Why is it that Runge's phenomenon gives rise to oscillations at the edges and not in other locations? What I don't understand is why the oscillations arises at the edges and not somewhere else? What seems weird to me is that the phenomenon seems to arise due to the sampling scheme of the interpolation points (or at least that is how its emphasized in the wikipedia article). Thus, if its a problem due to sampling why is it that the oscillations concentrate at the edges and not at other places?
I'd love to be able to go through the proof myself but with a weak background in basic analysis makes me doubt my capabilities of this. Therefore, intuitive/conceptual explanations are probably the ones I will most likely understand best, though I don't want to discourage people that want to provide a in depth answer, though don't forget to try to provide a high level one too if its not too much to ask.
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$\begingroup$...intuitive/conceptual explanations are probably the ones I will most likely understand best...
I guess seeing is believing:
These are plots of the real part, imaginary part, and magnitude of the Runge function $\dfrac1{1+25z^2}$ and its $n$-point polynomial interpolant over the points $z_k=\dfrac{2k}{n}-1;\quad k=1,\dots,n$, evaluated over complex arguments.
For reference, here's how the Runge function itself looks like in the complex plane:
Notice the poles jutting out of the surface. These two poles are precisely the reason why interpolating this function with polynomials is difficult; polynomials themselves don't ever have poles, so the poor polynomial interpolant oscillates grandly in a pathetic attempt to emulate the poles of the rational function it is trying to approximate. Both the functions are analytic (holomorphic for the polynomial interpolant; meromorphic for the Runge function), so any bad behavior in the complex plane gets passed along to the real line:
For those who want to play around with this in Mathematica:
Manipulate[GraphicsRow[ Table[Plot3D[Evaluate[{f[1/(1 + 25 (x + I y)^2)], f[InterpolatingPolynomial[ Table[With[{xx = 2 i/n - 1}, {xx, 1/(1 + 25 xx^2)}], {i, 0, n}], x + I y]]}], {x, -1, 1}, {y, -1, 1}, BoxRatios -> Automatic, ClippingStyle -> None, Mesh -> False, PlotLabel -> f, PlotRange -> {-3/2, 3/2}, PlotStyle -> {Opacity[3/4], Opacity[1/2]}], {f, {Re, Im, Abs}}]], {n, 2, 30, 1}]
