 
Summary: ERROR ESTIMATES IN SOBOLEV SPACES FOR MOVING LEAST
SQUARE APPROXIMATIONS
MAR A G. ARMENTANOy
Abstract. The aim of this paper is to obtain error estimatesfor movingleast square approxima
tionsin IR
N. We provethat, underappropriatehypothesisonthe weightfunctionandthe distribution
of points, the method produces optimal order error estimates in L1 and L2 for the approximations
of the function and its rst derivatives. These estimates are important in the analysis of Galerkin
approximations based on the moving least square method. In particular, our results provides error
estimates, optimal in order and regularity, for second order coercive problems.
Key words. error estimates, moving least square, meshless method
AMS subject classi cations. 65N15, 65N30, 65D10.
1. Introduction. The moving least square method, MLS ( 10], 8]), has been
used for the numerical solution of di erential equations in several papers ( 11], 2],
3], 1]). For this kind of application it is very important to obtain error estimates for
the function and its derivatives which are not known up to now for the Ndimensional
case.
In 9] Levin analyzes the MLS method for a particular weight function obtaining
error estimates in the uniform norm for the approximation of a regular function in N
dimensions. However, he does not obtain error estimates for the derivatives. In 1],
