The node polynomial ω(x)=(x-x₁)···(x-x_n)

We want to approximate a function f(x) on an interval [a,b]. We choose n nodes x₁,...,x_n on this interval, e.g. equidistant nodes. We then use the interpolating polynomial p(x).

The interpolation error f(x)-p(x) depends on the node polynomial   ω(x) = (x-x₁)···(x-x_n)
We want to choose the nodes so that     ω_max = max_[a,b]|ω(x)|     is as small as possible.

Drag the nodes on the x-axis with your finger/mouse and try to reduce max|ω(x)|!
Optimal choice of nodes: all oscillations have the same size and the graph is between the red lines (Chebyshev nodes).