Chebyshev polynomials

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In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, 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 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 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 an approximation that is close to the polynomial of best approximation to a continuous function under the maximum norm. This approximation leads directly to the method of Clenshaw-Curtis quadrature.

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

$(1-x^2)\,y'' - x\,y' + n^2\,y = 0 \,\!$

and

$(1-x^2)\,y'' - 3x\,y' + n(n+2)\,y = 0 \,\!$

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

1 Definition
2 Trigonometric definition
3 Pell equation definition
4 Mutual recurrence
5 Orthogonality
6 Minimal -norm
7 Relation between Chebyshev polynomials of the first and second kind
8 Other properties
9 Examples
10 Polynomial in Chebyshev form
11 Chebyshev roots
12 Spread polynomials
13 See also
14 References

[edit] Definition

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

$T_0(x) = 1 \,\!$

$T_1(x) = x \,\!$

$T_{n+1}(x) = 2xT_n(x) - T_{n-1}(x). \,\!$

One example of a generating function for T_n is

$\sum_{n=0}^{\infty}T_n(x) t^n = \frac{1-tx}{1-2tx+t^2}. \,\!$

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

$U_0(x) = 1 \,\!$

$U_1(x) = 2x \,\!$

$U_{n+1}(x) = 2xU_n(x) - U_{n-1}(x). \,\!$

One example of a generating function for U_n is

$\sum_{n=0}^{\infty}U_n(x) t^n = \frac{1}{1-2tx+t^2}. \,\!$

[edit] Trigonometric definition

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

$T_n(x)=\cos(n \arccos x)=\cosh(n\,\mathrm{arccosh}\,x) \,\!$

whence:

$T_n(\cos(\theta))=\cos(n\theta) \,\!$

for n = 0, 1, 2, 3, ..., while the polynomials of the second kind satisfy:

$U_n(\cos(\theta)) = \frac{\sin((n+1)\theta)}{\sin\theta} \,\!$

which is structurally quite similar to the Dirichlet kernel.

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.

This identity is extremely useful in conjunction with the recursive generating formula inasmuch as it enables one to calculate the cosine of any integral multiple of an angle solely in terms of the cosine of the base angle. Evaluating the first two Chebyshev polynomials:

$T_0(x)=\cos\ 0x\ =1 \,\!$

and:

$T_1(\cos(x))=\cos\ (x) \,\!$

one can straightforwardly determine that:

$\cos(2 \theta)=2\cos\theta \cos\theta - \cos(0 \theta) = 2\cos^{2}\,\theta - 1 \,\!$

$\cos(3 \theta)=2\cos\theta \cos(2\theta) - \cos\theta = 4\cos^3\,\theta - 3\cos\theta \,\!$

and so forth. To trivially check whether the results seem reasonable, sum the coefficients on both sides of the equals sign (that is, setting theta equal to zero, for which the cosine is unity), and one sees that 1 = 2 - 1 in the former expression and 1 = 4 - 3 in the latter.

An immediate corollary is the composition identity (or the "nesting property")

$T_n(T_m(x)) = T_{n\cdot m}(x).\,\!$

Written explicitly

$T_n(x) = \begin{cases} \cos(n\arccos(x)), & \ x \in [-1,1] \\ \cosh(n \, \mathrm{arccosh}(x)), & \ x \ge 1 \\ (-1)^n \cosh(n \, \mathrm{arccosh}(-x)), & \ x \le -1 \\ \end{cases} \,\!$

(not forgetting that the inverse hyperbolic cosines of x and −x differ by the constant π). From reasoning similar to that above, one can develop a closed-form generating formula for Chebyshev polynomials of the first kind:

$\cos(n \theta)=\frac{e^{i n \theta}+e^{-i n \theta}}{2}=\frac{(e^{i \theta})^n+(e^{i \theta})^{-n}}{2} \,\!$

which, combined with DeMoivre's formula:

$\! e^{i \theta}=\cos\theta+i \sin\theta=\cos\theta+i \sqrt{1-\cos^2\theta}=\cos\theta+\sqrt{\cos^2\theta-1} \,\!$

yields:

$\cos(n \theta)=\frac{\left(\cos\theta+ \sqrt{\cos^2\theta-1}\right)^n+\left(\cos\theta+ \sqrt{\cos^2\theta-1}\,\right)^{-n}}{2} \,\!$

which, of course, is far more expedient for determining the cosine of N times a given angle than is cranking through almost N rounds of the recursive generator calculation. Finally, if we replace $cos(θ)$ with x, we can alternatively write:

$T_n(x)=\frac{\left(x+ \sqrt{x^2-1}\right)^n+\left(x+ \sqrt{x^2-1}\right)^{-n}}{2} \,\!$

[edit] Pell equation definition

The Chebyshev polynomials can also be defined as the solutions to the Pell equation

$T_i^2 - (x^2-1) U_{i-1}^2 = 1 \,\!$

in a ring R[x] (e.g., see Demeyer (2007), p.70). Thus, they can be generated by the standard technique for Pell equations of taking powers of a fundamental solution:

$T_i + U_{i-1} \sqrt{x^2-1} = (x + \sqrt{x^2-1})^i. \,\!$

[edit] Mutual recurrence

Equivalently, the two sequences can also be defined at once from a pair of mutual recurrence equations:

$T_0(x) = 1\,\!$

$U_{-1}(x) = 0\,\!$

$T_{n+1}(x) = xT_n(x) - (1 - x^2)U_{n-1}(x)\,$

$U_n(x) = xU_{n-1}(x) + T_n(x)\,$

These can be derived from the trigonometric formulae; for example, if $x = \cos\vartheta$ , then

$T_{n+1}(x) = T_{n+1}(\cos\vartheta) = \cos((n + 1)\vartheta)\,\!$

$= \cos(n\vartheta)\cos\vartheta - \sin(n\vartheta)\sin\vartheta \,\!$

$= T_n(\cos\vartheta)\cos\vartheta - U_n(\cos\vartheta)\sin^2\vartheta \,\!$

$= xT_n(x) - (1 - x^2)U_n(x). \,\!$

(Both these equations and the trigonometric equations take a simpler form if we, like some works, follow the alternate convention of denoting our $U n$ (the polynomial of degree n) with $U n + 1$ instead.) we break knowledge's line(prof.Parand & his MS students)!

Integral Relation:

$\int\mathrm T_n(x)\,dx =1/(2(n+1))T_{n+1}(x)-1/(2(n-1))T_{n-1}(x) \,\!$

[edit] Orthogonality

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

$\frac{1}{\sqrt{1-x^2}}, \,\!$

on the interval [−1,1], i.e. we have:

$\int_{-1}^1 T_n(x)T_m(x)\,\frac{dx}{\sqrt{1-x^2}}=\left\{ \begin{matrix} 0 &: n\ne m~~~~~\\ \pi &: n=m=0\\ \pi/2 &: n=m\ne 0 \end{matrix} \right. \,\!$

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

$\sqrt{1-x^2} \,\!$

on the interval [−1,1], i.e. we have:

$\int_{-1}^1 U_n(x)U_m(x)\sqrt{1-x^2}\,dx = \begin{cases} 0 &: n\ne m\\ \pi/2 &: n=m \end{cases} \,\!$

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

[edit] Minimal $\infty$ -norm

For any given $1 \le n$ , among the polynomials of degree $n$ with leading coefficient 1, $f(x) = \frac1{2^{n-1}}T_n(x)$ is the one of which the maximal absolute value on the interval $[ - 1,1]$ is minimal. This maximal absolute value is $\frac1{2^{n-1}}$ and $| f (x) |$ reaches this maximum exactly $n + 1$ times: in $- 1$ and $1$ and the other $n - 1$ extremal points of $f$ .

[edit] Relation between Chebyshev polynomials of the first and second kind

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

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

$T_n(x) = \frac{1}{2} (U_n(x) - \, U_{n-2}(x)).$

$T_{n+1}(x) = xT_n(x) - (1 - x^2)U_{n-1}(x)\,$

$T_n(x) = U_n(x) - x \, U_{n-1}(x).$

The recurrence relationship of the derivative of Chebyshev polynomials can be derived from these relations

$2 T_n(x) = \frac{1}{n+1}\; \frac{d}{dx} T_{n+1}(x) - \frac{1}{n-1}\; \frac{d}{dx} T_{n-1}(x) \mbox{ , }\quad n=1,\ldots$

This relationship is used in the Chebyshev spectral method of solving differential equations.

[edit] Other properties

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

For every nonnegative integer $n$ , $T n (x)$ and $U n (x)$ are both polynomials of degree $n$ . They are even or odd functions of $x$ as $n$ is even or odd, so when written as polynomials of $x$ , it only has even or odd degree terms resp.

The leading coefficient of $T n$ is $2 n - 1$ if $1 \le n$ , but $1$ if $0 = n$ .

[edit] Examples

This image shows the first few Chebyshev polynomials of the first kind in the domain −1¼ < x < 1¼, −1¼ < y < 1¼; the flat T0, and T1, T2, T3, T4 and T5.

This image shows the first few Chebyshev polynomials of the first kind in the domain −1¼ < x < 1¼, −1¼ < y < 1¼; the flat T₀, and T₁, T₂, T₃, T₄ and T₅.

The first few Chebyshev polynomials of the first kind are

$T_0(x) = 1 \,$

$T_1(x) = x \,$

$T_2(x) = 2x^2 - 1 \,$

$T_3(x) = 4x^3 - 3x \,$

$T_4(x) = 8x^4 - 8x^2 + 1 \,$

$T_5(x) = 16x^5 - 20x^3 + 5x \,$

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

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

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

$T_9(x) = 256x^9 - 576x^7 + 432x^5 - 120x^3 + 9x. \,$

This image shows the first few Chebyshev polynomials of the second kind in the domain −1¼ < x < 1¼, −1¼ < y < 1¼; the flat U0, and U1, U2, U3, U4 and U5. Although not visible in the image, Un(1) = n + 1 and Un(−1) = (n + 1)(−1)n.

This image shows the first few Chebyshev polynomials of the second kind in the domain −1¼ < x < 1¼, −1¼ < y < 1¼; the flat U₀, and U₁, U₂, U₃, U₄ and U₅. Although not visible in the image, U_n(1) = n + 1 and U_n(−1) = (n + 1)(−1)ⁿ.

The first few Chebyshev polynomials of the second kind are

$U_0(x) = 1 \,$

$U_1(x) = 2x \,$

$U_2(x) = 4x^2 - 1 \,$

$U_3(x) = 8x^3 - 4x \,$

$U_4(x) = 16x^4 - 12x^2 + 1 \,$

$U_5(x) = 32x^5 - 32x^3 + 6x \,$

$U_6(x) = 64x^6 - 80x^4 + 24x^2 - 1. \,$

[edit] Polynomial in Chebyshev form

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

$p(x) = \sum_{n=0}^{N} a_n T_n(x)$

where T_n is the nth Chebyshev polynomial.

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

[edit] 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 definition and the fact that

$\cos\left(\frac{\pi}{2}\,(2k+1)\right)=0$

one can easily prove that the roots of T_n are

$x_k = \cos\left(\frac{\pi}{2}\,\frac{2k-1}{n}\right) \mbox{ , } k=1,\ldots,n.$

Similarly, the roots of U_n are

$x_k = \cos\left(\frac{k}{n+1}\pi\right) \mbox{ , } k=1,\ldots,n.$

[edit] Spread polynomials

The spread polynomials are in a sense equivalent to the Chebyshev polynomials of the first kind, but enable one to avoid square roots and conventional trigonometric functions in certain contexts, notably in rational trigonometry.

[edit] See also

[edit] References

Milton Abramowitz and Irene A. Stegun, eds. (1965). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover. ISBN 0-486-61272-4. (See chapter 22)

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Categories: Special hypergeometric functions | Orthogonal polynomials | Numerical analysis | Approximation theory

Chebyshev polynomials

From Wikipedia, the free encyclopedia

Contents

[edit] Definition

[edit] Trigonometric definition

[edit] Pell equation definition

[edit] Mutual recurrence

[edit] Orthogonality

[edit] Minimal $\infty$ -norm

[edit] Relation between Chebyshev polynomials of the first and second kind

[edit] Other properties

[edit] Examples

[edit] Polynomial in Chebyshev form

[edit] Chebyshev roots

[edit] Spread polynomials

[edit] See also

[edit] References

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Chebyshev polynomials

From Wikipedia, the free encyclopedia

Contents

[edit] Definition

[edit] Trigonometric definition

[edit] Pell equation definition

[edit] Mutual recurrence

[edit] Orthogonality

[edit] Minimal -norm

[edit] Relation between Chebyshev polynomials of the first and second kind

[edit] Other properties

[edit] Examples

[edit] Polynomial in Chebyshev form

[edit] Chebyshev roots

[edit] Spread polynomials

[edit] See also

[edit] References

Views

Navigation

interaction

Search

In other languages

[edit] Minimal $\infty$ -norm