Stable Moving Least-Squares

It is a common procedure for scattered data approximation to use local polynomial fitting in the least-squares sense. An important instance is the Moving Least-Squares where the corresponding weights of the data site vary smoothly, resulting in a smooth approximation. In this paper we build upon the techniques presented by Wendland and present a somewhat simpler error analysis of the MLS approximation. Then, we show by example that the N factor, which appears in the bound on the Lebesgue constant in [Holger Wendland, Local polynomial reproduction and moving least squares approximation, IMA J. Numer. Anal. 21 (1) (2001) 285–300], where NN is the number of points used in the approximation, can be realized. Hence, we devise a method for choosing the weights smoothly so that the corresponding Lebesgue constant can be bounded independently of NN. This is done by employing Voronoi weights. We conclude with some numerical examples exhibiting the effectiveness of the suggested method for highly irregular data sites.