Past exam of the options II course of the University of Oxford / 2011 / P / P3 / c

Define the reflected polynomial $\tilde{p}(x)=-p(-x)$. It belongs to ${\Pi }_n$. Since $u$ is odd, a change of variable and the evenness of the weight giveThus $\tilde{p}$ is another polynomial least-squares approximation. By uniqueness, $\tilde{p}=p$, soThe conclusion includes the zero polynomial, which is both odd and even.