approximating the $|x|$ function I'm familiar with Newman's 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?
 A: The Chebyshev series for $|x|$ on the interval $[-1,1]$ corresponds to the Fourier series for $|\cos(t)|$ on $[0, \pi]$.  This Fourier series is
$$ \frac{2}{\pi} + \sum_{k=1}^\infty \frac{4 (-1)^{k+1}}{\pi (4 k^2-1)} \cos(2 k t) $$
i.e. the Chebyshev series is
$$ |x| = \frac{2}{\pi} + \sum_{k=1}^\infty \frac{4(-1)^{k+1}}{\pi (4 k^2-1)} T_{2k}(x) $$
Since the coefficients are rational functions of $k$, truncations can't provide exponential error.
A: 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.

