Moving least squares

Moving least squares is a method of reconstructing continuous functions from a set of unorganized point samples via the calculation of a weighted least squares measure biased towards the region around the point at which the reconstructed value is requested.
In computer graphics, the moving least squares method is useful for reconstructing a surface from a set of points. Often it is used to create a 3D surface from a point cloud through either downsampling or upsampling.
Definition.
Consider a function formula_1 and a set of sample points formula_2. Then, the moving least square approximation of degree formula_3 at the point formula_4 is formula_5 where formula_6 minimizes the weighted least-square error 
over all polynomials formula_8 of degree formula_3 in formula_10. formula_11 is the weight and it tends to zero as formula_12.
In the example formula_13. The smooth interpolator of "order 3" is a quadratic interpolator.