Chebyshev polynomials

In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev (Пафнутий Чебышёв), are a sequence of orthogonal polynomials which are related to de Moivre's formula and which are easily defined recursively, like Fibonacci or Lucas numbers. One usually distinguishes between Chebyshev polynomials of the first kind which are denoted Tn and Chebyshev polynomials of the second kind which are denoted Un. The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebyshef or Tschebyscheff.

The Chebyshev polynomials Tn or Un are polynomials of degree n and the sequence of Chebyshev polynomials of either kind composes a polynomial sequence.

Chebyshev polynomials are important in approximation theory because the roots of the Chebyshev polynomials of the first kind, which are also called Chebyshev nodes, are used as nodes in polynomial interpolation. The resulting interpolation polynomial minimizes the problem of Runge's phenomenon and provides the best approximation to a continuous function under the maximum norm.

In the study of differential equations they arise as the solution to the Chebyshev differential equation

[itex](1-x^2)\,y'' - x\,y' + n^2\,y = 0 \mbox{ and } (1-x^2)\,y'' - 3x\,y' + n(n+2)\,y = 0[itex]

for the polynomials of the first and second kind, respectively. These equations are special cases of the Sturm-Liouville differential equation.

 Contents

Definition

The Chebyshev polynomials of the first kind are defined by the recurrence relation

[itex]T_0(x) = 1 \,[itex]
[itex]T_1(x) = x \,[itex]
[itex]T_{n+1}(x) = 2xT_n(x) - T_{n-1}(x). \,[itex]

One example of a generating function for this recurrence relation is

[itex]\sum_{n=0}^{\infty}T_n(x) t^n = \frac{1-tx}{1-2tx+t^2}.[itex]

The Chebyshev polynomials of the second kind are defined by the recurrence relation

[itex]U_0(x) = 1 \,[itex]
[itex]U_1(x) = 2x \,[itex]
[itex]U_{n+1}(x) = 2xU_n(x) - U_{n-1}(x). \,[itex]

One example of a generating function for this recurrence relation is

[itex]\sum_{n=0}^{\infty}U_n(x) t^n = \frac{1}{1-2tx+t^2}.[itex]

Trigonometric definition

The Chebyshev polynomials of the first kind can be defined by the trigonometric identity

[itex]T_n(\cos(\theta))=\cos(n\theta) \,[itex]

for n = 0, 1, 2, 3, .... . That cos(nx) is an nth-degree polynomial in cos(x) can be seen by observing that cos(nx) is the real part of one side of De Moivre's formula, and the real part of the other side is a polynomial in cos(x) and sin(x), in which all powers of sin(x) are even and thus replaceable via the identity cos²(x) + sin²(x) = 1.

Written explicitly

[itex]T_n(x) =

\left\{ \begin{matrix} & \cos(n\arccos(x)) & \mbox{ , } \ x \in [-1,1] \\ & \cosh(n \, \mathrm{arcosh}(x)) & \mbox{ , } \ x \ge 1 \\ & (-1)^n \cosh(n \, \mathrm{arcosh}(-x)) & \mbox{ , } \ x \le -1 \\ \end{matrix}\right. [itex]

Similarly, the polynomials of the second kind satisfy

[itex] U_n(\cos(\theta)) = \frac{\sin((n+1)\theta)}{\sin\theta}. [itex]

Orthogonality

Both the Tn and the Un form a sequence of orthogonal polynomials. The polynomials of the first kind are orthogonal with respect to the weight

[itex]\frac{1}{\sqrt{1-x^2}},[itex]

on the interval [−1,1], i.e. we have:

[itex]\int_{-1}^1 T_n(x)T_m(x)\,\frac{dx}{\sqrt{1-x^2}}=\left\{

\begin{matrix} 0 &: n\ne m~~~~~\\ \pi &: n=m=0\\ \pi/2 &: n=m\ne 0 \end{matrix} \right. [itex]

This can be proven by letting x= cos(θ) and using the identity Tn (cos(θ))=cos(nθ). Similarly, the polynomials of the second kind are orthogonal with respect to the weight

[itex]\sqrt{1-x^2}[itex]

on the interval [−1,1], i.e. we have:

[itex]\int_{-1}^1 U_n(x)U_m(x)\sqrt{1-x^2}\,dx=\left\{

\begin{matrix} 0 &: n\ne m\\ \pi/2 &: n=m \end{matrix} \right. [itex]

(which, when normalized to form a probability measure, is the Wigner semicircle distribution).

Other properties

The Chebyshev polynomials of the first and second kind are closely related by the following equations

[itex]\frac{d}{dx} \, T_n(x) = n U_{n-1}(x) \mbox{ , } n=1,\ldots[itex]
[itex]T_n(x) = U_n(x) - x \, U_{n-1}(x). [itex]

The Chebyshev polynomials are a special case of the ultraspherical or Gegenbauer polynomials, which themselves are a special case of the Jacobi polynomials.

Examples

Missing image
Chebyshev.png
This image shows the first few Chebyshev polynomials of the first kind in the domain -1¼<x<1¼, -1¼<y<1¼; the flat T0</font>, and T1</font>, T2</font>, T3</font>, T4</font> and T5</font>.

The first few Chebyshev polynomials of the first kind are

[itex] T_0(x) = 1 \,[itex]
[itex] T_1(x) = x \,[itex]
[itex] T_2(x) = 2x^2 - 1 \,[itex]
[itex] T_3(x) = 4x^3 - 3x \,[itex]
[itex] T_4(x) = 8x^4 - 8x^2 + 1 \,[itex]
[itex] T_5(x) = 16x^5 - 20x^3 + 5x \,[itex]
[itex] T_6(x) = 32x^6 - 48x^4 + 18x^2 - 1 \,[itex]
[itex] T_7(x) = 64x^7 - 112x^5 + 56x^3 - 7x \,[itex]
[itex] T_8(x) = 128x^8 - 256x^6 + 160x^4 - 32x^2 + 1 \,[itex]
[itex] T_9(x) = 256x^9 - 576x^7 + 432x^5 - 120x^3 + 9x. \,[itex]
Missing image
Chebyshev2.png
This image shows the first few Chebyshev polynomials of the second kind in the domain -1¼<x<1¼, -1¼<y<1¼; the flat U0</font>, and U1</font>, U2</font>, U3</font>, U4</font> and U5</font>. Although not visible in the image, Un(1)=n+1 and Un(-1)=(n+1)(-1)n.

The first few Chebyshev polynomials of the second kind are

[itex] U_0(x) = 1 \,[itex]
[itex] U_1(x) = 2x \,[itex]
[itex] U_2(x) = 4x^2 - 1 \,[itex]
[itex] U_3(x) = 8x^3 - 4x \,[itex]
[itex] U_4(x) = 16x^4 - 12x^2 + 1 \,[itex]
[itex] U_5(x) = 32x^5 - 32x^3 + 6x \,[itex]
[itex] U_6(x) = 64x^6 - 80x^4 + 24x^2 - 1. \,[itex]

Polynomial in Chebyshev form

A polynomial of degree N in Chebyshev form is a polynomial p(x) of the form

[itex]p(x) = \sum_{n=0}^{N} a_n T_n(x)[itex]

where Tn is the nth Chebyshev polynomial.

Polynomials in Chebyshev form can be evaluated using the Clenshaw algorithm.

Chebyshev roots

A Chebyshev polynomial of either kind with degree n has n different simple roots, called Chebyshev roots, in the interval [-1,1]. The roots are sometimes called Chebyshev nodes because they are used as nodes in polynomial interpolation. Using the trigonometric form one can easily prove that the roots of Tn are

[itex] x_i = \cos\left(\frac{2i-1}{2n}\pi\right) \mbox{ , } i=1,\ldots,n.[itex]

Similarly, the roots of Un are

[itex] x_i = \cos\left(\frac{i}{n+1}\pi\right) \mbox{ , } i=1,\ldots,n.[itex]

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