Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 77 / 5 / iv

Interpret polynomial reproduction as a guarantee for every polynomial of degree at most $d$, with the planar grid coordinates fixed by the affine-exact refinement. Constants are unchanged because every stencil sums to one. Linear height functions are unchanged because each stencil's weighted position is its refined grid location. Therefore the Loop subdivision limit reproduces all affine polynomials exactly.

Quadratic data already disprove the next degree. Use extruded data $g_j=j^2$ on a unit-spaced transverse row coordinate $y$. At spacing $h$, both even and odd rules from the induced subdivision mask give the value of $y^2$ at the refined location plus $h^2/4$. For the even stencil this follows fromand for the odd stencil from the average of the values at $y-h/2$ and $y+h/2$. Constants are reproduced, so the accumulated vertical displacement over all refinement levels isThe limit is $y^2+1/3$, not the original quadratic graph. HenceThe word “every” matters: some special higher-degree bivariate polynomials have cancelling second moments and can be reproduced. The conclusion concerns the entire polynomial space, which already fails at degree two.