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.

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.

Contents

1 Definition
2 Trigonometric definition
3 Mutual recurrence
4 Orthogonality
5 Basic properties
6 Minimal [\infty]-norm
7 Other properties
8 Examples
9 Polynomial in Chebyshev form
10 Chebyshev roots
11 Spread polynomials
12 See also
13 References

Definition

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

[T_0(x) = 1 \,]

[T_1(x) = x \,]

[T_(x) = 2xT_n(x) - T_(x). \,]

One example of a generating function for this recurrence relation is

[\sum_^T_n(x) t^n = \frac.]

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

[U_0(x) = 1 \,]

[U_1(x) = 2x \,]

[U_(x) = 2xU_n(x) - U_(x). \,]

One example of a generating function for this recurrence relation is

[\sum_^U_n(x) t^n = \frac.]

Trigonometric definition

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

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

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.

An immediate corollary is the composition identity

[T_n(T_m(x)) = T_(x).\,]

Written explicitly

[T_n(x) = \begin\cos(n\arccos(x)), & \ x \in [-1,1] \\\cosh(n \, \mathrm(x)), & \ x \ge 1 \\(-1)^n \cosh(n \, \mathrm(-x)), & \ x \le -1 \\\end]

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^}=\frac)^n+(e^)^}]

and since

[\! e^=\cos(\theta)+i \sin(\theta)]

and

[\sin(\theta)=\sqrt]

we have

[

e^=\cos(\theta)+ \sqrt]

which gives

[\cos(n \theta)=\frac\right)^n+\left(\cos(\theta)+ \sqrt\,\right)^}]

replacing [\cos(\theta)] with x, we have

[ T_n(x)=\frac\right)^n+\left(x+ \sqrt\right)^}]

Although the above form is the one most frequently encountered, if one uses definitions of cos(z), cosh(z), and their inverses for complex z, then

[\beginT_n(x) & = & \cos (n \arccos (x)) \\& = & \mathrm (n \, \mathrm (x))\end\ , \quad \forall x \in \mathbb.]

Similarly, the polynomials of the second kind satisfy

[ U_n(\cos(\theta)) = \frac. ]

This function is very similar to the Dirichlet kernel.

Mutual recurrence

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

[T_0(x) = 1\,]

[U_(x) = 1\,]

[T_(x) = xT_n(x) - (1 - x^2)U_(x)\,]

[U_n(x) = xU_(x) + T_n(x)\,]

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

[T_(x) = T_(\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_] instead.)

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},]

on the interval [−1,1], i.e. we have:

[\int_^1 T_n(x)T_m(x)\,\frac}=\left\0 &: n\ne m~~~~~\\\pi &: n=m=0\\\pi/2 &: n=m\ne 0\end\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]

on the interval [−1,1], i.e. we have:

[\int_^1 U_n(x)U_m(x)\sqrt\,dx = \begin0 &: n\ne m\\\pi/2 &: n=m\end]

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

Basic properties

For every nonnegative integer [n], [T_n(x)] and [U_n(x)] are both polynomials of degree [n]. They and are even or odd functions of [x] as [n] is even and 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^] if [1 \le n], but [1] if [0 = n].

Minimal [\infty]-norm

For any given [1 \le n], among the polynomials of degree [n] with leading coefficient 1, [f(x) = \frac1}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}] and [|f(x)|] reaches this maximum exactly [n+1] times: in [-1] and [1] and the other [n - 1] extremal points of [f].

Other properties

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

[\frac \, T_n(x) = n U_(x) \mbox n=1,\ldots]

[T_n(x) = U_n(x) - x \, U_(x). ]

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

The Chebyshev polynomials of the first kind satisfy the composition identity

[T_n(T_m(x)) = T_(x).\,]

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.

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, 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. \,]

Polynomial in Chebyshev form

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

[p(x) = \sum_^ a_n T_n(x)]

where T_n 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 T_n are

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

Similarly, the roots of U_n are