Newman's theorem on rational approximation of the absolute value

Remarks

For $0 \le x \le 1$ one has $p(x) > 0$ and $\abs{p(-x)} \le p(x)$, so the denominator is positive there, and by evenness on all of $[-1, 1]$. Then $x - r(x) = 2x\, \frac{p(-x)}{p(x) + p(-x)}$, and the proof bounds the ratio $\abs{p(-x)/p(x)} = \prod_k \abs{\xi^k - x}/(\xi^k + x)$ separately for $x$ near $0$ and for $x$ in $[\xi^n, 1]$, where some factor $\abs{\xi^k - x}/(\xi^k + x)$ is small. Newman also showed a lower bound of the form $e^{-c\sqrt{n}}$ for every rational approximation of type $(n, n)$, so the rate $e^{-c\sqrt{n}}$ is the true order. By contrast, the best uniform approximation of $\abs{x}$ on $[-1, 1]$ by polynomials of degree $n$ has error of order $1/n$, a result of S. N. Bernstein. Herbert Stahl later found the exact asymptotics of the best rational error, $8 e^{-\pi \sqrt{n}}$. Example: for $n = 100$ the bound is $3 e^{-10} \approx 1.4 \cdot 10^{-4}$, while the best polynomial approximation of degree $100$ has error about $2.8 \cdot 10^{-3}$, roughly $0.28/n$.

Going deeper

For a student

This theorem started the modern study of rational approximation of nonsmooth functions. It shows that rational functions are far stronger than polynomials near singularities, which is why rational methods are used for functions with corners, branch points and boundary layers. The geometric clustering of poles near the singularity reappears in many later constructions, for example in rational methods for problems on domains with corners.