Runge's phenomenon

Example

Let $f(x) = \dfrac{1}{1 + 25 x^2}$ on $[-1, 1]$. For $n \ge 1$ let $p_n$ be the interpolation polynomial of degree at most $n$ of $f$ at the equally spaced nodes $x_i = -1 + 2i/n$, $i = 0, \dots, n$. Then $$\max_{x \in [-1, 1]} \abs{f(x) - p_n(x)} \to \infty \quad \text{as } n \to \infty.$$ The error stays small in the middle of the interval but grows without bound near the end points, where $p_n$ oscillates more and more.

The function $f$ is smooth, in fact analytic on all of $\R$, so the failure is not due to a lack of differentiability. The interpolation error formula explains it. The derivatives $f^{(n+1)}$ grow roughly like $(n + 1)! \, 5^{n+1}$ on $[-1, 1]$, because $f$ has complex poles at $\pm i/5$, close to the interval. This growth outweighs the factor $\frac{1}{(n+1)!} \max \abs{\prod_i (x - x_i)}$ for equally spaced nodes, which is largest near the end points.

With Chebyshev nodes instead of equally spaced nodes, the interpolation polynomials of the same $f$ converge uniformly to $f$ on $[-1, 1]$.