Bernstein's theorem on analytic functions

Remarks

The region $\mathcal{E}_\rho$ is the image of the annulus $1 < \abs{z} < \rho$ under the map $z \mapsto (z + z^{-1})/2$, together with $[-1, 1]$. Proof idea for the direct half: for $1 < s < \rho$, the coefficients of the Chebyshev series of $f$ are $O(s^{-n})$, by a contour integral over the image of $\abs{z} = s$. Truncating the series gives $E_n(f) = O(s^{-n})$. For the converse, write $f = p_0 + \sum (p_{n+1} - p_n)$ with best approximations $p_n$ and bound each difference on $\mathcal{E}_s$ for $s < \rho$ by its size on $[-1, 1]$.

Example: for $a > 1$ and $f(x) = 1/(x - a)$, the pole $a$ lies on the boundary of $\mathcal{E}_\rho$ with $\rho = a + \sqrt{a^2 - 1}$. So $E_n(f)$ decays like $(a + \sqrt{a^2 - 1})^{-n}$, up to factors that grow slower than any exponential. For the Runge function $1/(1 + 25x^2)$, the poles $\pm i/5$ give $\rho = (1 + \sqrt{26})/5 \approx 1.22$.

Going deeper

For a student

This explains why Chebyshev methods converge so fast for analytic functions, and why nearby complex singularities slow them down even when $f$ looks harmless on the real line. It is an inverse and a direct theorem at once: geometric decay of $E_n(f)$ and analyticity in an ellipse are equivalent. The ellipses $\partial \mathcal{E}_\rho$ are the level curves of the Green function of the complement of $[-1, 1]$, which leads to the general theory of Walsh for other sets.