We study uniform approximation of differentiable or analytic functions of one or several variables on a compact set K by a sequence of discrete least squares polynomials. In particular, if K satisfies a Markov inequality and we use point evaluations on standard discretization grids with the number of points growing polynomially in the degree, these polynomials provide nearly optimal approximant...

We describe a new method for surface reconstruction based on unorganized point clouds without normals. We also present a new algorithm for refining the inital triangulation. The output of the method is a refined triangular mesh with points on the moving least squares surface of the original point cloud.

We introduce moving least squares approximation as an approximation scheme on the sphere. We prove error estimates and approximation orders. Finally, we show certain numerical results. x1. Introduction Recently, approximation on the sphere has become important because of its obvious applications to Meteorology, Oceanography and Geoscience and Geo-engineering in general. Over the last years seve...

Given integers N n 0, we consider the least squares problem of nding the vector of coeecients ~ P with respect to a polynomial basis P N j=0 wn(zj) 2 jf(zj) ? P (zj)j 2. Here a perturbation of the values f(zj) leads to some perturbation of the coeecient vector ~ P. We denote by n the maximal magniication of relative errors, i.e., the Euclidean condition number of the underlying weighted Vanderm...

We describe two experiments recently conducted with the approximate moving least squares (MLS) approximation method. On the one hand, the NFFT library of Kunis, Potts, and Steidl is coupled with the approximate MLS method to obtain a fast and accurate multivariate approximation method. The second experiment uses approximate MLS approximation in combination with a multilevel approximation algori...

Article history: Available online 29 March 2012

We present an efficient and reliable algorithm for discrete least squares approximation of a real-valued function given at arbitrary distinct nodes in [0, 2tt) by trigonometric polynomials. The algorithm is based on a scheme for the solution of an inverse eigenproblem for unitary Hessenberg matrices, and requires only O(mn) arithmetic operations as compared with 0(mn ) operations needed for alg...

A general method for near-best approximations to functionals on Rd, using scattered-data information is discussed. The method is actually the moving least-squares method, presented by the Backus-Gilbert approach. It is shown that the method works very well for interpolation, smoothing and derivatives’ approximations. For the interpolation problem this approach gives Mclain’s method. The method ...

Numerical solutions obtained by the Meshless Local Petrov-Galerkin (MLPG) method are presented for two dimensional steady-state heat conduction problems. The MLPG method is a truly meshless approach, and neither the nodal connectivity nor the background mesh is required for solving the initial-boundary-value problem. The penalty method is adopted to efficiently enforce the essential boundary co...