= Parity preservation by polynomial least-squares projection
On a symmetric interval with an even positive weight, the best <least-squares polynomial> approximation to an <odd> function is odd, and the approximation to an <even> function is even. Indeed, the even and odd <polynomial> subspaces are <orthogonal>. For odd $u$, write a candidate as $p=p_o+p_e$. Then
$$
\|u-p\|^2=\|u-p_o\|^2+\|p_e\|^2.
$$
Deleting a nonzero even component strictly reduces the error, so the minimizer has $p_e=0$. The same argument exchanges the parities for even $u$. This reduces the <normal equations> to the matching parity block and avoids computing coefficients known in advance to vanish.
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