曲面样条插值的惩罚勒贝格常数
The Penalized Lebesgue Constant for Surface Spline Interpolation
AI总结：
针对非均匀分布数据，利用类似流形上准均匀数据插值稳定性的技术，证明了曲面样条插值在Rd上具有更强的局部稳定性，并得到拉格朗日函数快速衰减的点态估计，结合勒贝格引理表明其收敛率与局部逼近方案相同。
摘要：
涉及准均匀分布散乱数据逼近的问题已通过RBF方法处理数十年。处理具有空间变化密度的数据尚未得到同等强度的研究，且远未得到充分理解。本文考虑非均匀排列数据的曲面样条插值（一种流行的RBF插值）的稳定性。使用类似于Hangelbroek、Narcowich和Ward最近用于证明流形上准均匀数据插值稳定性的技术，我们表明Rd上的曲面样条插值是稳定的，但具有更强的局部意义。我们还获得了点态估计，表明拉格朗日函数衰减非常快，且速率由数据点的局部间距决定。这些结果结合勒贝格引理表明，曲面样条插值享有与DeVore和Ron最近开发的局部逼近方案相同的收敛率。
英文摘要：
Problems involving approximation from scattered data where data is arranged quasi-uniformly have been treated by RBF methods for decades. Treating data with spatially varying density has not been investigated with the same intensity, and is far less well understood. In this article we consider the stability of surface spline interpolation (a popular type of RBF interpolation) for data with nonuniform arrangements. Using techniques similar to those recently employed by Hangelbroek, Narcowich and Ward to demonstrate the stability of interpolation from quasi-uniform data on manifolds, we show that surface spline interpolation on R^d is stable, but in a stronger, local sense. We also obtain pointwise estimates showing that the Lagrange function decays very rapidly, and at a rate determined by the local spacing of datasites. These results, in conjunction with a Lebesgue lemma, show that surface spline interpolation enjoys the same rates of convergence as those of the local approximation schemes recently developed by DeVore and Ron.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
