球面上的薄板样条
Thinplate Splines on the Sphere
摘要：
本文给出了球面上薄板样条插值的半再生核的显式封闭形式。${\mathbb R}^d$的多谐或薄板样条由Duchon引入，并已成为无数应用中广泛使用的工具。${\mathbb S}^{d-1}$的类似物是球面的薄板样条。Wahba在20世纪80年代初首次讨论了这个话题，讨论了${\mathbb S}^2$的案例。Wahba将相关的半再生核表示为无穷级数。这些半再生核在解决相关的样条插值和光滑问题的表达式中起着核心作用。本文的主要目的是给出半再生核的递推公式，并利用该递推公式得到许多核的显式封闭形式表达式。在许多情况下，闭合形式的表达式的计算速度要比级数展开式快得多。这将增强在计算中使用这些薄板样条用于球体的实用性。
英文摘要：
In this paper we give explicit closed forms for the semi-reproducing kernels associated with thinplate spline interpolation on the sphere. Polyharmonic or thinplate splines for ${\mathbb R}^d$ were introduced by Duchon and have become a widely used tool in myriad applications. The analogues for ${\mathbb S}^{d-1}$ are the thin plate splines for the sphere. The topic was first discussed by Wahba in the early 1980's, for the ${\mathbb S}^2$ case. Wahba presented the associated semi-reproducing kernels as infinite series. These semi-reproducing kernels play a central role in expressions for the solution of the associated spline interpolation and smoothing problems. The main aims of the current paper are to give a recurrence for the semi-reproducing kernels, and also to use the recurrence to obtain explicit closed form expressions for many of these kernels. The closed form expressions will in many cases be significantly faster to evaluate than the series expansions. This will enhance the practicality of using these thinplate splines for the sphere in computations.
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