Chebyshev Polynomials
The Chebyshev polynomials Tn(x) are defined by the remarkable identity Tn(cos θ) = cos(nθ). They oscillate equitably between −1 and +1, making them optimal for polynomial approximation and interpolation.
How it works
The Chebyshev polynomials of the first kind, Tn(x), are defined on [−1, 1] by the identity Tn(cos θ) = cos(nθ). This seemingly simple definition hides extraordinary depth. The first few are T0(x) = 1, T1(x) = x, T2(x) = 2x² − 1, and each subsequent one follows from the recurrence Tn+1(x) = 2xTn(x) − Tn−1(x).
The equioscillation property is what makes Chebyshev polynomials special among all polynomials. Among all monic polynomials of degree n (leading coefficient 1), the rescaled Chebyshev polynomial 21−nTn(x) has the smallest maximum absolute value on [−1, 1]. It achieves this by oscillating between +21−n and −21−n exactly n+1 times. No other polynomial can do better — this is the Chebyshev equioscillation theorem.
The roots of Tn(x) are xk = cos((2k−1)π/(2n)) for k = 1, …, n. These Chebyshev nodes have a beautiful geometric interpretation: they are the projections of equally-spaced points on the unit circle onto the x-axis. Notice how they cluster near the endpoints −1 and +1. This clustering is precisely what eliminates Runge’s phenomenon.
Runge’s phenomenon occurs when interpolating certain functions (like 1/(1+25x²)) at equally-spaced nodes: as the degree increases, the interpolant oscillates wildly near the endpoints. Using Chebyshev nodes instead, the interpolant converges smoothly. The extra node density near the endpoints counteracts the tendency for polynomial approximations to misbehave there. This is one of the most important practical applications of Chebyshev theory in numerical analysis.