ON GENERALIZED MOVING LEAST SQUARES AND DIFFUSE DERIVATIVES DAVOUD MIRZAEI † , ROBERT SCHABACK ‡,* , MEHDI DEHGHAN § Abstract. The Moving Least Squares (MLS) method provides an approxima- tion ˆ u of a function u based solely on values u(x j ) of u on scattered “meshless” nodes x j . Derivatives of u are usually approximated by derivatives of ˆ u. In contrast to this, we directly estimate derivatives of u from the data, without any detour via derivatives of ˆ u. This is a generalized Moving Least Squares technique, and we prove that it produces diffuse derivatives as introduced by Nyroles et. al. in 1992. Consequently, these turn out to be efficient direct estimates of the true derivatives, without anything “diffuse” about them, and we prove optimal rates of convergence towards the true derivatives. Numerical examples confirm this, and we finally show how the use of shifted and scaled polynomials as basis functions in the generalized and standard MLS approxi- mation stabilizes the algorithm. Keywords: Moving least squares (MLS) approximation; Local polynomial re- production; Full derivative; Diffuse derivative; Shifted scaled polynomials ba- sis; Meshless methods. 1. Introduction The Moving Least Squares (MLS) approximation has been introduced by [11] inspired by the pioneering work of [19] to approximate surfaces in multidimensional spaces. The MLS approximates the value u(x) of an unknown function u from given data u(x 1 ),...,u(x N ) at nodes x 1 ,...,x N near x by a value ˆ u(x)= N  j=1 a j (x)u(x j ) ≈ u(x), where the functions a j (x) are called shape functions. In the sense of [6], this is a meshless method, because it writes an approximate solution entirely in terms Date : August 12, 2011 . * Corresponding author. † Department of Mathematics, University of Isfahan, 81746-73441, Isfahan, Iran. d mirzaei@aut.ac.ir. ‡ Institut f¨ ur Numerische und Angewandte Mathematik, Universit¨at G¨ ottingen, Lotzestraße 16-18, D–37073 G¨ ottingen, Germany. schaback@math.uni-goettingen.de. § Department of Applied Mathematics, Faculty of Mathematics and Computer Science, Amirk- abir University of Technology, No. 424, Hafez Ave.,15914, Tehran, Iran. mdehghan@aut.ac.ir. 1