I want to find a series of polynomials with even powers $P_n : [0,1]\rightarrow\mathbb{C}$ such that is uniformly converging to the function $f(x)=x$.
I got a hint that says to first find a series $Q_n(x)$ that converges to $g(x)=\sqrt x$ and then set $P_n(x)=Q_n(x^2)$
I tried using fourier series but i got stuck when I was trying to solve the integral $$\hat g (n)= \frac{1}{2 \pi} \int_0^1 \sqrt x e^{-inx}$$ since this integral doesn't have a solution.