Chebyshev polynomials

In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev (Пафнутий Чебышёв), are special polynomials. One usually distinguishes between Chebyshev polynomials of the first kind which are denoted T_n and Chebyshev polynomials of the second kind which are denoted U_n. The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebyshef or Tschebyscheff.

The Chebyshev polynomials T_n or U_n are polynomials of degree n and the sequence of Chebyshev polynomial of either kind compose 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 differential equations they arise as the solution to the Chebyshev differential equation

<math>(1-x^2)\,y'' - x\,y' + n^2\,y = 0 \mbox{ and } (1-x^2)\,y'' - 3x\,y' + n(n+2)\,y = 0<math>

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

Definition

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

<math>T_0(x) = 1 \,<math>

<math>T_1(x) = x \,<math>

<math>T_{n+1}(x) = 2xT_n(x) - T_{n-1}(x) \,<math>

One example of a generating function for this recurrence relation is

<math>\sum_{n=0}^{\infty}T_n(x) t^n = \frac{1-tx}{1-2tx+t^2}.<math>

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

<math>U_0(x) = 1 \,<math>

<math>U_1(x) = 2x \,<math>

<math>U_{n+1}(x) = 2xU_n(x) - U_{n-1}(x) \,<math>

One example of a generating function for this recurrence relation is

<math>\sum_{n=0}^{\infty}U_n(x) t^n = \frac{1}{1-2tx+t^2}.<math>

Examples

The first few Chebyshev polynomials of the first kind are

<math> T_0(x) = 1 \,<math>

<math> T_1(x) = x \,<math>

<math> T_2(x) = 2x^2 - 1 \,<math>

<math> T_3(x) = 4x^3 - 3x \,<math>

<math> T_4(x) = 8x^4 - 8x^2 + 1 \,<math>

<math> T_5(x) = 16x^5 - 20x^3 + 5x \,<math>

<math> T_6(x) = 32x^6 - 48x^4 + 18x^2 - 1 \,<math>

<math> T_7(x) = 64x^7 - 112x^5 + 56x^3 - 7x \,<math>

<math> T_8(x) = 128x^8 - 256x^6 + 160x^4 - 32x^2 + 1 \,<math>

<math> T_9(x) = 256x^9 - 576x^7 + 432x^5 - 120x^3 + 9x \,<math>

The first few Chebyshev polynomials of the second kind are

<math> U_0(x) = 1 \,<math>

<math> U_1(x) = 2x \,<math>

<math> U_2(x) = 4x^2 - 1 \,<math>

<math> U_3(x) = 8x^3 - 4x \,<math>

<math> U_4(x) = 16x^4 - 12x^2 + 1 \,<math>

<math> U_5(x) = 32x^5 - 32x^3 + 6x \,<math>

<math> U_6(x) = 64x^6 - 80x^4 + 24x^2 - 1 \,<math>

Trigonometric definition

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

<math>T_n(\cos(\theta))=\cos(n\theta) \,<math>

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) + sin&sup2(x) = 1.

Written explicitly

<math>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. <math>

Similarly, the polynomials of the second kind satisfy

<math> U_n(\cos(\theta)) = \frac{\sin((n+1)\theta)}{\sin\theta} <math>

Notes

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

<math>\frac{d}{dx} \, T_n(x) = n U_{n-1}(x) \mbox{ , } n=1,\ldots<math>

<math>T_n(x) = U_n(x) - x \, U_{n-1}(x) <math>

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

<math>\frac{dx}{\sqrt{1-x^2}},<math>

on the interval [−1,1], i.e., we have

<math>\int_{-1}^1 T_n(x)T_m(x)\,\frac{dx}{\sqrt{1-x^2}}=0\quad\mbox{if}\ n\neq m.<math>

This is because (letting x = cos θ)

<math>\int_0^\pi\cos(n\theta)\cos(m\theta)\,d\theta=0\quad\mbox{if}\ n\neq m.<math>

Similarly, the polynomials of the second kind are orthogonal with respect to the weight

<math>\sqrt{1-x^2} \, dx.<math>

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

<math> x_i = \cos\left(\frac{2i-1}{2n}\pi\right) \mbox{ , } i=1,\ldots,n.<math>

Similarly, the roots of U_n are

<math> x_i = \cos\left(\frac{i}{n+1}\pi\right) \mbox{ , } i=1,\ldots,n.<math>

See also

• Chebyshev nodes
• Chebyshev filter
• Legendre polynomials
• Hermite polynomials

References

• M. Abramowitz and I. A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Chapter 22. New York: Dover, 1972.