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Chebyshev polynomials

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Polynomial sequence

Not to be confused withdiscrete Chebyshev polynomials.

Plot of the first fiveT_n Chebyshev polynomials (first kind)

Plot of the first fiveU_n Chebyshev polynomials (second kind)

TheChebyshev polynomials are two sequences oforthogonal polynomials related to thecosine and sine functions, notated as $T_{n}(x)$ and $U_{n}(x)$ . They can be defined in several equivalent ways, one of which starts withtrigonometric functions:

TheChebyshev polynomials of the first kind $T_{n}$ are defined by $T_{n}(\cos \theta )=\cos(n\theta ).$

Similarly, theChebyshev polynomials of the second kind $U_{n}$ are defined by $U_{n}(\cos \theta )\sin \theta =\sin {\big (}(n+1)\theta {\big )}.$

That these expressions define polynomials in $\cos \theta$ is not obvious at first sight but can be shown usingde Moivre's formula (seebelow).

The Chebyshev polynomialsT_n are polynomials with the largest possible leading coefficient whoseabsolute value on theinterval[−1, 1] is bounded by 1. They are also the "extremal" polynomials for many other properties.^[1]

In 1952,Cornelius Lanczos showed that the Chebyshev polynomials are important inapproximation theory for the solution of linear systems;^[2] theroots ofT_n(x), which are also calledChebyshev nodes, are used as matching points for optimizingpolynomial interpolation. The resulting interpolation polynomial minimizes the problem ofRunge's phenomenon and provides an approximation that is close to the best polynomial approximation to acontinuous function under themaximum norm, also called the "minimax" criterion. This approximation leads directly to the method ofClenshaw–Curtis quadrature.

These polynomials were named afterPafnuty Chebyshev.^[3] The letterT is used because of the alternativetransliterations of the nameChebyshev asTchebycheff,Tchebyshev (French) orTschebyschow (German).

Definitions

Recurrence definition

TheChebyshev polynomials of the first kind can be defined by the recurrence relation ${\begin{aligned}T_{0}(x)&=1,\\T_{1}(x)&=x,\\T_{n+1}(x)&=2x\,T_{n}(x)-T_{n-1}(x).\end{aligned}}$

TheChebyshev polynomials of the second kind can be defined by the recurrence relation ${\begin{aligned}U_{0}(x)&=1,\\U_{1}(x)&=2x,\\U_{n+1}(x)&=2x\,U_{n}(x)-U_{n-1}(x),\end{aligned}}$ which differs from the above only by the rule forn=1.

Trigonometric definition

The Chebyshev polynomials of the first and second kind can be defined as the unique polynomials satisfying $T_{n}(\cos \theta )=\cos(n\theta )$ and $U_{n}(\cos \theta )={\frac {\sin {\big (}(n+1)\theta {\big )}}{\sin \theta }},$ forn = 0, 1, 2, 3, ….

An equivalent way to state this is via exponentiation of acomplex number: given a complex numberz =a +bi with absolute value of one, $z^{n}=T_{n}(a)+ibU_{n-1}(a).$ Chebyshev polynomials can be defined in this form when studyingtrigonometric polynomials.^[4]

Thatcos nx is annth-degree polynomial incos x can be seen by observing thatcos nx is thereal part of one side ofde 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 incos x andsin x, in which all powers ofsin x areeven and thus replaceable through the identitycos² x + sin² x = 1. By the same reasoning,sin nx is theimaginary part of the polynomial, in which all powers ofsin x areodd and thus, if one factor ofsin x is factored out, the remaining factors can be replaced to create a(n − 1)st-degree polynomial incos x.

Forx outside the interval [-1,1], the above definition implies $T_{n}(x)={\begin{cases}\cos(n\arccos x)&{\text{ if }}~|x|\leq 1,\\\cosh(n\operatorname {arcosh} x)&{\text{ if }}~x\geq 1,\\(-1)^{n}\cosh(n\operatorname {arcosh} (-x))&{\text{ if }}~x\leq -1.\end{cases}}$

Commuting polynomials definition

Chebyshev polynomials can also be characterized by the following theorem:^[5]

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

Pell equation definition

The Chebyshev polynomials can also be defined as the solutions to thePell equation: $T_{n}(x)^{2}-\left(x^{2}-1\right)U_{n-1}(x)^{2}=1$ in aringR[x].^[6] Thus, they can be generated by the standard technique for Pell equations of taking powers of a fundamental solution: $T_{n}(x)+U_{n-1}(x)\,{\sqrt {x^{2}-1}}=\left(x+{\sqrt {x^{2}-1}}\right)^{n}~.$

Generating functions

Theordinary generating function forT_n is $\sum _{n=0}^{\infty }T_{n}(x)\,t^{n}={\frac {1-tx}{1-2tx+t^{2}}}.$

There are several othergenerating functions for the Chebyshev polynomials; theexponential generating function is ${\begin{aligned}\sum _{n=0}^{\infty }T_{n}(x){\frac {t^{n}}{n!}}&={\tfrac {1}{2}}{\Bigl (}{\exp }{\Bigl (}{\textstyle t{\bigl (}x-{\sqrt {x^{2}-1}}~\!{\bigr )}}{\Bigr )}+{\exp }{\Bigl (}{\textstyle t{\bigl (}x+{\sqrt {x^{2}-1}}~\!{\bigr )}}{\Bigr )}{\Bigr )}\\&=e^{tx}\cosh \left({\textstyle t{\sqrt {x^{2}-1}}}~\!\right).\end{aligned}}$

The generating function relevant for 2-dimensionalpotential theory andmultipole expansion is $\sum \limits _{n=1}^{\infty }T_{n}(x)\,{\frac {t^{n}}{n}}=\ln \left({\frac {1}{\sqrt {1-2tx+t^{2}}}}\right).$

The ordinary generating function forU_n is $\sum _{n=0}^{\infty }U_{n}(x)\,t^{n}={\frac {1}{1-2tx+t^{2}}},$ and the exponential generating function is $\sum _{n=0}^{\infty }U_{n}(x){\frac {t^{n}}{n!}}=e^{tx}{\biggl (}\!\cosh \left(t{\sqrt {x^{2}-1}}\right)+{\frac {x}{\sqrt {x^{2}-1}}}\sinh \left(t{\sqrt {x^{2}-1}}\right){\biggr )}.$

Relations between the two kinds of Chebyshev polynomials

The Chebyshev polynomials of the first and second kinds correspond to a complementary pair ofLucas sequencesṼ_n(P,Q) andŨ_n(P,Q) with parametersP = 2x andQ = 1: ${\begin{aligned}{\tilde {U}}_{n}(2x,1)&=U_{n-1}(x),\\{\tilde {V}}_{n}(2x,1)&=2\,T_{n}(x).\end{aligned}}$ It follows that they also satisfy a pair of mutual recurrence equations:^[7] ${\begin{aligned}T_{n+1}(x)&=x\,T_{n}(x)-(1-x^{2})\,U_{n-1}(x),\\U_{n+1}(x)&=x\,U_{n}(x)+T_{n+1}(x).\end{aligned}}$

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

Using this formula iteratively gives the sum formula: $U_{n}(x)={\begin{cases}2\sum _{{\text{ odd }}j>0}^{n}T_{j}(x)&{\text{ for odd }}n.\\2\sum _{{\text{ even }}j\geq 0}^{n}T_{j}(x)-1&{\text{ for even }}n,\end{cases}}$ while replacing $U_{n}(x)$ and $U_{n-2}(x)$ using thederivative formula for $T_{n}(x)$ gives the recurrence relationship for the derivative of $T_{n}$ :

$2\,T_{n}(x)={\frac {1}{n+1}}\,{\frac {\mathrm {d} }{\mathrm {d} x}}\,T_{n+1}(x)-{\frac {1}{n-1}}\,{\frac {\mathrm {d} }{\mathrm {d} x}}\,T_{n-1}(x),\qquad n=2,3,\ldots$

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

Turán's inequalities for the Chebyshev polynomials are:^[8] ${\begin{aligned}T_{n}(x)^{2}-T_{n-1}(x)\,T_{n+1}(x)&=1-x^{2}>0&&{\text{ for }}-1<x<1&&{\text{ and }}\\U_{n}(x)^{2}-U_{n-1}(x)\,U_{n+1}(x)&=1>0~.\end{aligned}}$

Theintegral relations are^[9]^[10] ${\begin{aligned}\int _{-1}^{1}{\frac {T_{n}(y)}{y-x}}\,{\frac {\mathrm {d} y}{\sqrt {1-y^{2}}}}&=\pi \,U_{n-1}(x)~,\\[1.5ex]\int _{-1}^{1}{\frac {U_{n-1}(y)}{y-x}}\,{\sqrt {1-y^{2}}}\mathrm {d} y&=-\pi \,T_{n}(x)\end{aligned}}$ where integrals are considered as principal value.

Explicit expressions

Using the complex number exponentiation definition of the Chebyshev polynomial, one can derive the following expression: $T_{n}(x)={\dfrac {1}{2}}{\bigg (}{\Big (}x-{\sqrt {x^{2}-1}}{\Big )}^{n}+{\Big (}x+{\sqrt {x^{2}-1}}{\Big )}^{n}{\bigg )}\quad {\text{ for }}x\in \mathbb {R} ,$ $T_{n}(x)={\dfrac {1}{2}}{\bigg (}{\Big (}x-{\sqrt {x^{2}-1}}{\Big )}^{n}+{\Big (}x-{\sqrt {x^{2}-1}}{\Big )}^{-n}{\bigg )}\quad {\text{ for }}x\in \mathbb {R} .$ 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 monomialsx^k follows fromde Moivre's formula: $T_{n}(\cos(\theta ))=\operatorname {Re} (\cos n\theta +i\sin n\theta )=\operatorname {Re} ((\cos \theta +i\sin \theta )^{n}),$ whereRe denotes thereal part of a complex number. Expanding the formula, one gets $(\cos \theta +i\sin \theta )^{n}=\sum \limits _{j=0}^{n}{\binom {n}{j}}i^{j}\sin ^{j}\theta \cos ^{n-j}\theta .$ 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: $\cos n\theta =\sum \limits _{j=0}^{\lfloor n/2\rfloor }{\binom {n}{2j}}(\cos ^{2}\theta -1)^{j}\cos ^{n-2j}\theta ,$ which in turn means that $T_{n}(x)=\sum \limits _{j=0}^{\lfloor n/2\rfloor }{\binom {n}{2j}}(x^{2}-1)^{j}x^{n-2j}.$ This can be written as a₂F₁hypergeometric function: ${\begin{aligned}T_{n}(x)&=\sum _{k=0}^{\left\lfloor {\frac {n}{2}}\right\rfloor }{\binom {n}{2k}}\left(x^{2}-1\right)^{k}x^{n-2k}\\&=x^{n}\sum _{k=0}^{\left\lfloor {\frac {n}{2}}\right\rfloor }{\binom {n}{2k}}\left(1-x^{-2}\right)^{k}\\&={\frac {n}{2}}\sum _{k=0}^{\left\lfloor {\frac {n}{2}}\right\rfloor }(-1)^{k}{\frac {(n-k-1)!}{k!(n-2k)!}}~(2x)^{n-2k}\quad {\text{ for }}n>0\\\\&=n\sum _{k=0}^{n}(-2)^{k}{\frac {(n+k-1)!}{(n-k)!(2k)!}}(1-x)^{k}\quad {\text{ for }}n>0\\\\&={}_{2}F_{1}\!\left(-n,n;{\tfrac {1}{2}};{\tfrac {1}{2}}(1-x)\right)\\\end{aligned}}$ with inverse^[11]^[12] $x^{n}=2^{1-n}\mathop {{\sum }'} _{j=0 \atop j\equiv n{\pmod {2}}}^{n}\!\!{\binom {n}{\tfrac {n-j}{2}}}\!\;T_{j}(x),$ where the prime at the summation symbol indicates that the contribution ofj = 0 needs to be halved if it appears.

A related expression forT_n as a sum of monomials with binomial coefficients and powers of two is $T_{n}(x)=\sum \limits _{m=0}^{\left\lfloor {\frac {n}{2}}\right\rfloor }(-1)^{m}\left({\binom {n-m}{m}}+{\binom {n-m-1}{n-2m}}\right)\cdot 2^{n-2m-1}\cdot x^{n-2m}.$

Similarly,U_n can be expressed in terms of hypergeometric functions: ${\begin{aligned}U_{n}(x)&={\frac {\left(x+{\sqrt {x^{2}-1}}\right)^{n+1}-\left(x-{\sqrt {x^{2}-1}}\right)^{n+1}}{2{\sqrt {x^{2}-1}}}}\\&=\sum _{k=0}^{\left\lfloor {n}/{2}\right\rfloor }{\binom {n+1}{2k+1}}\left(x^{2}-1\right)^{k}x^{n-2k}\\&=x^{n}\sum _{k=0}^{\left\lfloor {n}/{2}\right\rfloor }{\binom {n+1}{2k+1}}\left(1-x^{-2}\right)^{k}\\&=\sum _{k=0}^{\left\lfloor {n}/{2}\right\rfloor }{\binom {2k-(n+1)}{k}}~(2x)^{n-2k}&{\text{ for }}n>0\\&=\sum _{k=0}^{\left\lfloor {n}/{2}\right\rfloor }(-1)^{k}{\binom {n-k}{k}}~(2x)^{n-2k}&{\text{ for }}n>0\\&=\sum _{k=0}^{n}(-2)^{k}{\frac {(n+k+1)!}{(n-k)!(2k+1)!}}(1-x)^{k}&{\text{ for }}n>0\\&=(n+1)\,{}_{2}F_{1}{\big (}-n,n+2;{\tfrac {3}{2}};{\tfrac {1}{2}}(1-x){\big )}.\end{aligned}}$

Properties

Symmetry

${\begin{aligned}T_{n}(-x)&=(-1)^{n}\,T_{n}(x),\\[1ex]U_{n}(-x)&=(-1)^{n}\,U_{n}(x).\end{aligned}}$

That is, Chebyshev polynomials of even order haveeven symmetry and therefore contain only even powers ofx. Chebyshev polynomials of odd order haveodd symmetry and therefore contain only odd powers ofx.

Roots and extrema

A Chebyshev polynomial of either kind with degreen hasn differentsimple roots, calledChebyshev roots, in the interval[−1, 1]. The roots of the Chebyshev polynomial of the first kind are sometimes calledChebyshev nodes because they are used asnodes in polynomial interpolation. Using the trigonometric definition and the fact that: $\cos \left((2k+1){\frac {\pi }{2}}\right)=0$ one can show that the roots ofT_n are: $x_{k}=\cos \left({\frac {\pi (k+1/2)}{n}}\right),\quad k=0,\ldots ,n-1.$ Similarly, the roots ofU_n are: $x_{k}=\cos \left({\frac {k}{n+1}}\pi \right),\quad k=1,\ldots ,n.$ Theextrema ofT_n on the interval−1 ≤x ≤ 1 are located at: $x_{k}=\cos \left({\frac {k}{n}}\pi \right),\quad k=0,\ldots ,n.$

One unique property of the Chebyshev polynomials of the first kind is that on the interval−1 ≤x ≤ 1 all of theextrema have values that are either −1 or 1. Thus these polynomials have only two finitecritical values, the defining property ofShabat polynomials. Both the first and second kinds of Chebyshev polynomial have extrema at the endpoints, given by: ${\begin{aligned}T_{n}(1)&=1\\T_{n}(-1)&=(-1)^{n}\\U_{n}(1)&=n+1\\U_{n}(-1)&=(-1)^{n}(n+1).\end{aligned}}$

Theextrema of $T_{n}(x)$ on the interval $-1\leq x\leq 1$ where $n>0$ are located at $n+1$ values of $x {\displaystyle x}$ . 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., $k {\displaystyle k}$ and $d {\displaystyle d}$ are relatively prime numbers.

Specifically (Minimal polynomial of 2cos(2pi/n)^[13]^[14]) when $n {\displaystyle n}$ 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 $x {\displaystyle x}$ .
$T_{n}(x)=-1$ if $d>2$ and $2n/d$ is odd. There are $n/2$ such values of $x {\displaystyle x}$ .

When $n {\displaystyle n}$ 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 $x {\displaystyle x}$ .
$T_{n}(x)=-1$ if $x=-1$ , or $d>2$ and $2n/d$ is odd. There are $(n+1)/2$ such values of $x {\displaystyle x}$ .

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: ${\begin{aligned}{\frac {\mathrm {d} T_{n}}{\mathrm {d} x}}&=nU_{n-1}\\{\frac {\mathrm {d} U_{n}}{\mathrm {d} x}}&={\frac {(n+1)T_{n+1}-xU_{n}}{x^{2}-1}}\\{\frac {\mathrm {d} ^{2}T_{n}}{\mathrm {d} x^{2}}}&=n\,{\frac {nT_{n}-xU_{n-1}}{x^{2}-1}}=n\,{\frac {(n+1)T_{n}-U_{n}}{x^{2}-1}}.\end{aligned}}$

The last two formulas can be numerically troublesome due to the division by zero (⁠0/0⁠indeterminate form, specifically) atx = 1 andx = −1. ByL'Hôpital's rule: ${\begin{aligned}\left.{\frac {\mathrm {d} ^{2}T_{n}}{\mathrm {d} x^{2}}}\right|_{x=1}\!\!&={\frac {n^{4}-n^{2}}{3}},\\\left.{\frac {\mathrm {d} ^{2}T_{n}}{\mathrm {d} x^{2}}}\right|_{x=-1}\!\!&=(-1)^{n}{\frac {n^{4}-n^{2}}{3}}.\end{aligned}}$

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

Also, we have: ${\frac {\mathrm {d} ^{p}}{\mathrm {d} x^{p}}}\,T_{n}(x)=2^{p}\,n\mathop {{\sum }'} _{0\leq k\leq n-p \atop k\,\equiv \,n-p{\pmod {2}}}{\binom {{\frac {n+p-k}{2}}-1}{\frac {n-p-k}{2}}}{\frac {\left({\frac {n+p+k}{2}}-1\right)!}{\left({\frac {n-p+k}{2}}\right)!}}\,T_{k}(x),~\qquad p\geq 1,$ where the prime at the summation symbols means that the term contributed byk = 0 is to be halved, if it appears.

Concerning integration, the first derivative of theT_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 forn ≥ 2: $\int T_{n}\,\mathrm {d} x={\frac {1}{2}}\,\left({\frac {T_{n+1}}{n+1}}-{\frac {T_{n-1}}{n-1}}\right)={\frac {n\,T_{n+1}}{n^{2}-1}}-{\frac {x\,T_{n}}{n-1}}.$

The last formula can be further manipulated to express the integral ofT_n as a function of Chebyshev polynomials of the first kind only: ${\begin{aligned}\int T_{n}\,\mathrm {d} x&={\frac {n}{n^{2}-1}}T_{n+1}-{\frac {1}{n-1}}T_{1}T_{n}\\&={\frac {n}{n^{2}-1}}\,T_{n+1}-{\frac {1}{2(n-1)}}\,(T_{n+1}+T_{n-1})\\&={\frac {1}{2(n+1)}}\,T_{n+1}-{\frac {1}{2(n-1)}}\,T_{n-1}.\end{aligned}}$

Furthermore, we have: $\int _{-1}^{1}T_{n}(x)\,\mathrm {d} x={\begin{cases}{\frac {(-1)^{n}+1}{1-n^{2}}}&{\text{ if }}~n\neq 1\\0&{\text{ if }}~n=1.\end{cases}}$

Products of Chebyshev polynomials

The Chebyshev polynomials of the first kind satisfy the relation: $T_{m}(x)\,T_{n}(x)={\tfrac {1}{2}}\!\left(T_{m+n}(x)+T_{|m-n|}(x)\right)\!,\qquad \forall m,n\geq 0,$ which is easily proved from theproduct-to-sum formula for the cosine: $2\cos \alpha \,\cos \beta =\cos(\alpha +\beta )+\cos(\alpha -\beta ).$ Forn = 1 this results in the already known recurrence formula, just arranged differently, and withn = 2 it forms the recurrence relation for all even or all odd indexed Chebyshev polynomials (depending on the parity of the lowestm) which implies the evenness or oddness of these polynomials. Three more useful formulas for evaluating Chebyshev polynomials can be concluded from this product expansion: ${\begin{aligned}T_{2n}(x)&=2\,T_{n}^{2}(x)-T_{0}(x)&&=2T_{n}^{2}(x)-1,\\T_{2n+1}(x)&=2\,T_{n+1}(x)\,T_{n}(x)-T_{1}(x)&&=2\,T_{n+1}(x)\,T_{n}(x)-x,\\T_{2n-1}(x)&=2\,T_{n-1}(x)\,T_{n}(x)-T_{1}(x)&&=2\,T_{n-1}(x)\,T_{n}(x)-x.\end{aligned}}$

The polynomials of the second kind satisfy the similar relation: $T_{m}(x)\,U_{n}(x)={\begin{cases}{\frac {1}{2}}\left(U_{m+n}(x)+U_{n-m}(x)\right),&~{\text{ if }}~n\geq m-1,\\\\{\frac {1}{2}}\left(U_{m+n}(x)-U_{m-n-2}(x)\right),&~{\text{ if }}~n\leq m-2.\end{cases}}$ (with the definitionU₋₁ ≡ 0 by convention ). They also satisfy: $U_{m}(x)\,U_{n}(x)=\sum _{k=0}^{n}\,U_{m-n+2k}(x)=\sum _{\underset {\text{ step 2 }}{p=m-n}}^{m+n}U_{p}(x)~.$ form ≥n.Forn = 2 this recurrence reduces to: $U_{m+2}(x)=U_{2}(x)\,U_{m}(x)-U_{m}(x)-U_{m-2}(x)=U_{m}(x)\,{\big (}U_{2}(x)-1{\big )}-U_{m-2}(x)~,$ which establishes the evenness or oddness of the even or odd indexed Chebyshev polynomials of the second kind depending on whetherm starts with 2 or 3.

Composition and divisibility properties

The trigonometric definitions ofT_n andU_n imply the composition or nesting properties:^[15] ${\begin{aligned}T_{mn}(x)&=T_{m}(T_{n}(x)),\\U_{mn-1}(x)&=U_{m-1}(T_{n}(x))U_{n-1}(x).\end{aligned}}$ ForT_mn the order of composition may be reversed, making the family of polynomial functionsT_n acommutative semigroup under composition.

SinceT_m(x) is divisible byx ifm is odd, it follows thatT_mn(x) is divisible byT_n(x) ifm is odd. Furthermore,U_mn−1(x) is divisible byU_n−1(x), and in the case thatm is even, divisible byT_n(x)U_n−1(x).

Orthogonality

BothT_n andU_n form a sequence oforthogonal polynomials. The polynomials of the first kindT_n are orthogonal with respect to the weight: ${\frac {1}{\sqrt {1-x^{2}}}},$ on the interval[−1, 1], i.e. we have: $\int _{-1}^{1}T_{n}(x)\,T_{m}(x)\,{\frac {\mathrm {d} x}{\sqrt {1-x^{2}}}}={\begin{cases}0&~{\text{ if }}~n\neq m,\\[5mu]\pi &~{\text{ if }}~n=m=0,\\[5mu]{\frac {\pi }{2}}&~{\text{ if }}~n=m\neq 0.\end{cases}}$

This can be proven by lettingx = cosθ and using the defining identityT_n(cosθ) = cos(nθ).

Similarly, the polynomials of the second kindU_n are orthogonal with respect to the weight: ${\sqrt {1-x^{2}}}$ on the interval[−1, 1], i.e. we have: $\int _{-1}^{1}U_{n}(x)\,U_{m}(x)\,{\sqrt {1-x^{2}}}\,\mathrm {d} x={\begin{cases}0&~{\text{ if }}~n\neq m,\\[5mu]{\frac {\pi }{2}}&~{\text{ if }}~n=m.\end{cases}}$

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

These orthogonality properties follow from the fact that the Chebyshev polynomials solve theChebyshev differential equations: ${\begin{aligned}(1-x^{2})T_{n}''-xT_{n}'+n^{2}T_{n}&=0,\\[1ex](1-x^{2})U_{n}''-3xU_{n}'+n(n+2)U_{n}&=0,\end{aligned}}$ which areSturm–Liouville differential equations. It is a general feature of suchdifferential equations that there is a distinguished orthonormal set of solutions. (Another way to define the Chebyshev polynomials is as the solutions tothose equations.)

TheT_n also satisfy a discrete orthogonality condition: $\sum _{k=0}^{N-1}{T_{i}(x_{k})\,T_{j}(x_{k})}={\begin{cases}0&~{\text{ if }}~i\neq j,\\[5mu]N&~{\text{ if }}~i=j=0,\\[5mu]{\frac {N}{2}}&~{\text{ if }}~i=j\neq 0,\end{cases}}$ whereN is any integer greater thanmax(i,j),^[10] and thex_k are theNChebyshev nodes (see above) ofT_N(x): $x_{k}=\cos \left(\pi \,{\frac {2k+1}{2N}}\right)\quad ~{\text{ for }}~k=0,1,\dots ,N-1.$

For the polynomials of the second kind and any integerN >i +j with the same Chebyshev nodesx_k, there are similar sums: $\sum _{k=0}^{N-1}{U_{i}(x_{k})\,U_{j}(x_{k})\left(1-x_{k}^{2}\right)}={\begin{cases}0&{\text{ if }}~i\neq j,\\[5mu]{\frac {N}{2}}&{\text{ if }}~i=j,\end{cases}}$ and without the weight function: $\sum _{k=0}^{N-1}{U_{i}(x_{k})\,U_{j}(x_{k})}={\begin{cases}0&~{\text{ if }}~i\not \equiv j{\pmod {2}},\\[5mu]N\cdot (1+\min\{i,j\})&~{\text{ if }}~i\equiv j{\pmod {2}}.\end{cases}}$

For any integerN >i +j, based on theN zeros ofU_N(x): $y_{k}=\cos \left(\pi \,{\frac {k+1}{N+1}}\right)\quad ~{\text{ for }}~k=0,1,\dots ,N-1,$ one can get the sum: $\sum _{k=0}^{N-1}{U_{i}(y_{k})\,U_{j}(y_{k})(1-y_{k}^{2})}={\begin{cases}0&~{\text{ if }}i\neq j,\\[5mu]{\frac {N+1}{2}}&~{\text{ if }}i=j,\end{cases}}$ and again without the weight function: $\sum _{k=0}^{N-1}{U_{i}(y_{k})\,U_{j}(y_{k})}={\begin{cases}0&~{\text{ if }}~i\not \equiv j{\pmod {2}},\\[5mu]{\bigl (}\min\{i,j\}+1{\bigr )}{\bigl (}N-\max\{i,j\}{\bigr )}&~{\text{ if }}~i\equiv j{\pmod {2}}.\end{cases}}$

Minimal∞-norm

For any givenn ≥ 1, among the polynomials of degreen 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 exactlyn + 1 times at: $x=\cos {\frac {k\pi }{n}}\quad {\text{for }}0\leq k\leq n.$

Proof

Let's assume thatw_n(x) is a polynomial of degreen with leading coefficient 1 with maximal absolute value on the interval[−1, 1] less than1 / 2^n − 1.

Define $f_{n}(x)={\frac {1}{\,2^{n-1}\,}}\,T_{n}(x)-w_{n}(x)$

Because at extreme points ofT_n we have ${\begin{aligned}|w_{n}(x)|&<\left|{\frac {1}{2^{n-1}}}T_{n}(x)\right|\\f_{n}(x)&>0\qquad {\text{ for }}~x=\cos {\frac {2k\pi }{n}}~&&{\text{ where }}0\leq 2k\leq n\\f_{n}(x)&<0\qquad {\text{ for }}~x=\cos {\frac {(2k+1)\pi }{n}}~&&{\text{ where }}0\leq 2k+1\leq n\end{aligned}}$

From theintermediate value theorem,f_n(x) has at leastn roots. However, this is impossible, asf_n(x) is a polynomial of degreen − 1, so thefundamental theorem of algebra implies it has at mostn − 1 roots.

Remark

By theequioscillation theorem, among all the polynomials of degree≤ n, the polynomialf minimizes‖ f ‖_∞ on[−1, 1]if and only if there aren + 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 approximated by itself and minimizes the∞-norm.

Above, however,| f | reaches its maximum onlyn + 1 times because we are searching for the best polynomial of degreen ≥ 1 (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 orGegenbauer polynomials $C_{n}^{(\lambda )}(x)$ , which themselves are a special case of theJacobi polynomials $P_{n}^{(\alpha ,\beta )}(x)$ : ${\begin{aligned}T_{n}(x)&={\frac {n}{2}}\lim _{q\to 0}{\frac {1}{q}}\,C_{n}^{(q)}(x)\qquad ~{\text{ if }}~n\geq 1,\\&={\frac {1}{\binom {n-{\frac {1}{2}}}{n}}}P_{n}^{\left(-{\frac {1}{2}},-{\frac {1}{2}}\right)}(x)={\frac {2^{2n}}{\binom {2n}{n}}}P_{n}^{\left(-{\frac {1}{2}},-{\frac {1}{2}}\right)}(x)~,\\[2ex]U_{n}(x)&=C_{n}^{(1)}(x)\\&={\frac {n+1}{\binom {n+{\frac {1}{2}}}{n}}}P_{n}^{\left({\frac {1}{2}},{\frac {1}{2}}\right)}(x)={\frac {2^{2n+1}}{\binom {2n+2}{n+1}}}P_{n}^{\left({\frac {1}{2}},{\frac {1}{2}}\right)}(x)~.\end{aligned}}$

Chebyshev polynomials are also a special case ofDickson 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 $\alpha ={\tfrac {1}{2}}$ , they are related by $D_{n}(x,{\tfrac {1}{4}})=2^{1-n}T_{n}(x)$ and $E_{n}(x,{\tfrac {1}{4}})=2^{-n}U_{n}(x)$ .

Other properties

The curves given byy =T_n(x), or equivalently, by the parametric equationsy =T_n(cosθ) = cosnθ,x = cosθ, are a special case ofLissajous curves with frequency ratio equal ton.

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

Forx ≠ 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 forx =e^iθ.

There are relations betweenLegendre polynomials and Chebyshev polynomials

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

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

These identities can be proven using generating functions and discrete convolution

Chebyshev polynomials as determinants

From their definition by recurrence it follows that the Chebyshev polynomials can be obtained asdeterminants of specialtridiagonal matrices of size $k\times k$ : $T_{k}(x)=\det {\begin{bmatrix}x&1&0&\cdots &0\\1&2x&1&\ddots &\vdots \\0&1&2x&\ddots &0\\\vdots &\ddots &\ddots &\ddots &1\\0&\cdots &0&1&2x\end{bmatrix}},$ and similarly for $U_{k}$ .

Examples

First kind

The first few Chebyshev polynomials of the first kind in the domain−1 <x < 1: The flatT₀,T₁,T₂,T₃,T₄ andT₅.

The first few Chebyshev polynomials of the first kind areOEIS: A028297 ${\begin{aligned}T_{0}(x)&=1\\T_{1}(x)&=x\\T_{2}(x)&=2x^{2}-1\\T_{3}(x)&=4x^{3}-3x\\T_{4}(x)&=8x^{4}-8x^{2}+1\\T_{5}(x)&=16x^{5}-20x^{3}+5x\\T_{6}(x)&=32x^{6}-48x^{4}+18x^{2}-1\\T_{7}(x)&=64x^{7}-112x^{5}+56x^{3}-7x\\T_{8}(x)&=128x^{8}-256x^{6}+160x^{4}-32x^{2}+1\\T_{9}(x)&=256x^{9}-576x^{7}+432x^{5}-120x^{3}+9x\\T_{10}(x)&=512x^{10}-1280x^{8}+1120x^{6}-400x^{4}+50x^{2}-1\end{aligned}}$

Second kind

The first few Chebyshev polynomials of the second kind in the domain−1 <x < 1: The flatU₀,U₁,U₂,U₃,U₄ andU₅. Although not visible in the image,U_n(1) =n + 1 andU_n(−1) = (n + 1)(−1)ⁿ.

The first few Chebyshev polynomials of the second kind areOEIS: A053117 ${\begin{aligned}U_{0}(x)&=1\\U_{1}(x)&=2x\\U_{2}(x)&=4x^{2}-1\\U_{3}(x)&=8x^{3}-4x\\U_{4}(x)&=16x^{4}-12x^{2}+1\\U_{5}(x)&=32x^{5}-32x^{3}+6x\\U_{6}(x)&=64x^{6}-80x^{4}+24x^{2}-1\\U_{7}(x)&=128x^{7}-192x^{5}+80x^{3}-8x\\U_{8}(x)&=256x^{8}-448x^{6}+240x^{4}-40x^{2}+1\\U_{9}(x)&=512x^{9}-1024x^{7}+672x^{5}-160x^{3}+10x\\U_{10}(x)&=1024x^{10}-2304x^{8}+1792x^{6}-560x^{4}+60x^{2}-1\end{aligned}}$

As a basis set

The non-smooth function (top)y = −x³H(−x), whereH is theHeaviside 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 appropriateSobolev space, the set of Chebyshev polynomials form anorthonormal basis, so that a function in the same space can, on−1 ≤x ≤ 1, be expressed via the expansion:^[16] $f(x)=\sum _{n=0}^{\infty }a_{n}T_{n}(x).$

Furthermore, as mentioned previously, the Chebyshev polynomials form anorthogonal basis which (among other things) implies that the coefficientsa_n can be determined easily through the application of aninner product. This sum is called aChebyshev series or aChebyshev expansion.

Since a Chebyshev series is related to aFourier cosine series through a change of variables, all of the theorems, identities, etc. that apply toFourier series have a Chebyshev counterpart.^[16] These attributes include:

The Chebyshev polynomials form acomplete orthogonal system.
The Chebyshev series converges tof(x) if the function ispiecewise smooth andcontinuous. The smoothness requirement can be relaxed in most cases – as long as there are a finite number of discontinuities inf(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 fromFourier series make the Chebyshev polynomials important tools innumeric analysis; for example they are the most popular general purpose basis functions used in thespectral method,^[16] often in favor of trigonometric series due to generally faster convergence for continuous functions (Gibbs' phenomenon is still a problem).

TheChebfun software package supports function manipulation based on their expansion in the Chebysev basis.

Example 1

Consider the Chebyshev expansion oflog(1 + x). One can express: $\log(1+x)=\sum _{n=0}^{\infty }a_{n}T_{n}(x)~.$

One can find the coefficientsa_n either through the application of an inner product or by the discrete orthogonality condition. For the inner product: $\int _{-1}^{+1}\,{\frac {T_{m}(x)\,\log(1+x)}{\sqrt {1-x^{2}}}}\,\mathrm {d} x=\sum _{n=0}^{\infty }a_{n}\int _{-1}^{+1}{\frac {T_{m}(x)\,T_{n}(x)}{\sqrt {1-x^{2}}}}\,\mathrm {d} x,$ which gives: $a_{n}={\begin{cases}-\log 2&{\text{ for }}~n=0,\\{\frac {-2(-1)^{n}}{n}}&{\text{ for }}~n>0.\end{cases}}$

Alternatively, when the inner product of the function being approximated cannot be evaluated, the discrete orthogonality condition gives an often useful result forapproximate coefficients: $a_{n}\approx {\frac {\,2-\delta _{0n}\,}{N}}\,\sum _{k=0}^{N-1}T_{n}(x_{k})\,\log(1+x_{k}),$ whereδ_ij is theKronecker delta function and thex_k are theN Gauss–Chebyshev zeros ofT_N(x): $x_{k}=\cos \left({\frac {\pi \left(k+{\tfrac {1}{2}}\right)}{N}}\right).$ For anyN, these approximate coefficients provide an exact approximation to the function atx_k with a controlled error between those points. The exact coefficients are obtained withN = ∞, 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 coefficientsa_n very efficiently through thediscrete cosine transform: $a_{n}\approx {\frac {2-\delta _{0n}}{N}}\sum _{k=0}^{N-1}\cos \left({\frac {n\pi \left(\,k+{\tfrac {1}{2}}\right)}{N}}\right)\log(1+x_{k}).$

Example 2

To provide another example: ${\begin{aligned}\left(1-x^{2}\right)^{\alpha }&=-{\frac {1}{\sqrt {\pi }}}\,{\frac {\Gamma \left({\tfrac {1}{2}}+\alpha \right)}{\Gamma (\alpha +1)}}+2^{1-2\alpha }\,\sum _{n=0}\left(-1\right)^{n}\,{2\alpha \choose \alpha -n}\,T_{2n}(x)\\[1ex]&=2^{-2\alpha }\,\sum _{n=0}\left(-1\right)^{n}\,{2\alpha +1 \choose \alpha -n}\,U_{2n}(x).\end{aligned}}$

Partial sums

The partial sums of: $f(x)=\sum _{n=0}^{\infty }a_{n}T_{n}(x)$ are very useful in theapproximation of various functions and in the solution ofdifferential equations (seespectral method). Two common methods for determining the coefficientsa_n are through the use of theinner product as inGalerkin's method and through the use ofcollocation which is related tointerpolation.

As an interpolant, theN coefficients of the(N − 1)st partial sum are usually obtained on the Chebyshev–Gauss–Lobatto^[17] points (or Lobatto grid), which results in minimum error and avoidsRunge'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: $x_{k}=-\cos \left({\frac {k\pi }{N-1}}\right);\qquad k=0,1,\dots ,N-1.$

Polynomial in Chebyshev form

An arbitrary polynomial of degreeN can be written in terms of the Chebyshev polynomials of the first kind.^[10] Such a polynomialp(x) is of the form: $p(x)=\sum _{n=0}^{N}a_{n}T_{n}(x).$

Polynomials in Chebyshev form can be evaluated using theClenshaw 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:^[18] $C_{n}(x)=2T_{n}\left({\frac {x}{2}}\right),\qquad S_{n}(x)=U_{n}\left({\frac {x}{2}}\right)$ 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 $V_{n}(x)$ .^[19] Lists of both sets of polynomials are given inViète'sOpera Mathematica, Chapter IX, Theorems VI and VII.^[20] The Vieta–Lucas and Vieta–Fibonacci polynomials of real argument are, up to a power of $i {\displaystyle i}$ and a shift of index in the case of the latter, equal toLucas and Fibonacci polynomialsL_n andF_n of imaginary argument.

Shifted Chebyshev polynomials of the first and second kinds are related to the Chebyshev polynomials by:^[18] $T_{n}^{*}(x)=T_{n}(2x-1),\qquad U_{n}^{*}(x)=U_{n}(2x-1).$

When the argument of the Chebyshev polynomial satisfies2x − 1 ∈[−1, 1] the argument of the shifted Chebyshev polynomial satisfiesx ∈[0, 1]. Similarly, one can define shifted polynomials for generic intervals[a,b].

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 nameairfoil polynomials. According to J. C. Mason and G. H. Elliott, the terminology "third-kind" and "fourth-kind" is due toWalter Gautschi, "in consultation with colleagues in the field of orthogonal polynomials."^[21] TheChebyshev polynomials of the third kind are defined as: $V_{n}(x)={\frac {\cos \left(\left(n+{\frac {1}{2}}\right)\theta \right)}{\cos \left({\frac {\theta }{2}}\right)}}={\sqrt {\frac {2}{1+x}}}T_{2n+1}\left({\sqrt {\frac {x+1}{2}}}\right)$ and theChebyshev polynomials of the fourth kind are defined as: $W_{n}(x)={\frac {\sin \left(\left(n+{\frac {1}{2}}\right)\theta \right)}{\sin \left({\frac {\theta }{2}}\right)}}=U_{2n}\left({\sqrt {\frac {x+1}{2}}}\right),$ where $\theta =\arccos x$ .^[21]^[22]They coincide with theDirichlet kernel.

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: $\left(1-x^{2}\right)^{-1/2},\quad \left(1-x^{2}\right)^{1/2},\quad (1-x)^{-1/2}(1+x)^{1/2},\quad (1+x)^{-1/2}(1-x)^{1/2}$ and are proportional to Jacobi polynomials $P_{n}^{(\alpha ,\beta )}(x)$ with:^[22] $(\alpha ,\beta )=\left(-{\frac {1}{2}},-{\frac {1}{2}}\right),\quad (\alpha ,\beta )=\left({\frac {1}{2}},{\frac {1}{2}}\right),\quad (\alpha ,\beta )=\left(-{\frac {1}{2}},{\frac {1}{2}}\right),\quad (\alpha ,\beta )=\left({\frac {1}{2}},-{\frac {1}{2}}\right).$

All four families satisfy the recurrence $p_{n}(x)=2xp_{n-1}(x)-p_{n-2}(x)$ 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 $x {\displaystyle x}$ , $2x$ , $2x-1$ , or $2x+1$ .^[21]

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 orderChebyshev filter designs using equally terminated passive networks are an example of this.^[23] 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 theChebyshev nodes in the same manner as standard Chebyshev polynomials. $P_{N}=\prod _{i=1}^{N}(x-C_{i})$

where

$P_{N}$ is anN-th order Chebyshev polynomial
$C_{i}$ is thei-th Chebyshev node

In the case of even order modified Chebyshev polynomials, theeven order modified Chebyshev nodes are used to construct the even order modified Chebyshev polynomials. $Pe_{N}=\prod _{i=1}^{N}(x-Ce_{i})$

where

$Pe_{N}$ is anN-th order even order modified Chebyshev polynomial
$Ce_{i}$ is thei-th even order modified Chebyshev node

For example, the 4th order Chebyshev polynomial from theexample 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

References

^Rivlin, Theodore J. (1974). "Chapter 2, Extremal properties".The Chebyshev Polynomials. Pure and Applied Mathematics (1st ed.). New York-London-Sydney: Wiley-Interscience [John Wiley & Sons]. pp. 56–123.ISBN 978-047172470-4.
^Lanczos, C. (1952)."Solution of systems of linear equations by minimized iterations".Journal of Research of the National Bureau of Standards.49 (1): 33.doi:10.6028/jres.049.006.
^Chebyshev first presented his eponymous polynomials in a paper read before the St. Petersburg Academy in 1853:
Chebyshev, P. L. (1854)."Théorie des mécanismes connus sous le nom de parallélogrammes".Mémoires des Savants étrangers présentés à l'Académie de Saint-Pétersbourg (in French).7:539–586. Also published separately asChebyshev, P. L. (1853).Théorie des mécanismes connus sous le nom de parallélogrammes. St. Petersburg: Imprimerie de l'Académie Impériale des Sciences.doi:10.3931/E-RARA-120037.
^Schaeffer, A. C. (1941)."Inequalities of A. Markoff and S. Bernstein for polynomials and related functions".Bulletin of the American Mathematical Society.47 (8):565–579.doi:10.1090/S0002-9904-1941-07510-5.ISSN 0002-9904.
^Ritt, J. F. (1922)."Prime and Composite Polynomials".Trans. Amer. Math. Soc.23:51–66.doi:10.1090/S0002-9947-1922-1501189-9.
^Demeyer, Jeroen (2007).Diophantine Sets over Polynomial Rings and Hilbert's Tenth Problem for Function Fields(PDF) (Ph.D. thesis). p. 70. Archived fromthe original(PDF) on 2 July 2007.
^Bateman & Bateman Manuscript Project 1953,p. 184, eqs. 3–4.
^Beckenbach, E. F.; Seidel, W.; Szász, Otto (1951), "Recurrent determinants of Legendre and of ultraspherical polynomials",Duke Math. J.,18:1–10,doi:10.1215/S0012-7094-51-01801-7,MR 0040487
^Bateman & Bateman Manuscript Project 1953,p. 187, eqs. 47–48.
^^a ^b ^cMason & Handscomb 2002.
^Cody, W. J. (1970). "A survey of practical rational and polynomial approximation of functions".SIAM Review.12 (3):400–423.doi:10.1137/1012082.
^Mathar, Richard J. (2006)."Chebyshev series expansion of inverse polynomials".Journal of Computational and Applied Mathematics.196 (2):596–607.arXiv:math/0403344.doi:10.1016/j.cam.2005.10.013.
^Gürtaş, Y. Z. (2017). "Chebyshev Polynomials and the minimal polynomial of $\cos(2\pi /n)$ ".American Mathematical Monthly.124 (1):74–78.doi:10.4169/amer.math.monthly.124.1.74.S2CID 125797961.
^Wolfram, D. A. (2022). "Factoring Chebyshev polynomials of the first and second kinds with minimal polynomials of $\cos(2\pi /d)$ ".American Mathematical Monthly.129 (2):172–176.doi:10.1080/00029890.2022.2005391.S2CID 245808448.
^Rayes, M. O.; Trevisan, V.; Wang, P. S. (2005), "Factorization properties of chebyshev polynomials",Computers & Mathematics with Applications,50 (8–9):1231–1240,doi:10.1016/j.camwa.2005.07.003
^^a ^b ^cBoyd, John P. (2001).Chebyshev and Fourier Spectral Methods(PDF) (second ed.). Dover.ISBN 0-486-41183-4. Archived fromthe original(PDF) on 31 March 2010. Retrieved19 March 2009.
^"Chebyshev Interpolation: An Interactive Tour". Archived fromthe original on 18 March 2017. Retrieved2 June 2016.
^^a ^bHochstrasser 1972, p. 778.
^Horadam, A. F. (2002),"Vieta polynomials"(PDF),Fibonacci Quarterly,40 (3):223–232
^Viète, François (1646).Francisci Vietae Opera mathematica : in unum volumen congesta ac recognita / opera atque studio Francisci a Schooten(PDF). Bibliothèque nationale de France.
^^a ^b ^cMason, J. C.; Elliott, G. H. (1993), "Near-minimax complex approximation by four kinds of Chebyshev polynomial expansion",J. Comput. Appl. Math.,46 (1–2):291–300,doi:10.1016/0377-0427(93)90303-S
^^a ^bDesmarais, Robert N.; Bland, Samuel R. (1995),"Tables of properties of airfoil polynomials",NASA Reference Publication 1343, National Aeronautics and Space Administration
^Saal, Rudolf (January 1979).Handbook of Filter Design (in English and German) (1st ed.). Munich, Germany: Allgemeine Elektricitais-Gesellschaft. pp. 25, 26,56–61, 116, 117.ISBN 3-87087-070-2.

Sources

Hochstrasser, Urs W. (1972) [1964]."Orthogonal Polynomials". InAbramowitz, Milton;Stegun, Irene (eds.).Handbook of Mathematical Functions (10th printing, with corrections; first ed.). Washington D.C.: National Bureau of Standards. Ch. 22, pp. 771–792.LCCN 64-60036.MR 0167642. Reprint: 1983. New York: Dover.ISBN 978-0-486-61272-0.
Bateman, Harry; Bateman Manuscript Project (1953)."Tchebichef polynomials". InErdélyi, Arthur (ed.).Higher Transcendental Functions. Vol. 2. Research associates:W. Magnus,F. Oberhettinger [de],F. Tricomi (1st ed.). New York: McGraw-Hill. § 10.11, pp. 183–187.LCCN 53-5555. Caltech eprint43491. Reprint: 1981. Melbourne, FL: Krieger.ISBN 0-89874-069-X.
Mason, J. C.; Handscomb, D.C. (2002).Chebyshev Polynomials. Chapman and Hall/CRC.doi:10.1201/9781420036114.ISBN 978-1-4200-3611-4.

Further reading

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.
Elliott, David (1964)."The evaluation and estimation of the coefficients in the Chebyshev Series expansion of a function".Math. Comp.18 (86):274–284.doi:10.1090/S0025-5718-1964-0166903-7.MR 0166903.
Eremenko, A.; Lempert, L. (1994)."An Extremal Problem For Polynomials"(PDF).Proceedings of the American Mathematical Society.122 (1):191–193.doi:10.1090/S0002-9939-1994-1207536-1.MR 1207536.
Hernandez, M. A. (2001)."Chebyshev's approximation algorithms and applications".Computers & Mathematics with Applications.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. Vol. 1105. pp. 27–48.doi:10.1007/BFb0072398.ISBN 978-3-540-13899-0.
Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010),"Orthogonal Polynomials", inOlver, 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).
Salzer, Herbert E. (1976)."Converting interpolation series into Chebyshev series by recurrence formulas".Mathematics of Computation.30 (134):295–302.doi:10.1090/S0025-5718-1976-0395159-3.MR 0395159.
Scraton, R.E. (1969)."The Solution of integral equations in Chebyshev series".Mathematics of Computation.23 (108):837–844.doi:10.1090/S0025-5718-1969-0260224-4.MR 0260224.
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

Media related toChebyshev polynomials at Wikimedia Commons
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 340Numerical Analysis & Math 440Advanced Numerical Analysis. Fullerton, CA: California State University. Archived fromthe original on 29 May 2007. Retrieved17 August 2020.
"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.

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