Chebyshev polynomials

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. [1]). Both systems of Chebyshev polynomials are special cases of ultraspherical polynomials and Jacobi polynomials.

References

[1]	P.L. Chebyshev, , Collected works , 2 , Moscow-Leningrad (1947) pp. 23–51 (In Russian)
[2]	G. Szegö, "Orthogonal polynomials" , Amer. Math. Soc. (1975)