I understand that due to Runge's phenomenon, increasing the degree, $n$, of a polynomial interpolant can actually increase the error between the interpolant, $P_n$, and the function, $f$, you are trying to approximate, when using equispaced interpolation points.
Therefore I ask the question, is it possible to find some interval $[-a,a]$ over which $P_n$ does converge to $f$ as $n \to \infty$, while still using equispaced interpolation points?
In other words, does there exist some $a$, such that, $\underset{-a\leq x\leq a}\max|f(x)-P_n(x)| \to 0$ as $n \to \infty$, for equispaced interpolation points? Where $f$ is an arbitrary continuous function $\mathbb{R} \to \mathbb{R}$. For example, $f(x) = cos(\pi x)$.
If so, how does one compute $a$?