Polyharmonic and Related Kernels on Manifolds: Interpolation and Approximation
Polyharmonic and Related Kernels on Manifolds: Interpolation and Approximation
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流形上的多调和相关核：插值和逼近
DOI：
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发表时间：
2010
影响因子：
3
通讯作者：
J. Ward
中科院分区：
文献类型：
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作者：
T. Hangelbroek;F. Narcowich;J. Ward
这篇文章致力于发展一个理论，有效的核插值和近似的一般设置。对于广泛的一类紧致连通的C∞黎曼流形，包括球面和SO（3）的重要情形，利用微分几何和李群的技巧，我们建立了作为某些偏微分算子的基本解得到的核生成的拉格朗日函数是一致有界的，并且以代数速率（在某些情况下，是指数速率）从其中心衰减。一个直接的推论是，相应的Lebesgue常数插值以及L2最小化一致有界的一个常数，其唯一的依赖于一组数据的网站是反映在网格比，这措施的数据的均匀性。这里考虑的核包括球面上的限制曲面样条，以及SO（3）的曲面样条，两者都具有计算上可实现的基本封闭形式表示。除了获得有界Lebesgue常数在此设置中，我们还建立了一个“零引理”的域紧黎曼流形，一个持有尽可能多的一般性作为相应的欧几里德零引理（Lipschitz域满足内部锥条件）与常数，清楚地表明边界的几何形状的影响（通过锥参数），以及黎曼度量。
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 L2 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.