= Polynomial degree preservation under Chaikin subdivision
{title2=$n=0,1,2$}
For uniformly spaced abscissae, quarter and three-quarter averages applied to constant or affine samples leave the same polynomial. For $P(x)=ax^2+bx+c$ on spacing $h$, both new subgrids lie on $P(x)+3ah^2/16$. For degree $n\geq3$, their candidate polynomial formulas differ with leading term $(h^3/32)P'''$, which has a nonzero highest-degree coefficient. Thus preservation of polynomial degree for arbitrary sufficiently long data sequences holds exactly for degrees zero, one and two, despite quadratic values changing.
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