变尺度不连续权函数的移动最小二乘逼近
Moving Least Squares Approximation using Variably Scaled Discontinuous Weight Function
摘要：
在图像重建、信号处理、最优控制问题、界面问题、工程应用等领域中，经常会遇到具有间断的函数，因此对这些函数进行精确的逼近和插值是非常重要的。在本文中，我们设计了一个移动最小二乘方法的离散数据近似，其中包括不连续的权函数。这个想法是为了控制数据站点对近似的影响，不仅是关于它们与评估点的距离，而且关于基础函数的不连续性。在适当的{ \it分段}索伯列夫空间上给出了误差估计.数值实验结果与理论推导的收敛速度一致。
英文摘要：
Functions with discontinuities appear in many applications such as image reconstruction, signal processing, optimal control problems, interface problems, engineering applications and so on. Accurate approximation and interpolation of these functions are therefore of great importance. In this paper, we design a moving least-squares approach for scattered data approximation that incorporates the discontinuities in the weight functions. The idea is to control the influence of the data sites on the approximant, not only with regards to their distance from the evaluation point, but also with respect to the discontinuity of the underlying function. We also provide an error estimate on a suitable {\it piecewise} Sobolev Space. The numerical experiments are in compliance with the convergence rate derived theoretically.
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