The paper deals with the problem of extrapolating data derived from sampling a \(C^m\) function at scattered sites on a Lipschitz region \(\varOmega \) in \(\mathbb R^d\) to points outside of \(\varOmega \) in a computationally efficient way. While extrapolation problems go back to Whitney and many such problems have had successful theoretical resolutions, practical, computationally efficient implementations seem to be lacking. The goal here is to provide one way of obtaining such a method in a solid mathematical framework. The method utilized is a novel two-step moving least squares procedure (MLS) where the second step incorporates an error term obtained from the first MLS step. While the utility of the extrapolation degrades as a function of the distance to the boundary of \(\varOmega \), the method gives rise to improved meshfree approximation error estimates when using the local Lagrange kernels related to certain radial basis functions.

用外推的数据的问题的纸优惠从采样派生\（C ^ M \）功能在上一个李普希茨区域散射位点\（\ varOmega \）在\（\ mathbb R 1 d \）以外的点\（ \ varOmega \）以有效的计算方式。虽然外推问题可以追溯到惠特尼，并且许多此类问题都有成功的理论解决方案，但似乎缺少实用，计算效率高的实现。这里的目标是提供一种在固体数学框架中获得这种方法的方法。所使用的方法是新颖的两步移动最小二乘法（MLS），其中第二步包含从第一MLS步骤获得的误差项。虽然外推的实用性随与\（\ varOmega \）边界的距离而变差，但是当使用与某些径向基函数相关的局部Lagrange核时，该方法将产生改进的无网格近似误差估计。