The approximation power of moving least-squares

A general method for near-best approximations to functionals on $\mathbb{R}^d$, 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 is near-best in the sense that the local error is bounded in terms of the error of a local best polynomial approximation. The interpolation approximation in $\mathbb{R}^d$ is shown to be a $C^\infty$ function, and an approximation order result is proven for quasi-uniform sets of data points.