Proof that the Runge Phenomenon occurs

Question

Is there such a proof that states that the Runge Phenomena will always occur when interpolating with higher order polynomials or is this just observed empirically?

Answer 1

Runge Phenomena doesn't occur for all functions. For a detailed analysis on polynomial interpolation at equidistant points you can have a look here

Answer 2

The Runge Phenomenon does not always occur. You can interpolate, say, $f(x)= e^x$ using equally spaced nodes on any interval $[a,b]$, and the interpolating polynomials will converge to $f$ uniformly. (The same holds for any function whose Taylor series has infinite radius of convergence.)

When the Runge phenomenon does occur, it is a lot easier to observe empirically than to write down a rigorous proof that it happens. This is why the textbooks on numerical analysis tend to do the former and not the latter.

For the classical example of intepolating $f(x)=1/(1+x^2)$ on $[-5,5]$ by equally spaced nodes, David Speyer posted a complete proof of divergence at $x=4$.