论文标题
相同间隔的点最佳对于布朗桥核插值
Equally spaced points are optimal for Brownian Bridge kernel interpolation
论文作者
论文摘要
在本文中，我们展示了如何使用样条理论的想法来构建局部基础，以构建一般迭代的布朗桥核的翻译空间，$ k_ {β，\ varepsilon} $ for $β\ in \ mathbb {n} $，$ \ varepsilon \ geq 0 $。在简单的情况下，$β= 1 $，我们为相应的拉格朗日提供了一个明确的公式，这使我们能够解决插值问题而不反转任何线性系统。 我们使用此基础来证明与$ k_ {1，\ varepsilon} $进行插值均匀稳定，即，lebesgue常数与插值点的位置独立于数字界定，并且相同间隔的点是相关功率功能的唯一最小化点，因此是错误的功能，因此是错误的。在此推导中，我们研究了形状参数$ \ varepsilon> 0 $的作用，并讨论了其对这些误差和稳定性界限的影响。 本文讨论的一些想法可以扩展到更一般的绿色内核。
In this paper we show how ideas from spline theory can be used to construct a local basis for the space of translates of a general iterated Brownian Bridge kernel $k_{β,\varepsilon}$ for $β\in\mathbb{N}$, $\varepsilon\geq 0$. In the simple case $β=1$, we derive an explicit formula for the corresponding Lagrange basis, which allows us to solve interpolation problems without inverting any linear system. We use this basis to prove that interpolation with $k_{1,\varepsilon}$ is uniformly stable, i.e., the Lebesgue constant is bounded independently of the number an location of the interpolation points, and that equally spaced points are the unique minimizers of the associated power function, and are thus error optimal. In this derivation, we investigate the role of the shape parameter $\varepsilon>0$, and discuss its effect on these error and stability bounds. Some of the ideas discussed in this paper could be extended to more general Green kernels.
