Chebyshev polynomials

Chebyshev polynomials are two sequences of orthogonal polynomials, usually denoted Tₙ and Uₙ, characterized by cosine and hyperbolic-cosine identities. They are used in approximation, numerical analysis, and the study of polynomial roots.

Chebyshev recurrence: A pair of three-term recurrences that generate Chebyshev polynomials from their first two terms. The recurrences provide a direct way to compute both polynomial families.
Chebyshev approximation: Function approximation by finite expansions in Chebyshev polynomials. Their rapid coefficient decay often makes smooth functions economical to approximate.
Cosine: A periodic trigonometric function that gives the horizontal coordinate of a point on the unit circle. The defining identity expresses Tₙ as a multiple-angle cosine in disguise.
Legendre polynomials: Orthogonal polynomials on [−1,1] with constant weight, arising in mathematical physics and approximation. They share the interval but use a different weight from Chebyshev polynomials.
Chebyshev nodes: Interpolation points given by the extrema or roots of Chebyshev polynomials, mapped to a chosen interval. Their spacing reduces oscillatory interpolation errors near interval endpoints.
Polynomial interpolation: The construction of a polynomial matching prescribed values at selected points. Chebyshev polynomials guide the choice of interpolation points that limits worst-case error.
Hyperbolic cosine: The function cosh(x) = (eˣ + e⁻ˣ)/2, which grows exponentially for large positive x. The polynomial identities extend from cosine to cosh outside the interval from −1 to 1.
Hermite polynomials: Orthogonal polynomials on the real line under a Gaussian weight. Their unbounded domain and Gaussian weight contrast with Chebyshev polynomials' bounded interval.
Chebyshev polynomials of the first kind: The polynomial sequence Tₙ defined by Tₙ(cos θ) = cos(nθ). The cosine identity defines the best-known Chebyshev sequence.
Spectral method: A numerical method that represents solutions of differential equations using global basis functions. Chebyshev expansions provide a common basis for spectral calculations on bounded intervals.

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