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)