I'm familiar with [Newman's](https://projecteuclid.org/euclid.mmj/1028999029) rational approximation of the absolute value function via rational functions. Are there other explicit functions that approximate $|x|$ with exponential error? I was under the impression that Chebhyshev polynomials also can give such good approximations to $|x|$ but I could not find this proved anywhere. Can someone point to a book/paper where I can read more about this?



approximating the $|x|$ function
Denote the minimal approximation error to the function $f(x)=|x|$ in the uniform norm on $[-1,1]$ by 
$$
E\_{mn}(f,[-1,1])=\inf\_{r\in\mathcal{R}\_{mn}}\|f-r\|\_{\infty,[-1,1]},
$$
where $\mathcal{R}\_{mn}$ denotes the set of rational functions with numerators of degree at most $m$ and denominators of degree at most $n$.


For rational approximation, it is known, by a result of H.R. Stahl (1992), that
$$
\lim\_{n\to\infty}e^{\pi\sqrt{n}}E\_{nn}(f,[-1,1])=8,
$$
while for polynomial approximation, by a classical result of S.N. Bernstein, there exists a constant $\alpha$ such that
$$
\lim\_{n\to\infty}nE\_{n0}(f,[-1,1])=\alpha,
$$
(no explicit expression for the so-called Bernstein constant $0.27<\alpha<0.28$ is known). 


In particular, the rate of approximation by (any) polynomials can only decrease like $O(1/n)$. 


A possible reference is



> 
> Petrushev, P. P.; Popov, V. A. Rational approximation of real
>  functions. Encyclopedia of Mathematics and its Applications, 28.
>  Cambridge University Press, Cambridge, 1987.
> 
> 
> 


