The penalized Lebesgue constant for surface spline interpolation
The penalized Lebesgue constant for surface spline interpolation
复制标题
曲面样条插值的惩罚勒贝格常数
DOI：
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发表时间：
2009
期刊：
影响因子：
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通讯作者：
T. Hangelbroek
中科院分区：
文献类型：
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作者：
T. Hangelbroek
涉及数据准均匀排列的散乱数据的近似问题，几十年来一直由径向基函数法处理。用空间变化的密度处理数据还没有以同样的强度被研究过，也远没有被很好地理解。在这篇文章中，我们考虑了非均匀排列数据的曲面样条插值法(一种流行的径向基函数插值法)的稳定性。利用类似于Hangelbroek，Narcowich和Ward最近所使用的技巧来证明流形上的拟一致数据的插值法的稳定性，我们证明了R^d上的曲面样条插值法是稳定的，但具有更强的局部意义。我们还得到了逐点估计，表明拉格朗日函数衰减非常快，并且以由数据点的局部间距决定的速度衰减。结合勒贝格引理，这些结果表明，曲面样条插值法具有与DeVore和Ron新近提出的局部逼近格式相同的收敛速度。
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.