Chebyshev polynomials

The Chebyshev polynomials are two sequences of <a href="/facts/Orthogonal_polynomials/9cyRbDYU">orthogonal polynomials</a> related to the <a href="/facts/Trigonometric_functions/czej7ur0">cosine and sine functions</a>, notated as 
 
 
 
 
 T
 
 n
 
 
 (
 x
 )
 
 
 {\displaystyle T_{n}(x)}
 
 and 
 
 
 
 
 U
 
 n
 
 
 (
 x
 )
 
 
 {\displaystyle U_{n}(x)}
 
. They can be defined in several equivalent ways, one of which starts with <a href="/facts/Trigonometric_functions/czej7ur0">trigonometric functions</a>:
The Chebyshev polynomials of the first kind 
 
 
 
 
 T
 
 n
 
 
 
 
 {\displaystyle T_{n}}
 
 are defined by

T
          
            n
          
        
        (
        cos
        ⁡
        θ
        )
        =
        cos
        ⁡
        (
        n
        θ
        )
        .
      
    
    {\displaystyle T_{n}(\cos \theta )=\cos(n\theta ).}

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

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

That these expressions define polynomials in 
 
 
 
 cos
 ⁡
 θ
 
 
 {\displaystyle \cos \theta }
 
 is not obvious at first sight but can be shown using <a href="/facts/De_Moivre%27s_formula/uyG4k2uC">de Moivre's formula</a> (see below).
The Chebyshev polynomials Tn are polynomials with the largest possible leading coefficient whose <a href="/facts/Absolute_value/dwJBpEs5">absolute value</a> on the <a href="/facts/Interval_(mathematics)/e9se3vTe">interval</a> [−1, 1] is bounded by 1. They are also the "extremal" polynomials for many other properties.
In 1952, <a href="/facts/Cornelius_Lanczos/FRqrbJIb">Cornelius Lanczos</a> showed that the Chebyshev polynomials are important in <a href="/facts/Approximation_theory/87AaPXLp">approximation theory</a> for the solution of linear systems; the <a href="/facts/Root_of_a_polynomial/mlK3dhoT">roots</a> of Tn(x), which are also called <a href="/facts/Chebyshev_nodes/4onexv5e">Chebyshev nodes</a>, are used as matching points for optimizing <a href="/facts/Polynomial_interpolation/DmnuxaKT">polynomial interpolation</a>. The resulting interpolation polynomial minimizes the problem of <a href="/facts/Runge%27s_phenomenon/641beUAR">Runge's phenomenon</a> and provides an approximation that is close to the best polynomial approximation to a <a href="/facts/Continuous_function/sKbl02pB">continuous function</a> under the <a href="/facts/Maximum_norm/5yIkjNE3">maximum norm</a>, also called the "<a href="/facts/Minimax/VMjmmzGE">minimax</a>" criterion. This approximation leads directly to the method of <a href="/facts/Clenshaw%E2%80%93Curtis_quadrature/fFNmUf4G">Clenshaw–Curtis quadrature</a>.
These polynomials were named after <a href="/facts/Pafnuty_Chebyshev/LzRqLqRm">Pafnuty Chebyshev</a>. The letter T is used because of the alternative <a href="/facts/Transliteration/1eemsW6o">transliterations</a> of the name Chebyshev as Tchebycheff, Tchebyshev (French) or Tschebyschow (German).

Chebyshev polynomials open-in-new

Chebyshev polynomials