In moving least squares (see for example Wiki), you move your window throughout the domain. I have created a graph to illustrate my point further here.
Let $f(x)$ be the function of interest, and $g_1(x)$ and $g_2(x)$ are the least squares produced around the point $x_1$ and $x_2$, respectively. The shaded area in blue and green indicates the neighbors of $x_1$ and $x_2$ that are included in their corresponding least squares problems.
Let's say I have a point $x_m$ that lies in the intersection between two windows. I want to somehow evaluate how much of a difference the value from each least squares approximation would give me. With referring to the graph, I want to somehow bound:
$$ g_1(x_m) - g_2(x_m) $$ in some sense. Any suggestions with how I can start? I'm using a 1D example here to easily illustrate my points. Later on, how would I go and generalize it to 3D?