Error estimates in Sobolev spaces for moving least square approximations

The aim of this paper is to obtain error estimates for moving least square (MLS) approximations in R-N. We prove that, under appropriate hypotheses on the weight function and the distribution of points, the method produces optimal order error estimates in L-infinity and L-2 for the approximations of the function and its first derivatives. These estimates are important in the analysis of Galerkin approximations based on the MLS method. In particular, our results provide error estimates, optimal in order and regularity, for second order coercive problems.