases of spheres and SO (3), and using techniques involving differential geometry and Lie...

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Vydáno v:Foundations of computational mathematics Ročník 12; číslo 5; s. 625 - 670
Hlavní autoři: Hangelbroek, T., Narcowich, F. J., Ward, J. D.
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York Springer-Verlag 01.10.2012
Springer Nature B.V
Témata:
Applications of Mathematics
Computational mathematics
Computer Science
Constants
Differential geometry
Economics
Interpolation
Kernels
Linear and Multilinear Algebras
Manifolds
Math Applications in Computer Science
Mathematical analysis
Mathematical models
Mathematics
Mathematics and Statistics
Matrix Theory
Numerical Analysis
Splines
Topological manifolds
Manifold
Positive definite kernels
46E35
41A63
46E22
Sobolev spaces
Least squares approximation
41A05
ISSN:1615-3375, 1615-3383
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Shrnutí:This article is devoted to developing a theory for effective kernel interpolation and approximation in a general setting. For a wide class of compact, connected C ∞ Riemannian manifolds, including the important cases of spheres and SO (3), and using techniques involving differential geometry and Lie groups, we establish that the kernels obtained as fundamental solutions of certain partial differential operators generate Lagrange functions that are uniformly bounded and decay away from their center at an algebraic rate, and in certain cases, an exponential rate. An immediate corollary is that the corresponding Lebesgue constants for interpolation as well as for L 2 minimization are uniformly bounded with a constant whose only dependence on the set of data sites is reflected in the mesh ratio , which measures the uniformity of the data. The kernels considered here include the restricted surface splines on spheres, as well as surface splines for SO (3), both of which have elementary closed-form representations that are computationally implementable. In addition to obtaining bounded Lebesgue constants in this setting, we also establish a “zeros lemma” for domains on compact Riemannian manifolds—one that holds in as much generality as the corresponding Euclidean zeros lemma (on Lipschitz domains satisfying interior cone conditions) with constants that clearly demonstrate the influence of the geometry of the boundary (via cone parameters) as well as that of the Riemannian metric.
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ISSN:1615-3375
1615-3383
DOI:10.1007/s10208-011-9113-5