Moving least squares

Remarks

The minimiser is unique when the points $x_i$ with $\theta(\norm{x - x_i}) > 0$ include a unisolvent set for $\Pi_m$. In general $\hat f$ only approximates the data; singular weights, such as in the example below, make it interpolate. If the data come from a polynomial in $\Pi_m$, then $\hat f$ reproduces that polynomial.

Example: $m = 0$ and $\theta(r) = r^{-p}$ for $r > 0$, with $p > 0$, at a point $x$ that is not a data point. Then $p_x$ is the constant $\sum_i \theta_i f_i / \sum_i \theta_i$, where $\theta_i = \theta(\norm{x - x_i})$. Extended by the data values at the data points, this is inverse distance weighting.

Non-example: with constant weights $\theta = 1$ the fit is an ordinary least squares fit. It gives the same polynomial $p_x = p$ at every $x$, so the result is one global polynomial. The dependence of the weights on $x$, the "moving" feature, is lost.