Chebyshev polynomials

Plot of the Chebyshev polynomial of the first kind $T_n(x)$ with $n=5$ in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D

The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as $T_n(x)$ and $U_n(x)$. They can be defined in several equivalent ways, one of which starts with trigonometric functions:

The Chebyshev polynomials of the first kind $T_n$ are defined by: $$T_n(\cos \theta )=\cos (n\theta ).$$

Similarly, the Chebyshev polynomials of the second kind $U_n$ are defined by: $$U_n(\cos \theta )\sin \theta =\sin ((n+1)\theta ).$$

That these expressions define polynomials in $\cos \theta$ may not be obvious at first sight but follows by rewriting $\cos (n\theta )$ and $\sin ((n+1)\theta )$ using de Moivre's formula or by using the angle sum formulas for $\cos$ and $\sin$ repeatedly. For example, the double angle formulas, which follow directly from the angle sum formulas, may be used to obtain $T_2(\cos \theta )=\cos (2\theta )=2{\cos }^2\theta -1$ and $U_1(\cos \theta )\sin \theta =\sin (2\theta )=2\cos \theta \sin \theta$, which are respectively a polynomial in $\cos \theta$ and a polynomial in $\cos \theta$ multiplied by $\sin \theta$. Hence $T_2(x)=2x^2-1$ and $U_1(x)=2x$.

An important and convenient property of the Template:Math is that they are orthogonal with respect to the following inner product: $$⟨f,g⟩=\underset{-1}{\overset{1}{\int }}f(x)g(x)\frac{\mathrm{d}x}{\sqrt{1-x^2}},$$ and Template:Math are orthogonal with respect to another, analogous inner product, given below.

The Chebyshev polynomials Template:Math are polynomials with the largest possible leading coefficient whose absolute value on the interval Template:Closed-closed is bounded by 1. They are also the "extremal" polynomials for many other properties.

In 1952, Cornelius Lanczos showed that the Chebyshev polynomials are important in approximation theory for the solution of linear systems; the roots of Template:Math, 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 Template:Mvar is used because of the alternative transliterations of the name Chebyshev as Template:Lang, Template:Lang (French) or Template:Lang (German).

Definitions

Recurrence definition

Plot of the first five Template:Mvar Chebyshev polynomials (first kind)

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

$$\begin{array}{cc}{T}_0(x)& =1\\ T_1(x)& =x\\ T_{n+1}(x)& =2x{T}_n(x)-T_{n-1}(x).\end{array}$$ The recurrence also allows to represent them explicitly as the determinant of a tridiagonal matrix of size $k\times k$:

$$T_k(x)=\det \left[\begin{array}{ccccc}x& 1& 0& \cdots & 0\\ 1& 2x& 1& \ddots & ⋮\\ 0& 1& 2x& \ddots & 0\\ ⋮& \ddots & \ddots & \ddots & 1\\ 0& \cdots & 0& 1& 2x\end{array}\right]$$

The ordinary generating function for Template:Mvar is: $$\sum _{n=0}^{\mathrm{\infty }}{T}_n(x)t^n=\frac{1-tx}{1-2tx+t^2}.$$ 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 Template:Mvar Chebyshev polynomials (second kind)

The Chebyshev polynomials of the second kind are defined by the recurrence relation: $$\begin{array}{cc}{U}_0(x)& =1\\ U_1(x)& =2x\\ U_{n+1}(x)& =2x{U}_n(x)-U_{n-1}(x).\end{array}$$ Notice that the two sets of recurrence relations are identical, except for $T_1(x)=x$ vs. Template:Nowrap The ordinary generating function for Template:Mvar is: $$\sum _{n=0}^{\mathrm{\infty }}{U}_n(x)t^n=\frac{1}{1-2tx+t^2},$$ and 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: $$T_n(x)=\{\begin{array}{cc}\cos (n\arccos x)& \text{ if }|x|\le 1\\ \cosh (n\mathrm{arcosh}x)& \text{ if }x\ge 1\\ (-1{)}^n\cosh (n\mathrm{arcosh}(-x))& \text{ if }x\le -1\end{array}$$ or, in other words, as the unique polynomials satisfying: $$T_n(\cos \theta )=\cos (n\theta )$$ for Template:Math.

The polynomials of the second kind satisfy: $$U_{n-1}(\cos \theta )\sin \theta =\sin (n\theta ),$$ or $$U_n(\cos \theta )=\frac{\sin ((n+1)\theta )}{\sin \theta },$$ which is structurally quite similar to the Dirichlet kernel Template:Math: $$D_n(x)=\frac{\sin ((2n+1)\frac{x}{2})}{\sin \frac{x}{2}}=U_{2n}(\cos \frac{x}{2}).$$ (The Dirichlet kernel, in fact, coincides with what is now known as the Chebyshev polynomial of the fourth kind.)

An equivalent way to state this is via exponentiation of a complex number: given a complex number Template:Math with absolute value of one: $$z^n=T_n(a)+ib{U}_{n-1}(a).$$ Chebyshev polynomials can be defined in this form when studying trigonometric polynomials.

That Template:Math is an Template:Mvarth-degree polynomial in Template:Math can be seen by observing that Template:Math is the real part of one side of de Moivre's formula: $$\cos n\theta +i\sin n\theta =(\cos \theta +i\sin \theta {)}^n.$$ The real part of the other side is a polynomial in Template:Math and Template:Math, in which all powers of Template:Math are even and thus replaceable through the identity Template:Math. By the same reasoning, Template:Math is the imaginary part of the polynomial, in which all powers of Template:Math are odd and thus, if one factor of Template:Math is factored out, the remaining factors can be replaced to create a Template:Mathst-degree polynomial in Template:Math.

Commuting polynomials definition

Chebyshev polynomials can also be characterized by the following theorem:

If $F_n(x)$ is a family of monic polynomials with coefficients in a field of characteristic $0$ such that $\mathrm{deg}{F}_n(x)=n$ and $F_m(F_n(x))=F_n(F_m(x))$ for all $m$ and $n$, then, up to a simple change of variables, either $F_n(x)=x^n$ for all $n$ or $F_n(x)=2\cdot T_n(x/2)$ for all $n$.

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 Template:Math. Thus, they can be generated by the standard technique for Pell equations of taking powers of a fundamental solution: $$T_n(x)+U_{n-1}(x)\sqrt{x^2-1}={(x+\sqrt{x^2-1})}^n.$$

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 Template:Math and Template:Math with parameters Template:Math and Template:Math: $$\begin{array}{cc}{\tilde{U}}_n(2x,1)& =U_{n-1}(x),\\ {\tilde{V}}_n(2x,1)& =2T_n(x).\end{array}$$ It follows that they also satisfy a pair of mutual recurrence equations: $$\begin{array}{cc}{T}_{n+1}(x)& =x{T}_n(x)-(1-x^2)U_{n-1}(x),\\ U_{n+1}(x)& =x{U}_n(x)+T_{n+1}(x).\end{array}$$

The second of these may be rearranged using the recurrence definition for the Chebyshev polynomials of the second kind to give: $$T_n(x)=\frac{1}{2}(U_n(x)-U_{n-2}(x)).$$

Using this formula iteratively gives the sum formula: $$U_n(x)=\{\begin{array}{cc}2\sum _{\text{ odd }j>0}^n{T}_j(x)& \text{ for odd }n.\\ 2\sum _{\text{ even }j\ge 0}^n{T}_j(x)-1& \text{ for even }n,\end{array}$$ while replacing $U_n(x)$ and $U_{n-2}(x)$ using the derivative formula for $T_n(x)$ gives the recurrence relationship for the derivative of $T_n$:

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

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

Turán's inequalities for the Chebyshev polynomials are: $$\begin{array}{cccccc}{T}_n(x{)}^2-T_{n-1}(x)T_{n+1}(x)& =1-x^2>0& & \text{ for }-1<x<1& & \text{ and }\\ U_n(x{)}^2-U_{n-1}(x)U_{n+1}(x)& =1>0.\end{array}$$

The integral relations areTemplate:RTemplate:Sfn $$\begin{array}{cc}\underset{-1}{\overset{1}{\int }}\frac{T_n(y)}{y-x}\frac{\mathrm{d}y}{\sqrt{1-y^2}}& =\pi U_{n-1}(x),\\ \underset{-1}{\overset{1}{\int }}\frac{U_{n-1}(y)}{y-x}\sqrt{1-y^2}\mathrm{d}y& =-\pi T_n(x)\end{array}$$ where integrals are considered as principal value.

Explicit expressions

Different approaches to defining Chebyshev polynomials lead to different explicit expressions. The trigonometric definition gives an explicit formula as follows: $$\begin{array}{cc}{T}_n(x)& =\{\begin{array}{cc}\cos (n\arccos x)& \text{ for }-1\le x\le 1\\ \cosh (n\mathrm{arcosh}x)& \text{ for }1\le x\\ (-1{)}^n\cosh (n\mathrm{arcosh}(-x))& \text{ for }x\le -1\end{array}\end{array}$$ From this trigonometric form, the recurrence definition can be recovered by computing directly that the bases cases hold: $$T_0(\cos \theta )=\cos (0\theta )=1$$ and $$T_1(\cos \theta )=\cos \theta ,$$ and that the product-to-sum identity holds: $$2\cos n\theta \cos \theta =\cos [(n+1)\theta ]+\cos [(n-1)\theta ].$$

Using the complex number exponentiation definition of the Chebyshev polynomial, one can derive the following expression: $$T_n(x)=\frac{1}{2}\left(\right(x-\sqrt{x^2-1}{)}^n+(x+\sqrt{x^2-1}{)}^n)\text{ for }x\in ℝ$$ $$T_n(x)=\frac{1}{2}\left(\right(x-\sqrt{x^2-1}{)}^n+(x-\sqrt{x^2-1}{)}^{-n})\text{ for }x\in ℝ$$ The two are equivalent because $(x+\sqrt{x^2-1})(x-\sqrt{x^2-1})=1$.

An explicit form of the Chebyshev polynomial in terms of monomials Template:Math follows from de Moivre's formula: $$T_n(\cos (\theta ))=\mathrm{Re}(\cos n\theta +i\sin n\theta )=\mathrm{Re}((\cos \theta +i\sin \theta {)}^n),$$ where Template:Math denotes the real part of a complex number. Expanding the formula, one gets: $$(\cos \theta +i\sin \theta {)}^n=\sum _{j=0}^n(\genfrac{}{}{0ex}{}{n}{j})i^j{\sin }^j\theta {\cos }^{n-j}\theta .$$ The real part of the expression is obtained from summands corresponding to even indices. Noting $i^{2j}=(-1{)}^j$ and ${\sin }^{2j}\theta =(1-{\cos }^2\theta {)}^j$, one gets the explicit formula: $$\cos n\theta =\sum _{j=0}^{\lfloor n/2\rfloor }(\genfrac{}{}{0ex}{}{n}{2j})({\cos }^2\theta -1{)}^j{\cos }^{n-2j}\theta ,$$ which in turn means that: $$T_n(x)=\sum _{j=0}^{\lfloor n/2\rfloor }(\genfrac{}{}{0ex}{}{n}{2j})(x^2-1{)}^j{x}^{n-2j}.$$ This can be written as a Template:Math hypergeometric function: $$\begin{array}{cc}{T}_n(x)& =\sum _{k=0}^{\lfloor \frac{n}{2}\rfloor }(\genfrac{}{}{0ex}{}{n}{2k}){(x^2-1)}^k{x}^{n-2k}\\ & =x^n\sum _{k=0}^{\lfloor \frac{n}{2}\rfloor }(\genfrac{}{}{0ex}{}{n}{2k}){(1-x^{-2})}^k\\ & =\frac{n}{2}\sum _{k=0}^{\lfloor \frac{n}{2}\rfloor }(-1{)}^k\frac{(n-k-1)!}{k!(n-2k)!}(2x{)}^{n-2k}\text{ for }n>0\\ \\ & =n\sum _{k=0}^n(-2{)}^k\frac{(n+k-1)!}{(n-k)!(2k)!}(1-x{)}^k\text{ for }n>0\\ \\ & ={}_2{F}_1(-n,n;\frac{1}{2};\frac{1}{2}(1-x))\\ \end{array}$$ with inverse:

$$x^n=2^{1-n}{{\sum }^{\prime }}_{\genfrac{}{}{0ex}{}{j=0}{j\equiv n(\mathrm{mod}2)}}^n(\genfrac{}{}{0ex}{}{n}{\frac{n-j}{2}})T_j(x),$$ where the prime at the summation symbol indicates that the contribution of Template:Math needs to be halved if it appears.

A related expression for Template:Math as a sum of monomials with binomial coefficients and powers of two is $$T_n\left(x\right)=\sum _{m=0}^{\lfloor \frac{n}{2}\rfloor }{(-1)}^m((\genfrac{}{}{0ex}{}{n-m}{m})+(\genfrac{}{}{0ex}{}{n-m-1}{n-2m}))\cdot 2^{n-2m-1}\cdot x^{n-2m}.$$

Similarly, Template:Math can be expressed in terms of hypergeometric functions: $$\begin{array}{cc}{U}_n(x)& =\frac{{(x+\sqrt{x^2-1})}^{n+1}-{(x-\sqrt{x^2-1})}^{n+1}}{2\sqrt{x^2-1}}\\ & =\sum _{k=0}^{\lfloor n/2\rfloor }(\genfrac{}{}{0ex}{}{n+1}{2k+1}){(x^2-1)}^k{x}^{n-2k}\\ & =x^n\sum _{k=0}^{\lfloor n/2\rfloor }(\genfrac{}{}{0ex}{}{n+1}{2k+1}){(1-x^{-2})}^k\\ & =\sum _{k=0}^{\lfloor n/2\rfloor }(\genfrac{}{}{0ex}{}{2k-(n+1)}{k})(2x{)}^{n-2k}& \text{ for }n>0\\ & =\sum _{k=0}^{\lfloor n/2\rfloor }(-1{)}^k(\genfrac{}{}{0ex}{}{n-k}{k})(2x{)}^{n-2k}& \text{ for }n>0\\ & =\sum _{k=0}^n(-2{)}^k\frac{(n+k+1)!}{(n-k)!(2k+1)!}(1-x{)}^k& \text{ for }n>0\\ & =(n+1){}_2{F}_1(-n,n+2;\frac{3}{2};\frac{1}{2}(1-x)).\\ \end{array}$$

Properties

Symmetry

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

That is, Chebyshev polynomials of even order have even symmetry and therefore contain only even powers of Template:Mvar. Chebyshev polynomials of odd order have odd symmetry and therefore contain only odd powers of Template:Mvar.

Roots and extrema

A Chebyshev polynomial of either kind with degree Template:Mvar has Template:Mvar different simple roots, called Chebyshev roots, in the interval Template:Closed-closed. 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 Template:Mvar are: $$x_k=\cos \left(\frac{\pi (k+1/2)}{n}\right),k=0,\dots ,n-1.$$ Similarly, the roots of Template:Mvar are: $$x_k=\cos \left(\frac{k}{n+1}\pi \right),k=1,\dots ,n.$$ The extrema of Template:Mvar on the interval Template:Math 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 Template:Math 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: $$\begin{array}{cc}{T}_n(1)& =1\\ T_n(-1)& =(-1{)}^n\\ U_n(1)& =n+1\\ U_n(-1)& =(-1{)}^n(n+1).\end{array}$$

The extrema of $T_n(x)$ on the interval $-1\le x\le 1$ where $n>0$ are located at $n+1$ values of $x$. They are $\pm 1$, or $\cos \left(\frac{2\pi k}{d}\right)$ where $d>2$, $d|2n$, $0<k<d/2$ and $(k,d)=1$, i.e., $k$ and $d$ are relatively prime numbers.

Specifically, when $n$ is even:

• $T_n(x)=1$ if $x=\pm 1$, or $d>2$ and $2n/d$ is even. There are $n/2+1$ such values of $x$.
• $T_n(x)=-1$ if $d>2$ and $2n/d$ is odd. There are $n/2$ such values of $x$.

When $n$ is odd:

• $T_n(x)=1$ if $x=1$, or $d>2$ and $2n/d$ is even. There are $(n+1)/2$ such values of $x$.
• $T_n(x)=-1$ if $x=-1$, or $d>2$ and $2n/d$ is odd. There are $(n+1)/2$ such values of $x$.

This result has been generalized to solutions of $U_n(x)\pm 1=0$, and to $V_n(x)\pm 1=0$ and $W_n(x)\pm 1=0$ for Chebyshev polynomials of the third and fourth kinds, respectively.

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

The last two formulas can be numerically troublesome due to the division by zero (Template:Sfrac indeterminate form, specifically) at Template:Math and Template:Math. By L'Hôpital's rule: $$\begin{array}{cc}{\frac{{\mathrm{d}}^2{T}_n}{\mathrm{d}{x}^2}|}_{x=1}& =\frac{n^4-n^2}{3},\\ {\frac{{\mathrm{d}}^2{T}_n}{\mathrm{d}{x}^2}|}_{x=-1}& =(-1{)}^n\frac{n^4-n^2}{3}.\end{array}$$

More generally, $${\frac{{\mathrm{d}}^p{T}_n}{\mathrm{d}{x}^p}|}_{x=\pm 1}=(\pm 1{)}^{n+p}\prod _{k=0}^{p-1}\frac{n^2-k^2}{2k+1},$$ which is of great use in the numerical solution of eigenvalue problems.

Also, we have: $$\frac{{\mathrm{d}}^p}{\mathrm{d}{x}^p}{T}_n(x)=2^{p}n{{\sum }^{\prime }}_{\genfrac{}{}{0ex}{}{0\le k\le n-p}{k\equiv n-p(\mathrm{mod}2)}}(\genfrac{}{}{0ex}{}{\frac{n+p-k}{2}-1}{\frac{n-p-k}{2}})\frac{(\frac{n+p+k}{2}-1)!}{\left(\frac{n-p+k}{2}\right)!}{T}_k(x),p\ge 1,$$ where the prime at the summation symbols means that the term contributed by Template:Math is to be halved, if it appears.

Concerning integration, the first derivative of the Template:Mvar 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 Template:Math: $$\int T_n\mathrm{d}x=\frac{1}{2}(\frac{T_{n+1}}{n+1}-\frac{T_{n-1}}{n-1})=\frac{n{T}_{n+1}}{n^2-1}-\frac{x{T}_n}{n-1}.$$

The last formula can be further manipulated to express the integral of Template:Mvar as a function of Chebyshev polynomials of the first kind only: $$\begin{array}{cc}\int T_n\mathrm{d}x& =\frac{n}{n^2-1}{T}_{n+1}-\frac{1}{n-1}{T}_1{T}_n\\ & =\frac{n}{n^2-1}{T}_{n+1}-\frac{1}{2(n-1)}(T_{n+1}+T_{n-1})\\ & =\frac{1}{2(n+1)}{T}_{n+1}-\frac{1}{2(n-1)}{T}_{n-1}.\end{array}$$

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

Products of Chebyshev polynomials

The Chebyshev polynomials of the first kind satisfy the relation: $$T_m(x)T_n(x)=\frac{1}{2}(T_{m+n}(x)+T_{|m-n|}(x)),\mathrm{\forall }m,n\ge 0,$$ which is easily proved from the product-to-sum formula for the cosine: $$2\cos \alpha \cos \beta =\cos (\alpha +\beta )+\cos (\alpha -\beta ).$$ For Template:Math this results in the already known recurrence formula, just arranged differently, and with Template:Math it forms the recurrence relation for all even or all odd indexed Chebyshev polynomials (depending on the parity of the lowest Template:Mvar) which implies the evenness or oddness of these polynomials. Three more useful formulas for evaluating Chebyshev polynomials can be concluded from this product expansion: $$\begin{array}{cccc}{T}_{2n}(x)& =2T_n^2(x)-T_0(x)& & =2T_n^2(x)-1,\\ T_{2n+1}(x)& =2T_{n+1}(x)T_n(x)-T_1(x)& & =2T_{n+1}(x)T_n(x)-x,\\ T_{2n-1}(x)& =2T_{n-1}(x)T_n(x)-T_1(x)& & =2T_{n-1}(x)T_n(x)-x.\end{array}$$

The polynomials of the second kind satisfy the similar relation: $$T_m(x)U_n(x)=\{\begin{array}{cc}\frac{1}{2}(U_{m+n}(x)+U_{n-m}(x)),& \text{ if }n\ge m-1,\\ \\ \frac{1}{2}(U_{m+n}(x)-U_{m-n-2}(x)),& \text{ if }n\le m-2.\end{array}$$ (with the definition Template:Math by convention ). They also satisfy: $$U_m(x)U_n(x)=\sum _{k=0}^n{U}_{m-n+2k}(x)=\sum _{\underset{\text{ step 2 }}{p=m-n}}^{m+n}{U}_p(x).$$ for Template:Math. For Template:Math this recurrence reduces to: $$U_{m+2}(x)=U_2(x)U_m(x)-U_m(x)-U_{m-2}(x)=U_m(x)(U_2(x)-1)-U_{m-2}(x),$$ which establishes the evenness or oddness of the even or odd indexed Chebyshev polynomials of the second kind depending on whether Template:Mvar starts with 2 or 3.

Composition and divisibility properties

The trigonometric definitions of Template:Math and Template:Math imply the composition or nesting properties: $$\begin{array}{cc}{T}_{mn}(x)& =T_m(T_n(x)),\\ U_{mn-1}(x)& =U_{m-1}(T_n(x))U_{n-1}(x).\end{array}$$ For Template:Math the order of composition may be reversed, making the family of polynomial functions Template:Math a commutative semigroup under composition.

Since Template:Math is divisible by Template:Mvar if Template:Mvar is odd, it follows that Template:Math is divisible by Template:Math if Template:Mvar is odd. Furthermore, Template:Math is divisible by Template:Math, and in the case that Template:Mvar is even, divisible by Template:Math.

Orthogonality

Both Template:Mvar and Template:Mvar form a sequence of orthogonal polynomials. The polynomials of the first kind Template:Mvar are orthogonal with respect to the weight: $$\frac{1}{\sqrt{1-x^2}},$$ on the interval Template:Closed-closed, i.e. we have: $$\underset{-1}{\overset{1}{\int }}{T}_n(x)T_m(x)\frac{\mathrm{d}x}{\sqrt{1-x^2}}=\{\begin{array}{cc}0& \text{ if }n\ne m,\\ \pi & \text{ if }n=m=0,\\ \frac{\pi }{2}& \text{ if }n=m\ne 0.\end{array}$$

This can be proven by letting Template:Math and using the defining identity Template:Math.

Similarly, the polynomials of the second kind Template:Mvar are orthogonal with respect to the weight: $$\sqrt{1-x^2}$$ on the interval Template:Closed-closed, i.e. we have: $$\underset{-1}{\overset{1}{\int }}{U}_n(x)U_m(x)\sqrt{1-x^2}\mathrm{d}x=\{\begin{array}{cc}0& \text{ if }n\ne m,\\ \frac{\pi }{2}& \text{ if }n=m.\end{array}$$

(The measure Template:Math is, to within a normalizing constant, the Wigner semicircle distribution.)

These orthogonality properties follow from the fact that the Chebyshev polynomials solve the Chebyshev differential equations: $$\begin{array}{cc}(1-x^2){T_n}^{″}-x{T_n}^{\prime }+n^2{T}_n& =0,\\ (1-x^2){U_n}^{″}-3x{U_n}^{\prime }+n(n+2)U_n& =0,\end{array}$$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 Template:Mvar also satisfy a discrete orthogonality condition: $$\sum _{k=0}^{N-1}{T}_i(x_k)T_j(x_k)=\{\begin{array}{cc}0& \text{ if }i\ne j,\\ N& \text{ if }i=j=0,\\ \frac{N}{2}& \text{ if }i=j\ne 0,\end{array}$$ where Template:Mvar is any integer greater than Template:Math,Template:Sfn and the Template:Math are the Template:Mvar Chebyshev nodes (see above) of Template:Math: $$x_k=\cos \left(\pi \frac{2k+1}{2N}\right)\text{ for }k=0,1,\dots ,N-1.$$

For the polynomials of the second kind and any integer Template:Math with the same Chebyshev nodes Template:Math, there are similar sums: $$\sum _{k=0}^{N-1}{U}_i(x_k)U_j(x_k)(1-x_k^2)=\{\begin{array}{cc}0& \text{ if }i\ne j,\\ \frac{N}{2}& \text{ if }i=j,\end{array}$$ and without the weight function: $$\sum _{k=0}^{N-1}{U}_i(x_k)U_j(x_k)=\{\begin{array}{cc}0& \text{ if }i\equiv ̸j(\mathrm{mod}2),\\ N\cdot (1+\min \{i,j\})& \text{ if }i\equiv j(\mathrm{mod}2).\end{array}$$

For any integer Template:Math, based on the Template:Mvar zeros of Template:Math: $$y_k=\cos \left(\pi \frac{k+1}{N+1}\right)\text{ for }k=0,1,\dots ,N-1,$$ one can get the sum: $$\sum _{k=0}^{N-1}{U}_i(y_k)U_j(y_k)(1-y_k^2)=\{\begin{array}{cc}0& \text{ if }i\ne j,\\ \frac{N+1}{2}& \text{ if }i=j,\end{array}$$ and again without the weight function: $$\sum _{k=0}^{N-1}{U}_i(y_k)U_j(y_k)=\{\begin{array}{cc}0& \text{ if }i\equiv ̸j(\mathrm{mod}2),\\ (\min \{i,j\}+1)(N-\max \{i,j\})& \text{ if }i\equiv j(\mathrm{mod}2).\end{array}$$

Minimal Template:Math-norm

For any given Template:Math, among the polynomials of degree Template:Mvar 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 Template:Closed-closed is minimal.

This maximal absolute value is: $$\frac{1}{2^{n-1}}$$ and Template:Math reaches this maximum exactly Template:Math times at: $$x=\cos \frac{k\pi }{n}\text{for }0\le k\le n.$$

Remark

By the equioscillation theorem, among all the polynomials of degree Template:Math, the polynomial Template:Mvar minimizes Template:Math on Template:Closed-closed if and only if there are Template:Math points Template:Math such that Template:Math.

Of course, the null polynomial on the interval Template:Closed-closed can be approximated by itself and minimizes the Template:Math-norm.

Above, however, Template:Math reaches its maximum only Template:Math times because we are searching for the best polynomial of degree Template:Math (therefore the theorem evoked previously cannot be used).

Chebyshev polynomials as special cases of more general polynomial families

The Chebyshev polynomials are a special case of the ultraspherical or Gegenbauer polynomials $C_n^{(\lambda )}(x)$, which themselves are a special case of the Jacobi polynomials $P_n^{(\alpha ,\beta )}(x)$: $$\begin{array}{cc}{T}_n(x)& =\frac{n}{2}\underset{q\to 0}{\lim }\frac{1}{q}{C}_n^{(q)}(x)\text{ if }n\ge 1,\\ & =\frac{1}{(\genfrac{}{}{0ex}{}{n-\frac{1}{2}}{n})}{P}_n^{(-\frac{1}{2},-\frac{1}{2})}(x)=\frac{2^{2n}}{(\genfrac{}{}{0ex}{}{2n}{n})}{P}_n^{(-\frac{1}{2},-\frac{1}{2})}(x),\\ U_n(x)& =C_n^{(1)}(x)\\ & =\frac{n+1}{(\genfrac{}{}{0ex}{}{n+\frac{1}{2}}{n})}{P}_n^{(\frac{1}{2},\frac{1}{2})}(x)=\frac{2^{2n+1}}{(\genfrac{}{}{0ex}{}{2n+2}{n+1})}{P}_n^{(\frac{1}{2},\frac{1}{2})}(x).\end{array}$$

Chebyshev polynomials are also a special case of Dickson polynomials: $$D_n(2x\alpha ,{\alpha }^2)=2{\alpha }^n{T}_n(x)$$ $$E_n(2x\alpha ,{\alpha }^2)={\alpha }^n{U}_n(x).$$ In particular, when $\alpha =\frac{1}{2}$, they are related by $D_n(x,\frac{1}{4})=2^{1-n}{T}_n(x)$ and $E_n(x,\frac{1}{4})=2^{-n}{U}_n(x)$.

Other properties

The curves given by Template:Math, or equivalently, by the parametric equations Template:Math, Template:Math, are a special case of Lissajous curves with frequency ratio equal to Template:Mvar.

Similar to the formula: $$T_n(\cos \theta )=\cos (n\theta ),$$ we have the analogous formula: $$T_{2n+1}(\sin \theta )=(-1{)}^n\sin \left((2n+1)\theta \right).$$

For Template:Math: $$T_n\left(\frac{x+x^{-1}}{2}\right)=\frac{x^n+x^{-n}}{2}$$ 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 Template:Math.

There are relations between Legendre polynomials and Chebyshev polynomials

$\sum _{k=0}^n{P}_k\left(x\right)T_{n-k}\left(x\right)=(n+1)P_n\left(x\right)$

$\sum _{k=0}^n{P}_k\left(x\right)P_{n-k}\left(x\right)=U_n\left(x\right)$

These identities can be proven using generating functions and discrete convolution

Examples

First kind

The first few Chebyshev polynomials of the first kind in the domain Template:Math: The flat Template:Math, Template:Math, Template:Math, Template:Math, Template:Math and Template:Math.

The first few Chebyshev polynomials of the first kind are Template:OEIS2C $$\begin{array}{cc}{T}_0(x)& =1\\ T_1(x)& =x\\ T_2(x)& =2x^2-1\\ T_3(x)& =4x^3-3x\\ T_4(x)& =8x^4-8x^2+1\\ T_5(x)& =16x^5-20x^3+5x\\ T_6(x)& =32x^6-48x^4+18x^2-1\\ T_7(x)& =64x^7-112x^5+56x^3-7x\\ T_8(x)& =128x^8-256x^6+160x^4-32x^2+1\\ T_9(x)& =256x^9-576x^7+432x^5-120x^3+9x\\ T_{10}(x)& =512x^{10}-1280x^8+1120x^6-400x^4+50x^2-1\end{array}$$

Second kind

The first few Chebyshev polynomials of the second kind in the domain Template:Math: The flat Template:Math, Template:Math, Template:Math, Template:Math, Template:Math and Template:Math. Although not visible in the image, Template:Math and Template:Math.

The first few Chebyshev polynomials of the second kind are Template:OEIS2C $$\begin{array}{cc}{U}_0(x)& =1\\ U_1(x)& =2x\\ U_2(x)& =4x^2-1\\ U_3(x)& =8x^3-4x\\ U_4(x)& =16x^4-12x^2+1\\ U_5(x)& =32x^5-32x^3+6x\\ U_6(x)& =64x^6-80x^4+24x^2-1\\ U_7(x)& =128x^7-192x^5+80x^3-8x\\ U_8(x)& =256x^8-448x^6+240x^4-40x^2+1\\ U_9(x)& =512x^9-1024x^7+672x^5-160x^3+10x\\ U_{10}(x)& =1024x^{10}-2304x^8+1792x^6-560x^4+60x^2-1\end{array}$$

As a basis set

The non-smooth function (top) Template:Math, where Template:Mvar 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 Template:Math, 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 Template:Math 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 Template:Math if the function is piecewise smooth and continuous. The smoothness requirement can be relaxed in most casesTemplate:Snd as long as there are a finite number of discontinuities in Template:Math 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 Template:Math. One can express: $$\log (1+x)=\sum _{n=0}^{\mathrm{\infty }}{a}_n{T}_n(x).$$

One can find the coefficients Template:Math 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: $$a_n=\{\begin{array}{cc}-\log 2& \text{ for }n=0,\\ \frac{-2(-1{)}^n}{n}& \text{ for }n>0.\end{array}$$

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 Template:Mvar is the Kronecker delta function and the Template:Mvar are the Template:Mvar Gauss–Chebyshev zeros of Template:Math: $$x_k=\cos \left(\frac{\pi (k+\frac{1}{2})}{N}\right).$$ For any Template:Mvar, these approximate coefficients provide an exact approximation to the function at Template:Mvar with a controlled error between those points. The exact coefficients are obtained with Template:Math, thus representing the function exactly at all points in Template:Closed-closed. The rate of convergence depends on the function and its smoothness.

This allows us to compute the approximate coefficients Template:Mvar 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: $$\begin{array}{cc}{(1-x^2)}^{\alpha }& =-\frac{1}{\sqrt{\pi }}\frac{\mathrm{\Gamma }(\frac{1}{2}+\alpha )}{\mathrm{\Gamma }(\alpha +1)}+2^{1-2\alpha }\sum _{n=0}{(-1)}^n(\genfrac{}{}{0ex}{}{2\alpha }{\alpha -n})T_{2n}(x)\\ & =2^{-2\alpha }\sum _{n=0}{(-1)}^n(\genfrac{}{}{0ex}{}{2\alpha +1}{\alpha -n})U_{2n}(x).\end{array}$$

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 Template:Mvar 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 Template:Mvar coefficients of the Template:Mathst 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 Template:Mvar can be written in terms of the Chebyshev polynomials of the first kind.Template:Sfn Such a polynomial Template:Math 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.

Families of polynomials related to Chebyshev polynomials

Polynomials denoted $C_n(x)$ and $S_n(x)$ closely related to Chebyshev polynomials are sometimes used. They are defined by: $$C_n(x)=2T_n\left(\frac{x}{2}\right),S_n(x)=U_n\left(\frac{x}{2}\right)$$ and satisfy: $$C_n(x)=S_n(x)-S_{n-2}(x).$$ A. F. Horadam called the polynomials $C_n(x)$ Vieta–Lucas polynomials and denoted them $v_n(x)$. He called the polynomials $S_n(x)$ Vieta–Fibonacci polynomials and denoted them Template:Nowrap Lists of both sets of polynomials are given in Viète's Opera Mathematica, Chapter IX, Theorems VI and VII. The Vieta–Lucas and Vieta–Fibonacci polynomials of real argument are, up to a power of $i$ and a shift of index in the case of the latter, equal to Lucas and Fibonacci polynomials Template:Math and Template:Math of imaginary argument.

Shifted Chebyshev polynomials of the first and second kinds are related to the Chebyshev polynomials by: $$T_n^{*}(x)=T_n(2x-1),U_n^{*}(x)=U_n(2x-1).$$

When the argument of the Chebyshev polynomial satisfies Template:Math the argument of the shifted Chebyshev polynomial satisfies Template:Math. Similarly, one can define shifted polynomials for generic intervals Template:Closed-closed.

Around 1990 the terms "third-kind" and "fourth-kind" came into use in connection with Chebyshev polynomials, although the polynomials denoted by these terms had an earlier development under the name airfoil polynomials. According to J. C. Mason and G. H. Elliott, the terminology "third-kind" and "fourth-kind" is due to Walter Gautschi, "in consultation with colleagues in the field of orthogonal polynomials." The Chebyshev polynomials of the third kind are defined as: $$V_n(x)=\frac{\cos \left((n+\frac{1}{2})\theta \right)}{\cos \left(\frac{\theta }{2}\right)}=\sqrt{\frac{2}{1+x}}{T}_{2n+1}\left(\sqrt{\frac{x+1}{2}}\right)$$ and the Chebyshev polynomials of the fourth kind are defined as: $$W_n(x)=\frac{\sin \left((n+\frac{1}{2})\theta \right)}{\sin \left(\frac{\theta }{2}\right)}=U_{2n}\left(\sqrt{\frac{x+1}{2}}\right),$$ where $\theta =\arccos x$. In the airfoil literature $V_n(x)$ and $W_n(x)$ are denoted $t_n(x)$ and $u_n(x)$. The polynomial families $T_n(x)$, $U_n(x)$, $V_n(x)$, and $W_n(x)$ are orthogonal with respect to the weights: $${(1-x^2)}^{-1/2},{(1-x^2)}^{1/2},(1-x{)}^{-1/2}(1+x{)}^{1/2},(1+x{)}^{-1/2}(1-x{)}^{1/2}$$ and are proportional to Jacobi polynomials $P_n^{(\alpha ,\beta )}(x)$ with: $$(\alpha ,\beta )=(-\frac{1}{2},-\frac{1}{2}),(\alpha ,\beta )=(\frac{1}{2},\frac{1}{2}),(\alpha ,\beta )=(-\frac{1}{2},\frac{1}{2}),(\alpha ,\beta )=(\frac{1}{2},-\frac{1}{2}).$$

All four families satisfy the recurrence $p_n(x)=2x{p}_{n-1}(x)-p_{n-2}(x)$ with $p_0(x)=1$, where $p_n=T_n$, $U_n$, $V_n$, or $W_n$, but they differ according to whether $p_1(x)$ equals $x$, $2x$, $2x-1$, or Template:Nowrap

Even order modified Chebyshev polynomials

Some applications rely on Chebyshev polynomials but may be unable to accommodate the lack of a root at zero, which rules out the use of standard Chebyshev polynomials for these kinds of applications. Even order Chebyshev filter designs using equally terminated passive networks are an example of this. However, even order Chebyshev polynomials may be modified to move the lowest roots down to zero while still maintaining the desirable Chebyshev equi-ripple effect. Such modified polynomials contain two roots at zero, and may be referred to as even order modified Chebyshev polynomials. Even order modified Chebyshev polynomials may be created from the Chebyshev nodes in the same manner as standard Chebyshev polynomials. $$P_N=\prod _{i=1}^N(x-C_i)$$

where

• $P_N$ is an N-th order Chebyshev polynomial
• $C_i$ is the i-th Chebyshev node

In the case of even order modified Chebyshev polynomials, the even order modified Chebyshev nodes are used to construct the even order modified Chebyshev polynomials. $$P{e}_N=\prod _{i=1}^N(x-C{e}_i)$$

where

• $P{e}_N$ is an N-th order even order modified Chebyshev polynomial
• $C{e}_i$ is the i-th even order modified Chebyshev node

For example, the 4th order Chebyshev polynomial from the example above is $X^4-X^2+.125$, which by inspection contains no roots of zero. Creating the polynomial from the even order modified Chebyshev nodes creates a 4th order even order modified Chebyshev polynomial of $X^4-.828427X^2$, which by inspection contains two roots at zero, and may be used in applications requiring roots at zero.

See also

• Chebyshev filter
• Chebyshev cube root
• Dickson polynomials
• Legendre polynomials
• Laguerre polynomials
• Hermite polynomials
• Minimal polynomial of 2cos(2pi/n)
• Romanovski polynomials
• Chebyshev rational functions
• Approximation theory
• Function approximation
• The Chebfun system
• Discrete Chebyshev transform
• Markov brothers' inequality
• Clenshaw algorithm

Sources

• Template:Abramowitz Stegun ref
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• Template:Eom

External links

• Template:Commons category-inline
• Template:MathWorld
• Template:Cite web
• Template:Cite web – includes illustrative Java applet.
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• Template:Cite web