Past exam of the Probability, statistics and options II course of the University of Oxford / 2011 / P / P3 / c

For odd $u$, define $q(x)=-p_n(-x)$. It is another polynomial in ${\Pi }_n$. Changing variables $x\mapsto -x$, and using evenness of the weight $x^2$, gives $\Vert u-q\Vert =\Vert u-p_n\Vert$. Thus $q$ is also a best polynomial. Uniqueness from part (b) gives $p_n(x)=-p_n(-x)$, provingEquivalently, the even component of a candidate is orthogonal to the odd residual and can only increase the squared error.