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Chebyshev polynomial

The polynomial Tₙ defined by Tₙ(cos θ) = cos nθ, which swings between −1 and 1 exactly n + 1 times on the interval [−1, 1]. Scaled to lead with 1, it is the polynomial that stays closest to zero there, and it underlies near-best approximation.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

The Chebyshev maps T₂, T₃, T₄ and T₅. Four small square plots of the Chebyshev polynomials of degrees two to five on the interval from minus one to one, each with the diagonal drawn; the graph of degree n sweeps between the bottom and top of the square n times.

A solvable chaos of every degree

The logistic map at four is chaotic and, through a change of coordinates, completely solvable: its orbits are cosines of doubling angles. The trick is not a one-off. For every whole number n there is a polynomial of degree n that multiplies angles by n instead of 2, and every one of them is exactly as solvable, has exactly nᵏ points of period k, and preserves the same distribution — and any two of them commute, which almost no two polynomials do.

dynamics · Iteration The best degree-3 polynomial to eˣ: its error touches its largest size 5 times. The error curve of the best uniform polynomial approximation of degree 3 to eˣ on the interval from −1 to 1. It reaches its maximum size 5.528 × 10⁻³ at 5 points, alternately above and below, at x = -1.000, -0.682, 0.050, 0.732, 1.000.

The error that keeps coming back to its worst

Judge a polynomial by its largest error on an interval and there is exactly one best one of each degree. It is recognised without comparing it to anything else — its error rises to the same largest size, alternately above and below, one more time than there are coefficients.

analysis · Taylor series A disc of radius 0.2 and an ellipse of size 1.22 around the same interval. For 1/(1 + 25x²): the interval from −1 to 1 on the real axis, poles at 0 + 0.2i and 0 − 0.2i, the Taylor disc at 0 of radius 0.20, and the Bernstein ellipse with foci ±1 through the poles, with ρ = 1.2198.

An ellipse, not a disc

A Taylor series converges on a disc, and the disc's radius is the distance to the nearest singularity. Ask instead how well polynomials can follow a function on an interval, and the answer is an ellipse with the interval's ends as its foci — the largest one the function is smooth inside.

analysis · Taylor series

Named alongside it

The objects these essays reach for when they reach for this one.

ApproximationBoundChaosComplex planeConjugacyConvergenceEllipseError analysisExchange algorithmFourier seriesInterpolationInvariant measure

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