# Chebyshev polynomials

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of the first kind

Polynomials that are orthogonal on the interval $[ - 1 , 1 ]$ with the weight function

$$h _ {1} ( x) = \frac{1}{\sqrt {1 - x ^ {2} }} ,\ \ x \in ( - 1 , 1 ) .$$

For the standardized Chebyshev polynomials one has the formula

$$T _ {n} ( x) = \cos ( n \mathop{\rm arc} \cos x ) ,\ \ x \in [ - 1 , 1 ] ,$$

and the recurrence relation

$$T _ {n+} 1 ( x) = 2 x T _ {n} ( x) - T _ {n-} 1 ( x) ,$$

by which one can determine the sequence

$$T _ {0} ( x) = 1 ,\ T _ {1} ( x) = x ,\ \ T _ {2} ( x) = 2 x ^ {2} - 1 ,$$

$$T _ {3} ( x) = 4 x ^ {3} - 3 x ,\ T _ {4} ( x) = 8 x ^ {4} - 8 x ^ {2} + 1 ,$$

$$T _ {5} ( x) = 16 x ^ {5} - 20 x ^ {3} + 5 x , . . . .$$

The orthonormalized Chebyshev polynomials are:

$$\widehat{T} _ {0} ( x) = \frac{1}{\sqrt \pi } T _ {0} ( x) = \frac{1}{\sqrt \pi } ,$$

$$\widehat{T} _ {n} ( x) = \sqrt { \frac{2} \pi } T _ {n} ( x) = \sqrt { \frac{2} \pi } \cos ( n \mathop{\rm arc} \cos x ) ,\ n \geq 1 .$$

The leading coefficient of $T _ {n} ( x)$, for $n \geq 1$, is $2 ^ {n-} 1$. Hence Chebyshev polynomials with leading coefficient 1 are defined by the formula

$$\widetilde{T} _ {n} ( x) = \frac{1}{2 ^ {n-} 1 } T _ {n} ( x) = \ \frac{1}{2 ^ {n-} 1 } \cos ( n { \mathop{\rm arc} \cos } x ) ,\ \ n \geq 1 .$$

The zeros of $T _ {n} ( x)$, given by

$$x _ {k} ^ {(} n) = \cos \frac{2 k - 1 }{2n} \pi ,\ \ k = 1 \dots n ,$$

frequently occur as interpolation nodes in quadrature formulas. The polynomial $T _ {n} ( x)$ is a solution of the differential equation

$$( 1 - x ^ {2} ) y ^ {\prime\prime} - x y ^ \prime + n ^ {2} y = 0 .$$

The polynomials $\widetilde{T} _ {n} ( x)$ deviate as least as possible from zero on the interval $[ - 1 , 1 ]$, that is, for any other polynomial $\widetilde{F} _ {n} ( x)$ of degree $n$ with leading coefficient 1 one has the following condition

$$\max _ {x \in [ - 1 , 1 ] } | \widetilde{F} _ {n} ( x) | > \ \max _ {x \in [ - 1 , 1 ] } | \widetilde{T} _ {n} ( x) | = \frac{1}{2 ^ {n-} 1 } .$$

On the other hand, for any polynomial $Q _ {n} ( x)$ of degree $n$ or less and satisfying

$$\max _ {x \in [ - 1 , 1 ] } \ | Q _ {n} ( x) | = 1 ,$$

one has, for any $x _ {0} \in ( - \infty , - 1 ) \cup ( 1 , \infty )$, the inequality

$$| Q ( x _ {0} ) | \leq | T _ {n} ( x _ {0} ) | .$$

If a function $f$ is continuous on the interval $[ - 1 , 1 ]$ and if its modulus of continuity $\omega ( \delta , f )$ satisfies the Dini condition

$$\lim\limits _ {\delta \rightarrow 0 } \omega ( \delta , f ) \ \mathop{\rm ln} \frac{1} \delta = 0 ,$$

then this function can be expanded in a Fourier–Chebyshev series,

$$f ( x) = \sum _ { n= } 0 ^ \infty a _ {n} \widehat{T} _ {n} ( x) ,\ \ x \in [ - 1 , 1 ] ,$$

which converges uniformly on $[ - 1 , 1 ]$. The coefficients in this series are defined by the formula

$$a _ {n} = \int\limits _ { - } 1 ^ { 1 } f ( t) \widehat{T} _ {n} ( t) \frac{dt}{\sqrt {1- t ^ {2} } } .$$

If the function $f$ is $p$- times continuously differentiable on $[ - 1 , 1 ]$ and if its $p$- th derivative $f ^ {(} p)$ satisfies a Lipschitz condition of order $\alpha$, i.e. $f ^ {(} p) \in \mathop{\rm Lip} \alpha$, then one has the inequality

$$\left | f ( x) - \sum _ { k= } 0 ^ { n } a _ {k} \widehat{T} _ {k} ( x) \right | \leq \frac{c _ {1} \mathop{\rm ln} n }{n ^ {p + \alpha } } ,\ \ x \in [ - 1 , 1 ] ,$$

where the constant $c _ {1}$ does not depend on $n$ and $x$.

Chebyshev polynomials of the second kind are defined by

$$U _ {n} ( x) = \frac{1}{n+} 1 T _ {n+} 1 ^ { \prime } ( x) = \sin [ ( n + 1 ) { \mathop{\rm arc} \cos } x ] \frac{1}{\sqrt {1 - x ^ {2} } } .$$

These polynomials are orthogonal on the interval $[ - 1 , 1 ]$ with weight function

$$h _ {2} ( x) = \sqrt {1 - x ^ {2} } ,\ \ x \in [ - 1 , 1 ] .$$

For any polynomial $\widetilde{Q} _ {n} ( x)$ with leading coefficient 1 one has the inequality

$$\frac{1}{2 ^ {n-} 1 } = \int\limits _ { - } 1 ^ { 1 } | \widetilde{U} _ {n} ( x) | dx \leq \int\limits _ { - } 1 ^ { 1 } | \widetilde{Q} _ {n} ( x) | dx .$$

The Chebyshev polynomials were introduced in 1854 by P.L. Chebyshev (cf. ). Both systems of Chebyshev polynomials are special cases of ultraspherical polynomials and Jacobi polynomials.

How to Cite This Entry:
Chebyshev polynomials. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chebyshev_polynomials&oldid=46330
This article was adapted from an original article by P.K. Suetin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article