欧几里得空间中采样函数逼近与插值的补偿凸性方法：理论基础
Compensated Convexity Methods for Approximations and Interpolations of Sampled Functions in Euclidean Spaces: Theoretical Foundations
摘要：
利用补偿凸变换，给出了$\mathbb{R}^n$中紧集和有界开集补集上采样有界一致连续函数的Lipschitz连续和$C^{1,1}$几何逼近和插值方法.给出了有界一致连续函数、Lipschitz函数和$C^{1,1}$函数的逼近误差估计。我们还证明了我们的近似方法，这是微分和积分自由，不敏感的样本类型，是稳定的样本之间的Hausdorff距离。
英文摘要：
We introduce Lipschitz continuous and $C^{1,1}$ geometric approximation and interpolation methods for sampled bounded uniformly continuous functions over compact sets and over complements of bounded open sets in $\mathbb{R}^n$ by using compensated convex transforms. Error estimates are provided for the approximations of bounded uniformly continuous functions, of Lipschitz functions, and of $C^{1,1}$ functions. We also prove that our approximation methods, which are differentiation and integration free and not sensitive to sample type, are stable with respect to the Hausdorff distance between samples.
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