Remember that a function $f$ is called even if $f(−x) = f(x)$ and odd if $f(−x) = −f(x)$ for all $x$ in its domain. Let $w$ be an even weight function on the interval $(−a, a)$ and ${ϕ_0, ϕ_1, .., ϕ_n}$ be a system of orthogonal polynomials on $(−a, a)$ with respect to $w$, constructed using the Gram-Schmidt Orthogonalization.
(b) Let $f : [−a, a] \to \mathbb{R}$ and $p_n(x) = γ_0ϕ_0(x) + . . . + γnϕ_n(x)$ its best polynomial approximation of degree $n$ with respect to the weighted $2$-norm. Show that if $f$ is an even function, then all the odd coefficients $γ^2_j−1 $ are zero and if $f$ is an odd function, then all the even coefficients $γ^2_j$ are zero.