Interpolation Without Losing the Shape of the Problem

Interpolation is not simply “draw a polynomial through the points.” Node placement, smoothness, and the quantity we want to preserve all matter.

For nodes $x_0,\ldots,x_n$ , the Lagrange form is

p_n(x)=\sum_{j=0}^{n} f(x_j)\prod_{k\ne j}\frac{x-x_k}{x_j-x_k}.

Chebyshev nodes reduce the endpoint amplification that makes equally spaced high-degree interpolation unstable. Discontinuities create a different issue: oscillations near a jump persist even as more terms are added. That distinction—node instability versus limited smoothness—is the organizing idea for the original note series.