If $f$ is an even function in [-1,1], how do I show that it can be approximated by sequence of polynomials of $p_n(x^2)$?
[The question is followed by a hint saying we could consider $f(\sqrt{x})$ in $[0,1]$. I am not really sure how I can use this.]
I can see that since $f$ is even, we can take the $[0,1]$ interval and show that there exists a polynomial that can be approximated to $f$ and then do the same thing with $[-1,0]$, and show that the polynomial is also even and that an even polynomial would only have even powered $x$ terms. But I'm not sure if that is a correct way.
The core of the argument : If you have a function f in [0,1]. Let's define $g(x) = f(\sqrt{x})$
$g$ can be approximated by a polynom $p_n(x)$. So $f(\sqrt{x})$ can be approximated by $p_n(x)$, hence $f(y)$ can be approximated by $p_n(y^2)$ (because $x\mapsto \sqrt{x}$ is bijective over [0,1])