嵌入子流形上具有限制正定核的散乱数据插值：Sobolev误差估计
Scattered Data Interpolation on Embedded Submanifolds with Restricted Positive Definite Kernels: Sobolev Error Estimates
AI总结：
研究通过限制正定核（如径向基函数）在光滑紧致嵌入子流形上构造核插值，并给出本征空间的完整刻画及Sobolev型误差估计。
摘要：
本文研究了流形上核插值的逼近性质。我们考虑的核是通过将$\R^d$上的正定核（如径向基函数）限制到光滑紧致嵌入子流形$\M\subset \R^d$上得到的。对于具有有限光滑性的限制核，我们给出了$\M$上本征空间的完整刻画。在此及一些初步准备之后，我们给出了插值问题的Sobolev型误差估计。数值结果验证了理论，包括嵌入$\R^3$的一维曲线和二维环面。
英文摘要：
In this paper we investigate the approximation properties of kernel interpolants on manifolds. The kernels we consider will be obtained by the restriction of positive definite kernels on $\R^d$, such as radial basis functions (RBFs), to a smooth, compact embedded submanifold $\M\subset \R^d$. For restricted kernels having finite smoothness, we provide a complete characterization of the native space on $\M$. After this and some preliminary setup, we present Sobolev-type error estimates for the interpolation problem. Numerical results verifying the theory are also presented for a one-dimensional curve embedded in $\R^3$ and a two-dimensional torus.
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