ChebyshevPolynome

Chebyshev polynomials are a sequence of orthogonal polynomials that are closely related to the cosine function. They are named after the Russian mathematician Pafnuty Chebyshev. There are two types of Chebyshev polynomials: Chebyshev polynomials of the first kind, denoted by T_n(x), and Chebyshev polynomials of the second kind, denoted by U_n(x).

The Chebyshev polynomials of the first kind are defined by the recurrence relation T_0(x) = 1, T_1(x) = x, and T_{n+1}(x) = 2xT_n(x) - T_{n-1}(x) for n >= 1. An explicit form is given by T_n(cos(theta)) = cos(n*theta). This definition reveals a key property: on the interval [-1, 1], the absolute value of T_n(x) is always less than or equal to 1. This minimax property makes them very useful in approximation theory. They are the polynomials of degree n with the smallest possible maximum absolute value on the interval [-1, 1].

The Chebyshev polynomials of the second kind are defined by the recurrence relation U_0(x) = 1, U_1(x) = 2x, and U_{n+1}(x) = 2xU_n(x) - U_{n-1}(x) for n >= 1. An explicit form is given by U_n(cos(theta)) = sin((n+1)*theta) / sin(theta).

Chebyshev polynomials have numerous applications in numerical analysis, including polynomial interpolation, approximation of functions, numerical integration (quadrature rules), and solving differential equations. They are particularly favored for approximating functions because their roots (for interpolation) are distributed in a way that minimizes oscillations, leading to more stable approximations.