Runge's phenomenon on $f[x] = \frac{1}{1 + (5 x)^2}$

Question

I recently came across this problem when trying polynomial interpolation for the function $$f[x] = \frac{1}{1 + (5 x)^2}$$ using Mathematica. When another function $$g[x]=\frac{1}{1 + (\frac x5)^2}$$ used polynomial interpolation the results Mathematica gave were similar to the expansion series of $g[x]$. But the power series of $f[x]$ and it's polynomial interpolation blew up at the end points of $x=-1$ and $x=1$. Is there a complex singularity reason for this?

Answer 1

Runge's phenomenon is the consequence of two properties:

1. Fast growing Lebesgue constant with higher polynomial degree $P$.

2. The magnitude of the n-th order derivatives of the particular function grows quickly when $P$ increases.

The first function $f$ satisfies the seconde property on the intervall $[-1,+1]$. The second function $g$ does not satisfy the property on the same intervall, however on a wider range $[-25,+25]$.

Regards

Answer 2

A nice, more-or-less elementary, explanation of the Runge phenomenon is Epperson, "On the Runge example", AMM 94:4 (1987), pp. 329-341 (DOI 10.2307/2323093). On the real line the function is well-behaved, but it has poles nearby on the imaginary axis.