Local polynomial reproduction and moving least squares approximation

Local polynomial reproduction is a key ingredient in providing error estimates for several approximation methods. To bound the Lebesgue constants is a hard task especially in a multivariate setting. We provide a result which allows us to bound the Lebesgue constants uniformly and independently of the space dimension by oversampling. We get explicit and small bounds for the Lebesgue constants. Moreover, we use these results to establish error estimates for the moving least squares approximation scheme, also with special emphasis on the involved constants. We discuss the numerical treatment of the method and analyse its effort. Finally, we give large scale examples.