THE APPROXIMATION POWER OF MOVING LEAST-SQUARES

A general method for near-best approximations to functionals on R-d, u sing scattered-data information is discussed. The method is actually t he moving least-squares method, presented by the Backus-Gilbert approa ch. It is shown that the method works very well for interpolation, smo othing and derivatives' approximations. For the interpolation problem this approach gives Mclain's method. The method is near-best in the se nse that the local error is bounded in terms of the error of a local b est polynomial approximation. The interpolation approximation in R-d i s shown to be a C-infinity function, and an approximation order result is proven for quasi-uniform sets of data points.