The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as ${\displaystyle T_{n}(x)}$ and ${\displaystyle U_{n}(x)}$. They can be defined in several equivalent ways, one of which starts with trigonometric functions:

The Chebyshev polynomials of the first kind ${\displaystyle T_{n}}$ are defined by:

$${\displaystyle T_{n}(\cos \theta )=\cos(n\theta ).}$$

Similarly, the Chebyshev polynomials of the second kind ${\displaystyle U_{n}}$ are defined by:

$${\displaystyle U_{n}(\cos \theta )\sin \theta =\sin {\big (}(n+1)\theta {\big )}.}$$

That these expressions define polynomials in ${\displaystyle \cos \theta }$ may not be obvious at first sight but follows by rewriting ${\displaystyle \cos(n\theta )}$ and ${\displaystyle \sin {\big (}(n+1)\theta {\big )}}$ using de Moivre's formula or by using the angle sum formulas for ${\displaystyle \cos }$ and ${\displaystyle \sin }$ repeatedly. For example, the double angle formulas, which follow directly from the angle sum formulas, may be used to obtain ${\displaystyle T_{2}(\cos \theta )=\cos(2\theta )=2\cos ^{2}\theta -1}$ and ${\displaystyle U_{1}(\cos \theta )\sin \theta =\sin(2\theta )=2\cos \theta \sin \theta }$, which are respectively a polynomial in ${\displaystyle \cos \theta }$ and a polynomial in ${\displaystyle \cos \theta }$ multiplied by ${\displaystyle \sin \theta }$. Hence ${\displaystyle T_{2}(x)=2x^{2}-1}$ and ${\displaystyle U_{1}(x)=2x}$.

An important and convenient property of the T_n(x) is that they are orthogonal with respect to the inner product:

$${\displaystyle \langle f,g\rangle =\int _{-1}^{1}f(x)\,g(x)\,{\frac {\mathrm {d} x}{\sqrt {1-x^{2}}}},}$$

The Chebyshev polynomials T_n are polynomials with the largest possible leading coefficient whose absolute value on the interval [−1, 1] is bounded by 1. They are also the "extremal" polynomials for many other properties.^[1]

In 1952, Cornelius Lanczos showed that the Chebyshev polynomials are important in approximation theory for the solution of linear systems;^[2] the roots of T_n(x), which are also called Chebyshev nodes, are used as matching points for optimizing polynomial interpolation. The resulting interpolation polynomial minimizes the problem of Runge's phenomenon and provides an approximation that is close to the best polynomial approximation to a continuous function under the maximum norm, also called the "minimax" criterion. This approximation leads directly to the method of Clenshaw–Curtis quadrature.

These polynomials were named after Pafnuty Chebyshev.^[3] The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebycheff, Tchebyshev (French) or Tschebyschow (German).

Definitions

Recurrence definition

The Chebyshev polynomials of the first kind are obtained from the recurrence relation:

$${\displaystyle {\begin{aligned}T_{0}(x)&=1\\T_{1}(x)&=x\\T_{n+1}(x)&=2x\,T_{n}(x)-T_{n-1}(x).\end{aligned}}}$$

${\displaystyle k\times k}$