Jackson's theorem states that if a function $f$ satisfies $$|f(x)-f(y)|\le M|x-y|,$$ then it can be uniformly approximated at the rate $$|f(x)-P_n(x)|\le \frac{M}{2(n+1)} = O(1/n),$$ where $P_n(x)$ is the best uniform approximation polynomial of $f(x)$ on $[-1,1]$.
My question is when $f(x) = |x|$, can the convergence rate $O(1/n)$ be exceeded? In other words, can one construct a series of polynomials $Q_n(x)$ such that $$|f(x)-Q_n(x)| < O(1/n),$$ (such as $O(1/n^2)$)?
I tried to use the truncated Taylor expansion of $$(1-(1-x^2))^{1/2} = 1-\sum_{n=1}^{\infty}\frac{(2n-3)!!}{2^nn!}(1-x^2)^n,$$ but it seems this method can only obtain an $O(1/\sqrt{n})$ rate, even worse than $O(1/n)$. The Bernstein approximation of $|x|$ also seems to be $O(1/\sqrt{n})$. I have no idea of how to construct alternative good approximations.