Chebyshev polynomials

Not to be confused with discrete Chebyshev polynomials.

The Chebyshev polynomials are two sequences of polynomials related to the sine and cosine functions, notated as T_n(x) and U_n(x) . They can be defined several ways that have the same end result; in this article the polynomials are defined by starting with trigonometric functions:

The Chebyshev polynomials of the first kind (T_n) are given by

T_n( cos(θ) ) = cos(n θ) .

Similarly, define the Chebyshev polynomials of the second kind (U_n) as

U_n( cos(θ) ) sin(θ) = sin((n + 1)θ) .

These definitions are not polynomials as such, but using various trig identities they can be converted to polynomial form. For example, for n = 2 the T₂ formula can be converted into a polynomial with argument x = cos(θ) , using the double angle formula:

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

Replacing the terms in the formula with the definitions above, we get

T₂(x) = 2 x² − 1 .

The other T_n(x) are defined similarly, where for the polynomials of the second kind (U_n) we must use de Moivre's formula to get sin(n θ) as sin(θ) times a polynomial in cos(θ) . For instance,

$\sin (3\theta )=(4{\cos }^{}(\theta )-1)\sin (\theta )$

gives

U₂(x) = 4x² − 1 .

Once converted to polynomial form, T_n(x) and U_n(x) are called Chebyshev polynomials of the first and second kind, respectively.

An important and convenient property of the T_n(x) is that they are orthogonal with respect to the inner product

$⟨f(x),g(x)⟩=\underset{-1}{\overset{1}{\int }}f(x)g(x)\frac{\mathrm{d}x}{\sqrt{1-x^2}},$

and U_n(x) are orthogonal with respect to another, analogous inner product product, given below. This follows from the fact that the Chebyshev polynomials solve the Chebyshev differential equations

$(1-x^2)y^{″}-x{y}^{\prime }+n^{2}y=0,$

$(1-x^2)y^{″}-3x{y}^{\prime }+n(n+2)y=0,$

which are Sturm–Liouville differential equations. It is a general feature of such differential equations that there is a distinguished orthonormal set of solutions. (Another way to define the Chebyshev polynomials is as the solutions to those equations.)

The Chebyshev polynomials T_n are polynomials with the largest possible leading coefficient, whose absolute value on the interval [−1, 1] is bounded by 1. They are also the "extremal" polynomials for many other properties.

Chebyshev polynomials are important in approximation theory because the roots of T_n(x) , which are also called Chebyshev nodes, are used as matching-points for optimizing polynomial interpolation. The resulting interpolation polynomial minimizes the problem of Runge's phenomenon, and provides an approximation that is close to the best polynomial approximation to a continuous function under the maximum norm, also called the "minimax" criterion. This approximation leads directly to the method of Clenshaw–Curtis quadrature.

These polynomials were named after Pafnuty Chebyshev. The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebycheff, Tchebyshev (French) or Tschebyschow (German).

Definition

Plot of the first five T_n Chebyshev polynomials of the first kind

The Chebyshev polynomials of the first kind are obtained from the recurrence relation

$\begin{array}{rl}{T}_0(x)& =1\\ T_1(x)& =x\\ T_{n+1}(x)& =2x{T}_n(x)-T_{n-1}(x).\end{array}$

The ordinary generating function for T_n is

$\sum _{n=0}^{\mathrm{\infty }}{T}_n(x)t^n=\frac{1-tx}{1-2tx+t^2}.$

Proof —

$\begin{array}{rl}\text{Define }G& \equiv \sum _{n=0}^{\mathrm{\infty }}{T}_n(x)t^n\\ & =T_0(x)+t{T}_1(x)+\sum _{n=2}^{\mathrm{\infty }}{T}_n(x)t^n\\ & =1+tx+\sum _{n=0}^{\mathrm{\infty }}{T}_{n+2}(x)t^{n+2}\\ & =1+tx+\sum _{n=0}^{\mathrm{\infty }}(2x{T}_{n+1}(x)-T_n(x))t^{n+2}\\ & =1+tx+\sum _{n=0}^{\mathrm{\infty }}2x{T}_{n+1}(x)t^{n+2}-\sum _{n=0}^{\mathrm{\infty }}{T}_n(x)t^{n+2}\\ & =1+tx+2tx\sum _{n=0}^{\mathrm{\infty }}{T}_{n+1}(x)t^{n+1}-t^2\sum _{n=0}^{\mathrm{\infty }}{T}_n(x)t^n\\ & =1+tx+2tx(\sum _{n=0}^{\mathrm{\infty }}{T}_n(x)t^n-1)-t^2\sum _{n=0}^{\mathrm{\infty }}{T}_n(x)t^n\\ & =1+tx+2tx(G-1)-t^{2}G\\ & =1+tx+2txG-2tx-t^{2}G\\ G-2txG+t^{2}G& =1+tx-2tx\\ G& =\frac{1-tx}{1-2tx+t^2}\end{array}$

There are several other generating functions for the Chebyshev polynomials; the exponential generating function is

$\sum _{n=0}^{\mathrm{\infty }}{T}_n(x)\frac{t^n}{n!}=\frac{1}{2}(e^{t(x-\sqrt{x^2-1})}+e^{t(x+\sqrt{x^2-1})})=e^{tx}\cosh \left(t\sqrt{x^2-1}\right).$

The generating function relevant for 2-dimensional potential theory and multipole expansion is

$\sum _{n=1}^{\mathrm{\infty }}{T}_n(x)\frac{t^n}{n}=\ln \left(\frac{1}{\sqrt{1-2tx+t^2}}\right).$

Plot of the first five U_n Chebyshev polynomials of the second kind

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

$\begin{array}{rl}{U}_0(x)& =1\\ U_1(x)& =2x\\ U_{n+1}(x)& =2x{U}_n(x)-U_{n-1}(x).\end{array}$

Notice that the two sets of recurrence relations are identical, except for $T_1(x)=x$ vs. $U_1(x)=2x.$ The ordinary generating function for U_n is

$\sum _{n=0}^{\mathrm{\infty }}{U}_n(x)t^n=\frac{1}{1-2tx+t^2};$

the exponential generating function is

$\sum _{n=0}^{\mathrm{\infty }}{U}_n(x)\frac{t^n}{n!}=e^{tx}(\cosh \left(t\sqrt{x^2-1}\right)+\frac{x}{\sqrt{x^2-1}}\sinh \left(t\sqrt{x^2-1}\right)).$

Trigonometric definition

As described in the introduction, the Chebyshev polynomials of the first kind can be defined as the unique polynomials satisfying

$T_n(x)=\{\begin{array}{ll}\cos (n\arccos x)& \text{ if }\left|x\right|\le 1\\ \cosh (n\mathrm{arcosh}x)& \text{ if }x\ge 1\\ (-1{)}^n\cosh (n\mathrm{arcosh}(-x))& \text{ if }x\le -1\end{array}$

or, in other words, as the unique polynomials satisfying

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

for n = 0, 1, 2, 3, ... which as a technical point is a variant (equivalent transpose) of Schröder's equation. That is, T_n(x) is functionally conjugate to n x, codified in the nesting property below. Further compare to the spread polynomials, in the section below.

The polynomials of the second kind satisfy:

$U_{n-1}(\cos \theta )\cdot \sin \theta =\sin (n\theta ),$

or

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

which is structurally quite similar to the Dirichlet kernel D_n(x):

$D_n(x)=\frac{\sin ((2n+1)\frac{x}{2})}{\sin \frac{x}{2}}=U_{2n}(\cos \frac{x}{2}).$

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. 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 through the identity cos² x + sin² x = 1. By the same reasoning, sin nx is the imaginary part of the polynomial, in which all powers of sin x are odd and thus, if one is factored out, the remaining can be replaced to create a (n-1)th-degree polynomial in cos x.

The identity is quite 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,

$T_0(\cos \theta )=\cos 0\theta =1$

and

$T_1(\cos \theta )=\cos \theta ,$

one can straightforwardly determine that

$\begin{array}{rl}\cos 2\theta & =2\cos \theta \cos \theta -1=2{\cos }^{}\theta -1\\ \cos 3\theta & =2\cos \theta \cos 2\theta -\cos \theta =4{\cos }^{}\theta -3\cos \theta ,\end{array}$

and so forth.

Two immediate corollaries are the composition identity (or nesting property specifying a semigroup)

$T_n(T_m(x))=T_{nm}(x);$

and the expression of complex exponentiation in terms of Chebyshev polynomials: given z = a + bi,

$\begin{array}{rl}{z}^n& =|z{|}^n(\cos (n\arccos \frac{a}{\left|z\right|})+i\sin (n\arccos \frac{a}{\left|z\right|}))\\ & =|z{|}^n{T}_n\left(\frac{a}{\left|z\right|}\right)+ib|z{|}^{n-1}{U}_{n-1}\left(\frac{a}{\left|z\right|}\right).\end{array}$

Pell equation definition

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

$T_n(x{)}^2-(x^2-1)U_{n-1}(x{)}^2=1$

in a ring R[x]. Thus, they can be generated by the standard technique for Pell equations of taking powers of a fundamental solution:

$T_n(x)+U_{n-1}(x)\sqrt{x^2-1}={(x+\sqrt{x^2-1})}^n.$

Products of Chebyshev polynomials

When working with Chebyshev polynomials quite often products of two of them occur. These products can be reduced to combinations of Chebyshev polynomials with lower or higher degree and concluding statements about the product are easier to make. It shall be assumed that in the following the index m is greater than or equal to the index n and n is not negative. For Chebyshev polynomials of the first kind the product expands to

$2T_m(x)T_n(x)=T_{m+n}(x)+T_{|m-n|}(x)$

which is an analogy to the addition theorem

$2\cos \alpha \cos \beta =\cos (\alpha +\beta )+\cos (\alpha -\beta )$

with the identities

$\alpha \equiv m\arccos x\text{ and }\beta \equiv n\arccos x.$

For n = 1 this results in the already known recurrence formula, just arranged differently, and with n = 2 it forms the recurrence relation for all even or all odd Chebyshev polynomials (depending on the parity of the lowest m) which allows to design functions with prescribed symmetry properties. Three more useful formulas for evaluating Chebyshev polynomials can be concluded from this product expansion:

$\begin{array}{rlrl}{T}_{2n}(x)& =2T_n^2(x)-T_0(x)& & =2T_n^2(x)-1\\ T_{2n+1}(x)& =2T_{n+1}(x)T_n(x)-T_1(x)& & =2T_{n+1}(x)T_n(x)-x\\ T_{2n-1}(x)& =2T_{n-1}(x)T_n(x)-T_1(x)& & =2T_{n-1}(x)T_n(x)-x\end{array}$

For Chebyshev polynomials of the second kind, products may be written as:

$U_m(x)U_n(x)=\sum _{k=0}^n{U}_{m-n+2k}(x)=\sum _{\underset{\text{ step 2 }}{p=m-n}}^{m+n}{U}_p(x).$

for m ≥ n.

By this, like above, with n = 2 the recurrence formula for Chebyshev polynomials of the second kind reduces for both types of symmetry to

$U_{m+2}(x)=U_2(x)U_m(x)-U_m(x)-U_{m-2}(x)=U_m(x)(U_2(x)-1)-U_{m-2}(x),$

depending on whether m starts with 2 or 3.

Relations between the two kinds of Chebyshev polynomials

The Chebyshev polynomials of the first and second kinds correspond to a complementary pair of Lucas sequences Ṽ_n(P,Q) and Ũ_n(P,Q) with parameters P = 2x and Q = 1:

$\begin{array}{rl}{\tilde{U}}_n(2x,1)& =U_{n-1}(x),\\ {\tilde{V}}_n(2x,1)& =2T_n(x).\end{array}$

It follows that they also satisfy a pair of mutual recurrence equations:

$\begin{array}{rl}{T}_{n+1}(x)& =x{T}_n(x)-(1-x^2)U_{n-1}(x),\\ U_{n+1}(x)& =x{U}_n(x)+T_{n+1}(x).\end{array}$

The Chebyshev polynomials of the first and second kinds are also connected by the following relations:

$\begin{array}{rlrl}{T}_n(x)& =\frac{1}{2}(U_n(x)-U_{n-2}(x)).& & \\ T_n(x)& =U_n(x)-x{U}_{n-1}(x).& & \\ U_n(x)& =2\sum _{\text{ odd }j}^n{T}_j(x)& & \text{ for odd }n.\\ U_n(x)& =2\sum _{\text{ even }j}^n{T}_j(x)-1& & \text{ for even }n.\end{array}$

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

$2T_n(x)=\frac{1}{n+1}\frac{\mathrm{d}}{\mathrm{d}x}{T}_{n+1}(x)-\frac{1}{n-1}\frac{\mathrm{d}}{\mathrm{d}x}{T}_{n-1}(x)n=2,3,\dots$

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

Turán's inequalities for the Chebyshev polynomials are

$\begin{array}{rlrlrl}{T}_n(x{)}^2-T_{n-1}(x)T_{n+1}(x)& =1-x^2>0& & \text{ for }-1<x<1& & \text{ and }\\ U_n(x{)}^2-U_{n-1}(x)U_{n+1}(x)& =1>0.\end{array}$

The integral relations are

$\begin{array}{rl}\underset{-1}{\overset{1}{\int }}\frac{T_n(y)\mathrm{d}y}{(y-x)\sqrt{1-y^2}}& =\pi U_{n-1}(x),\\ \underset{-1}{\overset{1}{\int }}\frac{\sqrt{1-y^2}{U}_{n-1}(y)\mathrm{d}y}{y-x}& =-\pi T_n(x)\end{array}$

where integrals are considered as principal value.

Explicit expressions

Different approaches to defining Chebyshev polynomials lead to different explicit expressions such as:

$\begin{array}{rl}{T}_n(x)& =\{\begin{array}{ll}\cos (n\arccos x)& \text{ for }\left|x\right|\le 1\\ \\ \frac{1}{2}\left(\right(x-\sqrt{x^2-1}{)}^n+(x+\sqrt{x^2-1}{)}^n)& \text{ for }\left|x\right|\ge 1\\ \end{array}\\ \\ & =\{\begin{array}{ll}\cos (n\arccos x)& \text{ for }-1\le x\le 1\\ \\ \cosh (n\mathrm{arcosh}x)& \text{ for }1\le x\\ \\ (-1{)}^n\cosh (n\mathrm{arcosh}(-x))& \text{ for }x\le -1\\ \end{array}\\ \\ \\ T_n(x)& =\sum _{k=0}^{\lfloor \frac{n}{2}\rfloor }(\genfrac{}{}{0ex}{}{n}{2k}){(x^2-1)}^k{x}^{n-2k}\\ & =x^n\sum _{k=0}^{\lfloor \frac{n}{2}\rfloor }(\genfrac{}{}{0ex}{}{n}{2k}){(1-x^{-2})}^k\\ & =\frac{n}{2}\sum _{k=0}^{\lfloor \frac{n}{2}\rfloor }(-1{)}^k\frac{(n-k-1)!}{k!(n-2k)!}(2x{)}^{n-2k}\text{ for }n>0\\ \\ & =n\sum _{k=0}^n(-2{)}^k\frac{(n+k-1)!}{(n-k)!(2k)!}(1-x{)}^k\text{ for }n>0\\ \\ & ={}_2{F}_1(-n,n;\frac{1}{2};\frac{1}{2}(1-x))\\ \end{array}$

with inverse

$x^n=2^{1-n}{{{\sum }^{\prime }}^{\prime }}_{j=0,n-j\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}}^n(\genfrac{}{}{0ex}{}{n}{\frac{n-j}{2}})T_j(x),$

where the prime at the sum symbol indicates that the contribution of j = 0 needs to be halved if it appears.

$\begin{array}{rl}{U}_n(x)& =\frac{{(x+\sqrt{x^2-1})}^{n+1}-{(x-\sqrt{x^2-1})}^{n+1}}{2\sqrt{x^2-1}}\\ & =\sum _{k=0}^{\lfloor \frac{n}{2}\rfloor }(\genfrac{}{}{0ex}{}{n+1}{2k+1}){(x^2-1)}^k{x}^{n-2k}\\ & =x^n\sum _{k=0}^{\lfloor \frac{n}{2}\rfloor }(\genfrac{}{}{0ex}{}{n+1}{2k+1}){(1-x^{-2})}^k\\ & =\sum _{k=0}^{\lfloor \frac{n}{2}\rfloor }(\genfrac{}{}{0ex}{}{2k-(n+1)}{k})(2x{)}^{n-2k}& \text{ for }n>0\\ & =\sum _{k=0}^{\lfloor \frac{n}{2}\rfloor }(-1{)}^k(\genfrac{}{}{0ex}{}{n-k}{k})(2x{)}^{n-2k}& \text{ for }n>0\\ & =\sum _{k=0}^n(-2{)}^k\frac{(n+k+1)!}{(n-k)!(2k+1)!}(1-x{)}^k& \text{ for }n>0\\ & =(n+1){}_2{F}_1(-n,n+2;\frac{3}{2};\frac{1}{2}(1-x))\\ \end{array}$

where ₂F₁ is a hypergeometric function.

Properties

Symmetry

$\begin{array}{rl}{T}_n(-x)& =(-1{)}^n{T}_n(x)=\{\begin{array}{ll}{T}_n(x)& \text{ for }n\text{ even}\\ \\ -T_n(x)& \text{ for }n\text{ odd}\end{array}\\ \\ \\ U_n(-x)& =(-1{)}^n{U}_n(x)=\{\begin{array}{ll}{U}_n(x)& \text{ for }n\text{ even}\\ \\ -U_n(x)& \text{ for }n\text{ odd}\end{array}\\ \end{array}$

That is, Chebyshev polynomials of even order have even symmetry and contain only even powers of x. Chebyshev polynomials of odd order have odd symmetry and contain only odd powers of x.

Roots and extrema

A Chebyshev polynomial of either kind with degree n has n different simple roots, called Chebyshev roots, in the interval [−1, 1] . The roots of the Chebyshev polynomial of the first kind are sometimes called Chebyshev nodes because they are used as nodes in polynomial interpolation. Using the trigonometric definition and the fact that

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

one can show that the roots of T_n are

$x_k=\cos \left(\frac{\pi (k+1/2)}{n}\right),k=0,\dots ,n-1.$

Similarly, the roots of U_n are

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

The extrema of T_n on the interval −1 ≤ x ≤ 1 are located at

$x_k=\cos \left(\frac{k}{n}\pi \right),k=0,\dots ,n.$

One unique property of the Chebyshev polynomials of the first kind is that on the interval −1 ≤ x ≤ 1 all of the extrema have values that are either −1 or 1. Thus these polynomials have only two finite critical values, the defining property of Shabat polynomials. Both the first and second kinds of Chebyshev polynomial have extrema at the endpoints, given by:

$T_n(1)=1$

$T_n(-1)=(-1{)}^n$

$U_n(1)=n+1$

$U_n(-1)=(-1{)}^n(n+1).$

Differentiation and integration

The derivatives of the polynomials can be less than straightforward. By differentiating the polynomials in their trigonometric forms, it can be shown that:

$\begin{array}{rl}\frac{\mathrm{d}{T}_n}{\mathrm{d}x}& =n{U}_{n-1}\\ \frac{\mathrm{d}{U}_n}{\mathrm{d}x}& =\frac{(n+1)T_{n+1}-x{U}_n}{x^2-1}\\ \frac{{\mathrm{d}}^2{T}_n}{\mathrm{d}{x}^2}& =n\frac{n{T}_n-x{U}_{n-1}}{x^2-1}=n\frac{(n+1)T_n-U_n}{x^2-1}.\end{array}$

The last two formulas can be numerically troublesome due to the division by zero (0/0 indeterminate form, specifically) at x = 1 and x = −1. It can be shown that:

$\begin{array}{rl}{\frac{{\mathrm{d}}^2{T}_n}{\mathrm{d}{x}^2}|}_{x=1}& =\frac{n^4-n^2}{3},\\ {\frac{{\mathrm{d}}^2{T}_n}{\mathrm{d}{x}^2}|}_{x=-1}& =(-1{)}^n\frac{n^4-n^2}{3}.\end{array}$

Proof —

The second derivative of the Chebyshev polynomial of the first kind is

$T^{\prime }{\text{'}}_n=n\frac{n{T}_n-x{U}_{n-1}}{x^2-1}$

which, if evaluated as shown above, poses a problem because it is indeterminate at x = ±1. Since the function is a polynomial, (all of) the derivatives must exist for all real numbers, so the taking to limit on the expression above should yield the desired value:

$T^{\prime }{\text{'}}_n(1)=\underset{x\to 1}{\lim }n\frac{n{T}_n-x{U}_{n-1}}{x^2-1}$

where only x = 1 is considered for now. Factoring the denominator:

$T^{\prime }{\text{'}}_n(1)=\underset{x\to 1}{\lim }n\frac{n{T}_n-x{U}_{n-1}}{(x+1)(x-1)}=\underset{x\to 1}{\lim }n\frac{\frac{n{T}_n-x{U}_{n-1}}{x-1}}{x+1}.$

Since the limit as a whole must exist, the limit of the numerator and denominator must independently exist, and

$T^{\prime }{\text{'}}_n(1)=n\frac{\underset{x\to 1}{\lim }\frac{n{T}_n-x{U}_{n-1}}{x-1}}{\underset{x\to 1}{\lim }(x+1)}=\frac{n}{2}\underset{x\to 1}{\lim }\frac{n{T}_n-x{U}_{n-1}}{x-1}.$

The denominator (still) limits to zero, which implies that the numerator must be limiting to zero, i.e. U_{n − 1}(1) = nT_n(1) = n which will be useful later on. Since the numerator and denominator are both limiting to zero, L'Hôpital's rule applies:

$\begin{array}{rl}{T}^{\prime }{\text{'}}_n(1)& =\frac{n}{2}\underset{x\to 1}{\lim }\frac{\frac{\mathrm{d}}{\mathrm{d}x}(n{T}_n-x{U}_{n-1})}{\frac{\mathrm{d}}{\mathrm{d}x}(x-1)}\\ & =\frac{n}{2}\underset{x\to 1}{\lim }\frac{\mathrm{d}}{\mathrm{d}x}(n{T}_n-x{U}_{n-1})\\ & =\frac{n}{2}\underset{x\to 1}{\lim }(n^2{U}_{n-1}-U_{n-1}-x\frac{\mathrm{d}}{\mathrm{d}x}\left(U_{n-1}\right))\\ & =\frac{n}{2}(n^2{U}_{n-1}(1)-U_{n-1}(1)-\underset{x\to 1}{\lim }x\frac{\mathrm{d}}{\mathrm{d}x}\left(U_{n-1}\right))\\ & =\frac{n^4}{2}-\frac{n^2}{2}-\frac{1}{2}\underset{x\to 1}{\lim }\frac{\mathrm{d}}{\mathrm{d}x}\left(n{U}_{n-1}\right)\\ & =\frac{n^4}{2}-\frac{n^2}{2}-\frac{T^{\prime }{\text{'}}_n(1)}{2}\\ T^{\prime }{\text{'}}_n(1)& =\frac{n^4-n^2}{3}.\\ \end{array}$

The proof for x = −1 is similar, with the fact that T_n(−1) = (−1)ⁿ being important.

Indeed, the following, more general formula holds:

${\frac{d^p{T}_n}{d{x}^p}|}_{x=\pm 1}=(\pm 1{)}^{n+p}\prod _{k=0}^{p-1}\frac{n^2-k^2}{2k+1}.$

This latter result is of great use in the numerical solution of eigenvalue problems.

$\frac{{\mathrm{d}}^p}{\mathrm{d}{x}^p}{T}_n(x)=2^{p}n{{{\sum }^{\prime }}^{\prime }}_{0\le k\le n-p,n-p-k\text{ even }}(\genfrac{}{}{0ex}{}{\frac{n+p-k}{2}-1}{\frac{n-p-k}{2}})\frac{(\frac{n+p+k}{2}-1)!}{\left(\frac{n-p+k}{2}\right)!}{T}_k(x),p\ge 1,$

where the prime at the summation symbols means that the term contributed by k = 0 is to be halved, if it appears.

Concerning integration, the first derivative of the T_n implies that

$\int U_n\mathrm{d}x=\frac{T_{n+1}}{n+1}$

and the recurrence relation for the first kind polynomials involving derivatives establishes that for n ≥ 2

$\int T_n\mathrm{d}x=\frac{1}{2}(\frac{T_{n+1}}{n+1}-\frac{T_{n-1}}{n-1})=\frac{n{T}_{n+1}}{n^2-1}-\frac{x{T}_n}{n-1}.$

The latter formula can be further manipulated to express the integral of T_n as a function of Chebyshev polynomials of the first kind only:

$\int T_n\mathrm{d}x=\frac{n}{n^2-1}{T}_{n+1}-\frac{1}{n-1}{T}_1{T}_n=\frac{n}{n^2-1}{T}_{n+1}-\frac{1}{2(n-1)}(T_{n+1}+T_{n-1})=\frac{1}{2(n+1)}{T}_{n+1}-\frac{1}{2(n-1)}{T}_{n-1}.$

Furthermore, we have

$\underset{-1}{\overset{1}{\int }}{T}_n(x)\mathrm{d}x=\{\begin{array}{ll}\frac{(-1{)}^n+1}{1-n^2}& \text{ if }n\ne 1\\ 0& \text{ if }n=1\end{array}.$

Orthogonality

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

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

on the interval [−1, 1], i.e. we have:

$\underset{-1}{\overset{1}{\int }}{T}_n(x)T_m(x)\frac{\mathrm{d}x}{\sqrt{1-x^2}}=\{\begin{array}{ll}0& \text{ if }n\ne m,\\ \\ \pi & \text{ if }n=m=0,\\ \\ \frac{\pi }{2}& \text{ if }n=m\ne 0.\end{array}$

This can be proven by letting x = cos θ and using the defining identity T_n(cos θ) = cos nθ.

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

$\sqrt{1-x^2}$

on the interval [−1, 1], i.e. we have:

$\underset{-1}{\overset{1}{\int }}{U}_n(x)U_m(x)\sqrt{1-x^2}\mathrm{d}x=\{\begin{array}{ll}0& \text{ if }n\ne m,\\ \frac{\pi }{2}& \text{ if }n=m.\end{array}$

(The measure √1 − x² dx is, to within a normalizing constant, the Wigner semicircle distribution.)

The T_n also satisfy a discrete orthogonality condition:

$\sum _{k=0}^{N-1}{T}_i(x_k)T_j(x_k)=\{\begin{array}{ll}0& \text{ if }i\ne j,\\ N& \text{ if }i=j=0,\\ \frac{N}{2}& \text{ if }i=j\ne 0,\end{array}$

where N is any integer greater than i+j, and the x_k are the N Chebyshev nodes (see above) of T_N(x):

$x_k=\cos \left(\pi \frac{2k+1}{2N}\right)\text{ for }k=0,1,\dots ,N-1.$

For the polynomials of the second kind and any integer N>i+j with the same Chebyshev nodes x_k, there are similar sums:

$\sum _{k=0}^{N-1}{U}_i(x_k)U_j(x_k)(1-x_k^2)=\{\begin{array}{ll}0& \text{ if }i\ne j,\\ \frac{N}{2}& \text{ if }i=j,\end{array}$

and without the weight function:

$\sum _{k=0}^{N-1}{U}_i(x_k)U_j(x_k)=\{\begin{array}{ll}0& \text{ if }i\equiv ̸j(\mathrm{mod}2),\\ N\cdot (1+\min \{i,j\})& \text{ if }i\equiv j(\mathrm{mod}2).\end{array}$

For any integer N>i+j, based on the N zeros of U_N(x):

$y_k=\cos \left(\pi \frac{k+1}{N+1}\right)\text{ for }k=0,1,\dots ,N-1,$

one can get the sum:

$\sum _{k=0}^{N-1}{U}_i(y_k)U_j(y_k)(1-y_k^2)=\{\begin{array}{ll}0& \text{ if }i\ne j,\\ \frac{N+1}{2}& \text{ if }i=j,\end{array}$

and again without the weight function:

$\sum _{k=0}^{N-1}{U}_i(y_k)U_j(y_k)=\{\begin{array}{ll}0& \text{ if }i\equiv ̸j(\mathrm{mod}2),\\ (\min \{i,j\}+1)(N-\max \{i,j\})& \text{ if }i\equiv j(\mathrm{mod}2).\end{array}$

Minimal ∞-norm

For any given n ≥ 1, among the polynomials of degree n with leading coefficient 1 (monic polynomials),

$f(x)=\frac{1}{2^{n-1}}{T}_n(x)$

is the one of which the maximal absolute value on the interval [−1, 1] is minimal.

This maximal absolute value is

$\frac{1}{2^{n-1}}$

and |f(x)| reaches this maximum exactly n + 1 times at

$x=\cos \frac{k\pi }{n}\text{for }0\le k\le n.$

Proof —

Let's assume that w_n(x) is a polynomial of degree n with leading coefficient 1 with maximal absolute value on the interval [−1,1] less than 1 / 2^{n − 1}.

Define

$f_n(x)=\frac{1}{2^{n-1}}{T}_n(x)-w_n(x)$

Because at extreme points of T_n we have

$\begin{array}{rl}|w_n(x)|& <|\frac{1}{2^{n-1}}{T}_n(x)|\\ f_n(x)& >0\text{ for }x=\cos \frac{2k\pi }{n}& & \text{ where }0\le 2k\le n\\ f_n(x)& <0\text{ for }x=\cos \frac{(2k+1)\pi }{n}& & \text{ where }0\le 2k+1\le n\end{array}$

From the intermediate value theorem, f_n(x) has at least n roots. However, this is impossible, as f_n(x) is a polynomial of degree n − 1, so the fundamental theorem of algebra implies it has at most n − 1 roots.

Remark: By the Equioscillation theorem, among all the polynomials of degree ≤ n, the polynomial f minimizes ||f||_∞ on [−1,1] if and only if there are n + 2 points −1 ≤ x₀ < x₁ < ... < x_{n + 1} ≤ 1 such that |f(x_i)| = ||f||_∞.

Of course, the null polynomial on the interval [−1,1] can be approach by itself and minimizes the ∞-norm.

Above, however, |f| reaches its maximum only n + 1 times because we are searching for the best polynomial of degree n ≥ 1 (therefore the theorem evoked previously cannot be used).

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:

$\begin{array}{rl}{T}_n(x)& =\frac{n}{2}\underset{q\to 0}{\lim }\frac{1}{q}{C}_n^{(q)}(x)\text{ if }n\ge 1,\\ \\ U_n(x)& =\frac{n+1}{(\genfrac{}{}{0ex}{}{n+\frac{1}{2}}{n})}{P}_n^{(\frac{1}{2},\frac{1}{2})}(x)=\frac{n+1}{(\genfrac{}{}{0ex}{}{n+\frac{1}{2}}{n})}{C}_n^{(1)}(x).\end{array}$

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 respectively. In fact,

$T_{2n}(x)=T_n(2x^2-1)=2T_n(x{)}^2-1$

and

$2x{U}_n(1-2x^2)=(-1{)}^n{U}_{2n+1}(x).$

The leading coefficient of T_n is 2^{n − 1} if 1 ≤ n, but 1 if 0 = n .

T_n are a special case of Lissajous curves with frequency ratio equal to n.

Several polynomial sequences like Lucas polynomials (L_n), Dickson polynomials (D_n), Fibonacci polynomials (F_n) are related to Chebyshev polynomials T_n and U_n.

The Chebyshev polynomials of the first kind satisfy the relation

$T_j(x)T_k(x)=\frac{1}{2}(T_{j+k}(x)+T_{|k-j|}(x)),\mathrm{\forall }j,k\ge 0,$

which is easily proved from the product-to-sum formula for the cosine. The polynomials of the second kind satisfy the similar relation

$T_j(x)U_k(x)=\{\begin{array}{ll}\frac{1}{2}(U_{j+k}(x)+U_{k-j}(x)),& \text{ if }k\ge j-1,\\ \\ \frac{1}{2}(U_{j+k}(x)-U_{j-k-2}(x)),& \text{ if }k\le j-2.\end{array}$

(with the definition U₋₁ ≡ 0 by convention ).

Similar to the formula

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

we have the analogous formula

$T_{2n+1}(\sin \theta )=(-1{)}^n\sin ((2n+1)\theta ).$

For x ≠ 0,

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

and

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

which follows from the fact that this holds by definition for x = e^iθ.

Define

$C_n(x)\equiv 2T_n\left(\frac{x}{2}\right).$

Then C_n(x) and C_m(x) are commuting polynomials:

$C_n(C_m(x))=C_m(C_n(x)),$

as is evident in the Abelian nesting property specified above.

Generalized Chebyshev polynomials

The generalized Chebyshev polynomials T_a are defined by

$T_a(\cos x)={}_2{F}_1(a,-a;\frac{1}{2};\frac{1}{2}(1-\cos x))=\cos ax,x\in (-\pi ,\pi ),$

where a is not necessarily an integer, and ₂F₁(a, b; c; z) is the Gaussian hypergeometric function; as an example $T_{1/2}(x)=\sqrt{\frac{1+x}{2}}$. The power series expansion

$T_a(x)=\cos \left(\frac{\pi a}{2}\right)+a\sum _{j=1}\frac{(2x{)}^j}{2j}\cos \left(\frac{\pi (a-j)}{2}\right)(\genfrac{}{}{0ex}{}{\frac{a+j-2}{2}}{j-1})$

converges for $x\in [-1,1].$

Examples

First kind

The first few Chebyshev polynomials of the first kind in the domain −1 < x < 1: The flat T₀, T₁, T₂, T₃, T₄ and T₅.

The first few Chebyshev polynomials of the first kind are OEIS: A028297

$\begin{array}{rl}{T}_0(x)& =1\\ T_1(x)& =x\\ T_2(x)& =2x^2-1\\ T_3(x)& =4x^3-3x\\ T_4(x)& =8x^4-8x^2+1\\ T_5(x)& =16x^5-20x^3+5x\\ T_6(x)& =32x^6-48x^4+18x^2-1\\ T_7(x)& =64x^7-112x^5+56x^3-7x\\ T_8(x)& =128x^8-256x^6+160x^4-32x^2+1\\ T_9(x)& =256x^9-576x^7+432x^5-120x^3+9x\\ T_{10}(x)& =512x^{10}-1280x^8+1120x^6-400x^4+50x^2-1\\ T_{11}(x)& =1024x^{11}-2816x^9+2816x^7-1232x^5+220x^3-11x\end{array}$

Second kind

The first few Chebyshev polynomials of the second kind in the domain −1 < x < 1: The flat U₀, 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 OEIS: A053117

$\begin{array}{rl}{U}_0(x)& =1\\ U_1(x)& =2x\\ U_2(x)& =4x^2-1\\ U_3(x)& =8x^3-4x\\ U_4(x)& =16x^4-12x^2+1\\ U_5(x)& =32x^5-32x^3+6x\\ U_6(x)& =64x^6-80x^4+24x^2-1\\ U_7(x)& =128x^7-192x^5+80x^3-8x\\ U_8(x)& =256x^8-448x^6+240x^4-40x^2+1\\ U_9(x)& =512x^9-1024x^7+672x^5-160x^3+10x\end{array}$

As a basis set

The non-smooth function (top) y = −x³H(−x), where H is the Heaviside step function, and (bottom) the 5th partial sum of its Chebyshev expansion. The 7th sum is indistinguishable from the original function at the resolution of the graph.

In the appropriate Sobolev space, the set of Chebyshev polynomials form an orthonormal basis, so that a function in the same space can, on −1 ≤ x ≤ 1 be expressed via the expansion:

$f(x)=\sum _{n=0}^{\mathrm{\infty }}{a}_n{T}_n(x).$

Furthermore, as mentioned previously, the Chebyshev polynomials form an orthogonal basis which (among other things) implies that the coefficients a_n can be determined easily through the application of an inner product. This sum is called a Chebyshev series or a Chebyshev expansion.

Since a Chebyshev series is related to a Fourier cosine series through a change of variables, all of the theorems, identities, etc. that apply to Fourier series have a Chebyshev counterpart. These attributes include:

• The Chebyshev polynomials form a complete orthogonal system.
• The Chebyshev series converges to f(x) if the function is piecewise smooth and continuous. The smoothness requirement can be relaxed in most cases – as long as there are a finite number of discontinuities in f(x) and its derivatives.
• At a discontinuity, the series will converge to the average of the right and left limits.

The abundance of the theorems and identities inherited from Fourier series make the Chebyshev polynomials important tools in numeric analysis; for example they are the most popular general purpose basis functions used in the spectral method, often in favor of trigonometric series due to generally faster convergence for continuous functions (Gibbs' phenomenon is still a problem).

Example 1

Consider the Chebyshev expansion of log(1 + x). One can express

$\log (1+x)=\sum _{n=0}^{\mathrm{\infty }}{a}_n{T}_n(x).$

One can find the coefficients a_n either through the application of an inner product or by the discrete orthogonality condition. For the inner product,

$\underset{-1}{\overset{+1}{\int }}\frac{T_m(x)\log (1+x)}{\sqrt{1-x^2}}\mathrm{d}x=\sum _{n=0}^{\mathrm{\infty }}{a}_n\underset{-1}{\overset{+1}{\int }}\frac{T_m(x)T_n(x)}{\sqrt{1-x^2}}\mathrm{d}x,$

which gives

$a_n=\{\begin{array}{ll}-\log 2& \text{ for }n=0,\\ \frac{-2(-1{)}^n}{n}& \text{ for }n>0.\end{array}$

Alternatively, when the inner product of the function being approximated cannot be evaluated, the discrete orthogonality condition gives an often useful result for approximate coefficients,

$a_n\approx \frac{2-{\delta }_{0n}}{N}\sum _{k=0}^{N-1}{T}_n(x_k)\log (1+x_k),$

where δ_ij is the Kronecker delta function and the x_k are the N Gauss–Chebyshev zeros of T_N(x):

$x_k=\cos \left(\frac{\pi (k+\frac{1}{2})}{N}\right).$

For any N, these approximate coefficients provide an exact approximation to the function at x_k with a controlled error between those points. The exact coefficients are obtained with N = ∞, thus representing the function exactly at all points in [−1,1]. The rate of convergence depends on the function and its smoothness.

This allows us to compute the approximate coefficients a_n very efficiently through the discrete cosine transform

$a_n\approx \frac{2-{\delta }_{0n}}{N}\sum _{k=0}^{N-1}\cos \left(\frac{n\pi (k+\frac{1}{2})}{N}\right)\log (1+x_k).$

Example 2

To provide another example:

$\begin{array}{rl}(1-x^2{)}^{\alpha }& =-\frac{1}{\sqrt{\pi }}\frac{\mathrm{\Gamma }(\frac{1}{2}+\alpha )}{\mathrm{\Gamma }(\alpha +1)}+2^{1-2\alpha }\sum _{n=0}(-1{)}^n(\genfrac{}{}{0ex}{}{2\alpha }{\alpha -n})T_{2n}(x)\\ & =2^{-2\alpha }\sum _{n=0}(-1{)}^n(\genfrac{}{}{0ex}{}{2\alpha +1}{\alpha -n})U_{2n}(x).\end{array}$

Partial sums

The partial sums of

$f(x)=\sum _{n=0}^{\mathrm{\infty }}{a}_n{T}_n(x)$

are very useful in the approximation of various functions and in the solution of differential equations (see spectral method). Two common methods for determining the coefficients a_n are through the use of the inner product as in Galerkin's method and through the use of collocation which is related to interpolation.

As an interpolant, the N coefficients of the (N − 1)th partial sum are usually obtained on the Chebyshev–Gauss–Lobatto points (or Lobatto grid), which results in minimum error and avoids Runge's phenomenon associated with a uniform grid. This collection of points corresponds to the extrema of the highest order polynomial in the sum, plus the endpoints and is given by:

$x_k=-\cos \left(\frac{k\pi }{N-1}\right);k=0,1,\dots ,N-1.$

Polynomial in Chebyshev form

An arbitrary polynomial of degree N can be written in terms of the Chebyshev polynomials of the first kind. Such a polynomial p(x) is of the form

$p(x)=\sum _{n=0}^N{a}_n{T}_n(x).$

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

Shifted Chebyshev polynomials

Shifted Chebyshev polynomials of the first kind are defined as

$T_n^{*}(x)=T_n(2x-1).$

When the argument of the Chebyshev polynomial is in the range of 2x − 1 ∈ [−1, 1] the argument of the shifted Chebyshev polynomial is x ∈ [0, 1]. Similarly, one can define shifted polynomials for generic intervals [a,b].

Spread polynomials

The spread polynomials are a rescaling of the shifted Chebyshev polynomials of the first kind so that the range is also [0, 1]. That is,

$S_n(x)=\frac{1-T_n(1-2x)}{2}.$

See also

• Chebyshev filter
• Chebyshev cube root
• Dickson polynomials
• Legendre polynomials
• Hermite polynomials
• Romanovski polynomials
• Chebyshev rational functions
• Approximation theory
• The Chebfun system
• Discrete Chebyshev transform
• Markov brothers' inequality

Sources

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• Dette, Holger (1995). "A note on some peculiar nonlinear extremal phenomena of the Chebyshev polynomials". Proceedings of the Edinburgh Mathematical Society. 38 (2): 343–355. arXiv:math/9406222. doi:10.1017/S001309150001912X. S2CID 16703489.
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• Hernandez, M. A. (2001). "Chebyshev's approximation algorithms and applications". Comp. Math. Applic. 41 (3–4): 433–445. doi:10.1016/s0898-1221(00)00286-8.
• Mason, J. C. (1984). "Some properties and applications of Chebyshev polynomial and rational approximation". Rational Approximation and Interpolation. Lecture Notes in Mathematics. 1105. pp. 27–48. doi:10.1007/BFb0072398. ISBN 978-3-540-13899-0.
• Mason, J. C.; Handscomb, D.C. (2002). Chebyshev Polynomials. Taylor & Francis.
• Mathar, R. J. (2006). "Chebyshev series expansion of inverse polynomials". J. Comput. Appl. Math. 196 (2): 596–607. arXiv:math/0403344. Bibcode:2006JCoAM.196..596M. doi:10.1016/j.cam.2005.10.013. S2CID 16476052.
• Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), "Orthogonal Polynomials", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248
• Remes, Eugene. "On an Extremal Property of Chebyshev Polynomials" (PDF).
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• Smith, Lyle B. (1966). "Computation of Chebyshev series coefficients". Comm. ACM. 9 (2): 86–87. doi:10.1145/365170.365195. S2CID 8876563. Algorithm 277.
• Suetin, P. K. (2001) [1994], "Chebyshev polynomials", Encyclopedia of Mathematics, EMS Press

External links

• Weisstein, Eric W. "Chebyshev polynomial[s] of the first kind". MathWorld.
• Mathews, John H. (2003). "Module for Chebyshev polynomials". Department of Mathematics. Course notes for Math 340 Numerical Analysis & Math 440 Advanced Numerical Analysis. Fullerton, CA: California State University. Archived from the original on 29 May 2007. Retrieved 17 August 2020.
• "Chebyshev interpolation: An interactive tour". Mathematical Association of America (MAA) – includes illustrative Java applet.
• "Numerical computing with functions". The Chebfun Project.
• "Is there an intuitive explanation for an extremal property of Chebyshev polynomials?". Math Overflow. Question 25534.
• "Chebyshev polynomial evaluation and the Chebyshev transform". Boost. Math.