It is known that if $f\in C[-1, 1]$ is odd/even, then the best polynomial approximation (in $L^{\infty}$ norm) of degree $n$, denoted by $p_n$, must also be odd/even. This has been asked on MSE before.
In particular, if $f\in C[-1,1]$ is odd, then all best polynomial approximations $p_n$ vanish at $0$. And I am wondering about the converse claim:
If all best polynomial approximation $p_n$ to $f\in C[-1,1]$ vanishes at $0$, can we say $f$ is odd?
It seems not simple to answer, I am wondering if there is a proof or disproof for it.
If the approximation is in $L^2$ sense, then the zeros of orthogonal polynomials have the interlacing property, so it seems easier to deal with.