Local polynomial reproduction and moving least squares approximation

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