Runge's Phenomenon
Polynomial interpolation through equally-spaced points seems like it should improve as you add more points. Instead, the interpolant develops wild oscillations near the edges. This counterintuitive failure — Runge's phenomenon — teaches us that where you place your points matters more than how many you use. Toggle Chebyshev nodes to see the cure.
In 1901, Carl Runge demonstrated that polynomial interpolation through equally-spaced points on f(x) = 1/(1 + 25x²) does not converge to the function as the number of points increases. Instead, the interpolation error grows exponentially near the endpoints of the interval.
The root cause is that equally-spaced nodes cluster too few points near the edges of [−1, 1], where the polynomial needs the most control. The Lebesgue constant — which measures how much an interpolation can amplify small perturbations — grows exponentially for equally-spaced points (roughly as 2n/n) but only logarithmically for Chebyshev points.
Chebyshev nodes xk = cos((2k−1)π/2n) are the zeros of Chebyshev polynomials of the first kind. They cluster densely near the endpoints, providing exactly the edge control that equally-spaced points lack. For any continuous function on [−1, 1], Chebyshev interpolation converges if the function has bounded variation, and converges exponentially fast if the function is analytic.
This has profound practical implications: if you must interpolate data, use Chebyshev-spaced nodes (or better yet, Chebyshev series). Equally-spaced interpolation is safe only for low-degree polynomials or on short intervals. Splines — piecewise low-degree polynomials — offer another practical escape from Runge's phenomenon.
Try dragging the interpolation points to see how the polynomial responds. Moving a point near the edge has a much larger effect than moving one near the center. This is the Lebesgue function in action.