Chebyshev polynomials

Of all the orthogonal polynomials, one family is secretly just disguised cosines, and that disguise gives it a magic property: among all polynomials of a given degree with leading coefficient fixed, it stays as small as possible across the whole interval. The Chebyshev polynomials are this minimal-oscillation family, the gold standard for polynomial approximation and numerical stability.

The first kind is defined by the strikingly simple identity T_n(cos theta) = cos(n theta), so on the interval minus 1 to 1, with x = cos theta, T_n is literally a cosine in disguise: T_0 = 1, T_1 = x, T_2 = 2 x^2 minus 1, T_3 = 4 x^3 minus 3 x, with the recurrence T_{n+1} = 2 x T_n minus T_{n-1}. They are orthogonal on minus 1 to 1 with the weight 1 over square root of (1 minus x^2). Their crucial feature is equioscillation: T_n stays within minus 1 and 1 and touches those bounds n+1 times, spreading its error perfectly evenly. A second kind U_n, with U_n(cos theta) = sin((n+1) theta)/sin theta, accompanies them.

Because T_n minimises the worst-case error, Chebyshev polynomials are the foundation of near-optimal function approximation (Chebyshev and minimax approximation), of spectral methods for differential equations, of Clenshaw-Curtis numerical integration, and of designing filters with flat or equal-ripple response. Interpolating at the Chebyshev nodes (the projected cosine points) cures the wild edge oscillations that ruin equally spaced polynomial interpolation. Catalogued here as functions; their generating ODE lives in the series-solutions topic.