薄板样条径向基函数最优恢复的初步证明

Moran Kim, Chohong Min
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引用次数: 1

摘要

在许多实际应用中,我们面临着在一些数据点上采样的未知函数的重建问题。在无限多种可能的重构中,当数据点在r2中采样时,已知薄板样条插值是Beppo-Levi半范数中振荡最小的重构。支持这一论点的传统证据相当冗长和复杂,使学生和研究人员难以理解。在本文中,我们介绍了最优重构的一个简单而简短的证明。我们的证明是独一无二的,只需要初等的数学背景。
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AN ELEMENTARY PROOF OF THE OPTIMAL RECOVERY OF THE THIN PLATE SPLINE RADIAL BASIS FUNCTION
In many practical applications, we face the problem of reconstruction of an un- known function sampled at some data points. Among infinitely many possible reconstructions, the thin plate spline interpolation is known to be the least oscillatory one in the Beppo-Levi semi norm, when the data points are sampled in R 2 . The traditional proofs supporting the argu- ment are quite lengthy and complicated, keeping students and researchers off its understanding. In this article, we introduce a simple and short proof for the optimal reconstruction. Our proof is unique and reguires only elementary mathematical background.
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