체비셰프 다항식

체비셰프 다항식은 코사인 및 사인 함수와 관련된 두 개의 다항식 시퀀스로, ${\displaystyle T_{}}$ n$x$) 및$$ ${\displaystyle U_{}}$ n${\displaystyle {}_{}x}$) {$displaystyle U_{n}($$x$$)\}$으로 $표기$됩니다.이러한 값은 여러 가지 동등한 방법으로 정의할 수 있으며, 그 중 하나는 삼각함수로 시작합니다.

$제1종{\displaystyle {}_{}}$ $(\$의 체비셰프 다항식은 다음과 같이 정의된다.

$\displaystyle T_{n}(\cos \theta)=\cos(n\theta)}$

마찬가지로, 두 번째 $종류{\displaystyle {}_{}}$ ${\displaystyle U_{}}$n {\$displaystyle U_{$n$$}의 체비셰프 다항식을 정의합니다.

$(\displaystyle U_{n}(\cos \theta)\sin \theta =\sin(n+1)\theta.}$

이러한 표현은 ${\displaystyle \cos }$ $⁡(\displaystyle \cos \theta)$에서 다항식을 정의하는 것은 언뜻 보기에는 명확하지 않을 수 있지만, 그 뒤에 cos$((\displaystyle \cos(n\theta)$ ${\displaystyle 및}$ sin${\displaystyle （n}$ +${\displaystyle 1}$) {$displaystyle \sin$ ( $n+1)\${displaystyle$\sin$ (\$cos$(\ta$})}$의 공식으로 고쳐 ${\displaystyle \mathrm{씁니다}}$.mulas는 cos${\displaystyle$ \cos$}$와$$sin {\$displaystyle \sin}$을 ${\displaystyle \mathrm{반복}}$하여 나타냅니다.예를 들어, 각도합 공식에서 직접 이어지는 이중 각도 공식은 ${\displaystyle {T\mathrm{}}_{}}$2 (${\displaystyle \cos \theta }$) ${\displaystyle =}$ ${\displaystyle \cos }$θ (${\displaystyle 2}$ ) ${\displaystyle =}$ $2$ ${\displaystyle {cos\mathrm{}}^{}}$ 2${\displaystyle \theta }$ -$$1 (\$displaystyle T_{2$} (\$cos \theta$) $= \cos ^{2\ta$} - $1$ 및${\displaystyle {}_{}}1{\displaystyle {}_{}}$ (${\displaystyle 1}$ ) cos ${\displaystyle =}$ 1 cos = cos = cos ^{2\ta } cos {2\ta } cos ⁡ 1 (1 ) ${\displaystyle t_{}}$ ) ⁡ )를 구하는데 사용할 수 있습니다.$\theta$) \$sin \theta$= \$sin (2\theta )=$ 2$\cos \theta$ \$sin$ \theta$$}는 각각 cos$$${\$ \$\theta$$$${\displaystyle \cos }$의 다항식이고 cos $⁡$의 다항식이고 cos${\displaystyle }$\$theta }$는 sin$$θ { $display style$ \$theta$$}$의 곱이다.} $및$ U ${\displaystyle 1}$ (${\displaystyle }$x )${\displaystyle }$=$2$ x \ displaystyle$U_{1$} ($x) =$ 2x$$}$$。

T(x)의_n 중요하고 편리한 특성은 내부 곱에 대해 직교한다는 것입니다.

${\displaystyle \displaystyle f,g\rangle =\int _{-1}^{f(x),g(x),{\frac {mathrm {d}x}{\displaystyle {1-x^{2}}}},$

및_n U(x)는 아래에 제시된 또 다른 유사한 내적물에 대해 직교한다.

체비셰프 다항식_n T는 구간 [-1, 1]의 절대값이 1로 제한되는 가능한 가장 큰 선행 계수를 갖는 다항식입니다.또한 다른 많은 ^[1]성질에 대한 "초급" 다항식입니다.

체비셰프 다항식은 체비셰프 노드라고도 불리는 T(x)의_n 루트가 다항식 보간 최적화를 위한 일치점으로 사용되기 때문에 근사 이론에서 중요하다.결과 보간 다항식은 룽게 현상의 문제를 최소화하고 "미니맥스" 기준이라고도 불리는 최대 노름 하의 연속 함수에 대한 최상의 다항식 근사치에 가까운 근사치를 제공한다.이 근사치는 클렌쇼-커티스 직교법(Clensshaw-Curtis quadrature)으로 직접 이어진다.

이 다항식들은 파프누티 체비셰프의 ^[2]이름을 따서 명명되었다.T는 체비셰프라는 이름을 체비셰프, 체비셰프(프랑스어) 또는 체비쇼프(독일어)로 대체 번역하기 때문에 사용됩니다.

정의들

반복 정의

첫 번째 종류의 체비셰프 다항식은 반복 관계에서 구한다.

$디스플레이 스타일T_{0}(x)&=1\T_{1}(x)&=x\T_{n+1}(x)&=2x,T_{n}(x)-T_{n-1}(x)\end { aligned}}$

T의 일반적인_n 생성 함수는

${\displaystyle \sum _{n=0}^{\infty}T_{n}(x),t^{n}=blac{1-blac}{1-2blac+t^{2}}}}.}$

체비셰프 다항식에는 몇 가지 다른 생성 함수가 있다; 지수 생성 함수는

${\displaystyle \sum _{n=0}^{\infty }T_{n}(x){\frac {t^{n}}{n!}}=black{1}{2}}\!\left(e^{t\left(x-{\paramrt {x^{2}-1}}\right)}+e^{t\paramrt {x^{2}-1}\right)=e^{param}\cosh \left(t {x^{2}-1}\right)\right).}$

2차원 전위 이론과 다극 팽창에 관련된 생성 함수는

$\displaystyle \sum \sum _{n=1}^{\infty }T_{n}(x),{\frac {t^{n}}{n}=\ln \leftfac {1}{\displayrt {1-2g+t^{2}(}}}}}\right).}$

두 번째 종류의 체비셰프 다항식은 반복 관계에 의해 정의된다.

$디스플레이 스타일U_{0}(x)&=1\U_{1}(x)&=2x\\U_{n+1}(x)&=2x,U_{n}(x)-U_{n-1}(x)\end { aligned}}$

${\displaystyle {T\mathrm{}}_{}}$ 1 (${\displaystyle x}$ )$=$ { { $displaystyle T_{1}(x$)=$x$$}$ vs 를 $제외$하고, 2 세트의 반복 관계는 동일합니다.${\displaystyle {U\mathrm{}}_{}}$ 1${\displaystyle {}_{}}$(${\displaystyle x}$ ) ${\displaystyle =}$ ${\displaystyle 2}$ { $displaystyle U_{1}(x$) $= 2x$ 。U의 일반적인_n 생성 함수는

${\displaystyle \sum _{n=0}^{\infty}U_{n}(x),t^{n}=black{1}{1-2g+t^{2}}}},}$

지수 생성 함수는

${\displaystyle \sum _{n=0}^{\infty}}U_{n}(x){\frac {t^{n}}{n!}=e^{filename}\!\left(\!\cosh \left(tcosh {x^2}-1})+{\frac {x}{\fracrt {x^2}-1}}\sinh \left(tcosh {x2}-1}\right)\!\right).}$

삼각 정의

서문에 설명된 것처럼, 첫 번째 종류의 체비셰프 다항식은 다음을 만족시키는 고유한 다항식으로 정의될 수 있다.

${\displaystyle T_{n}(x)=cosbegin{cases}\cos(n\arccos x)&{\text{ if }x\leq 1\\cosh}x}&{\text{ if }x\geq 1\(-1)^{n\cosh(n\opername {\cosh})}$

다른 말로 하자면, 다음을 만족시키는 고유한 다항식으로서

$(\displaystyle T_{n}(\cos \theta)=\cos(n\theta)}$

n = 0, 1, 2, 3, …의 경우, 이는 기술적 점으로서 슈뢰더 방정식의 변형(등가 전치)이다.즉_n, T(x)는 함수적으로 n x에 공역하며, 아래의 내포 속성에 코드화되어 있습니다.

두 번째 종류의 다항식은 다음을 만족한다.

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

또는

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

이것은 구조적으로 Dirichlet_n 커널 D(x)와 매우 유사합니다.

${\displaystyle D_{n}(x)=sin\left((2n+1){\dfrac {x}{2}},\right)}{\sin(\dfrac {x}{2}}}}=U_{2n}\!\!\left(\cos {frac {x}{2}}\오른쪽).}$

(사실 디리클레 커널은 현재 제4종 체비셰프 다항식으로 알려진 것과 일치합니다.)

그 cos nx는 cos x의 n차 다항식이다. cos nx는 de Moivre 공식의 한 변의 실측 부분이라는 것을 관찰함으로써 알 수 있다.다른 변의 실수는 cos x와 sin x의 다항식이며, 여기서 sin x의 모든 거듭제곱은 짝수이고 따라서 등식² cos x + sin² x = 1을 통해 치환될 수 있다.같은 추론에 의해, sin nx는 sin x의 모든 거듭제곱이 홀수인 다항식의 허수 부분이며, 따라서 sin x의 한 인자를 인수분해하면 나머지 인자를 치환하여 cos x의 (n-1)차 다항식을 만들 수 있다.

동일성은 재귀 생성 공식과 함께 매우 유용하며, 이는 베이스 각도의 코사인만을 기준으로 각도의 정수 배수의 코사인 값을 계산할 수 있게 해준다.

첫 번째 종류의 처음 두 개의 체비셰프 다항식은 다음과 같은 정의에서 직접 계산된다.

${\displaystyle T_{0}(\cos \theta)=\cos(0\theta)=1}$

그리고.

${\displaystyle T_{1}(\cos \theta)=\cos \theta,}$

나머지는 곱-합 동일성의 전문화를 사용하여 평가할 수 있다.

$(\displaystyle 2\cos n\theta \cos \theta =\cos \lbrack (n+1)\cos \rbrack +\cos \lbrack (n-1)\theta \rbrack }$

예를 들면,

$디스플레이 스타일T_{2}(\cos \theta)&=\cos 2\theta = 2\cos \theta \cos \theta -1=2\cos ^{2}\theta -1.\cos\T_{3}(\cos \theta)&=\cos 3\theta = 2\cos \theta -\cos \theta = 4\cos ^{3}\cos - 3\cos \theta .\end{aligned}}$

반대로 삼각함수의 임의의 정수승은 체비셰프 다항식을 사용하여 삼각함수의 선형 조합으로 표현될 수 있다.

$\displaystyle \cos ^{n}\theta = 2^{1-n}\!\mathop {{\sum }}_{j=0}^{n}_{n-j,{\text{even}}}\!\!{\binom {n}{\tfrac {n-j}{2}},T_{j}(\cos \theta )$

여기서 소수는 j = 0의 기여가 나타나면 반감해야 함을 나타내고, ${\displaystyle T_{}}$j ( ${\displaystyle \cos }$ ${\displaystyle )}$ )$$ ${\displaystyle \cos }$ ${\displaystyle }$ ( ${\displaystyle j\theta }$ ) \ $displaystyle T_{j}( cos$ \$theta$ ) = \$cos$($j \theta$ )$$}$$}

직접적인 결과는 체비셰프 다항식의 관점에서 복소수 지수의 표현이다: 주어진 z = a + bi,

$(\displaystyle\displaystyle\signed\z^{n}&=z^{n}\!\left(\cos \left(n\arccos {a}{z }}\right)+i440frac {b}{b}}\sin \left(n\arccos {a}{z}}}\right)\\&= z^{n},T_{n}\!\!\;\left({\frac {a}{z }}\오른쪽)+ib z^{n-1}\U_{n-1}\!!\!\;\leftflac{a}{z }}\right).\end { aligned}}$

통근 다항식 정의

체비셰프 다항식은 또한 다음과 같은 ^[3]정리로 특징지을 수 있다.

${\displaystyle F}$ n${\displaystyle {}_{}}$(${\displaystyle x}$ )$${$displaystyle$ ${\displaystyle {F\_}_{}}$${n}($${\displaystyle {}_{}}$) } {$displaystyle$ 0${\displaystyle {}_{}}$ ${\displaystyle =}$ {$displaystyle$ F_{$n}(x$) $= n$ {${\displaystyle {displaystyle}_{}}$$F_$${\displaystyle {}_{}}$$n$}(${\displaystyle x)}$ ${\displaystyle =}$ ${\displaystyle F_{}{}_{}}$m (${\displaystyle x_{}}$ )${\displaystyle }$= ${\displaystyle F_{}}$m${\displaystyle {}_{}}$(${\displaystyle x}$ ${\displaystyle )}$ （ F ) ） （ F .=$F_{$n}($F_{m}($x$$)) 모든$m$ {\$displaystyle$m$$}$및{\displaystyle }$ n {\$displaystyle$n}에 대해 $F{\displaystyle }$${\displaystyle {}_{}n(}$x${\displaystyle )}$ = ${\displaystyle x^{}}$ $n($x)$$= 모든$n$ {$displaystyle$$F_$$$${\displaystyle {}_{}}$${$n}(x) = ${\displaystyle x}$$^{$n$$} = F${\displaystyle (}$x${\displaystyle ){}_{}}$= 2/${\displaystyle {displaystyle}_{}}$ ${\displaystyle N}$(x ) 중 ${\displaystyle {}_{}}$의 간단한 변수 $변경까지$입니다${\displaystyle .}$$T\_{\displaystyle }$${n}(x/2)$(모든$$ n$\displaystyle$ n$)$에 대해

펠 방정식의 정의

체비셰프 다항식은 또한 펠 방정식의 해로 정의될 수 있다.

${\displaystyle T_{n}(x)^{2}-\left(x^{2}-1\오른쪽)U_{n-1}(x)^{2}=1}$

링 R[x]^[4]로따라서 기본 솔루션의 거듭제곱을 취하는 Pell 방정식의 표준 기법에 의해 생성될 수식은 다음과 같습니다.

${\displaystyle T_{n}(x)+U_{n-1}(x),{\sqrt {x^2}-1}=\left(x+{\sqrt {x^{2}-1}}\right)^{n}~.}$

두 종류의 체비셰프 다항식 사이의 관계

첫 번째와 두 번째 종류의 체비셰프 다항식은 P = 2x 및 Q = 1인 루카스 시퀀스 δ_n(P, Q)와 δ_n(P, Q)의 보완 쌍에 해당한다.

${\displaystyle {\tilde {U}_{n}(2x,1)&=U_{n-1}(x),\{\tilde {V}}_{n}(2x,1)&=2,T_{n}(x).\end { aligned}}$

따라서 이들 쌍은 상호 반복 ^[5]방정식 쌍도 충족합니다.

$디스플레이 스타일T_{n+1}(x)&={n}(x)-(1-x^{2}),U_{n-1}(x),\U_{n+1}(x)&=x,U_{n}(x)+T_{n+1}(x)\end { aligned}}$

이들 중 두 번째는 체비셰프 다항식의 반복 정의를 사용하여 재배치할 수 있다.

${\displaystyle T_{n}(x)=snfrac {1}{2}}{\big (}U_{n}(x)-U_{n-2)(x){\big}}.}$

이 공식을 반복적으로 사용하면 합 공식은 다음과 같습니다.

${\displaystyle U_{n}(x)=sum begin{case}2\sum _{\text{홀수}}j}^{n}홀수 }}n의 경우 T_{j}(x)&{\text{.\\2\sum _{\text{짝수}}j}^{n}T_{j}(x)-1&{\text{짝수}}n,\end{case}}}$

${\displaystyle T_{}}$n (${\displaystyle x}$)의 도함수식을 사용하여 U ${\displaystyle n{}_{}}$( x $){$$displaystyle$$$U_${n$$}($$x$) ${\displaystyle {및\mathrm{U}}_{}}$ n${\displaystyle {}_{}}$- 2${\displaystyle }$(x$)$를 $대체$하면서$T{\displaystyle {}_{}}$ n {$displaystyle$T_${n$$}(x$)의 도함수에 대한 반복 ${\displaystyle {관계}_{}}$를 제공합니다.

${\displaystyle 2,T_{n}(x)=snfrac {1}{n+1},{\frac {d} {\mathrm {d} {\mathrm {d} x},T_{n+1}-{\frac {1},{\frac {n-1},{\mathrm} {d} {d} {d},$

이 관계는 미분 방정식을 푸는 체비셰프 스펙트럼 방법에 사용됩니다.

체비셰프 다항식에^[6] 대한 투란의 부등식은 다음과 같다.

$디스플레이 스타일T_{n}(x)^{n2}-T_{n-1}(x), T_{n+1}(x)&=1-x^{2}>0&{\text{for }-1<x<1&{\text{및}}}\U_{n}(x)^{2}-U_{n-1}(x),U_{n+1}(x)&=1>0~\end { aligned}}$

일체적 관계는^{[5]: 187(47)(48) [7]}

${{displaystyle {naligned}\int _{-1}^{n}(y)}{(y-x){\sqrt {1-y^{2}}}}}},\mathrm {d} y=\pi,U_{n-1}(x)~,\\int {-1}{frac}{fr}{{-frc}}}}}{{-fr}}}}}}}}}}$

여기서 적분이 주요 값으로 간주됩니다.

명시적 표현

체비셰프 다항식을 정의하는 방법은 다음과 같은 다양한 명시적 표현으로 이어진다.

$디스플레이 스타일T_ᆭ())&, =ᆮ\cos(n\arccos))\qquad&{\text{에}}일)\leq 1\\\\{\dfrac{1}{2}}{\bigg(}{\Big(}X좌표{\sqrt{x^{2}-1}}{\Big)}^{n}+{\Big(}x+{\sqrt{x^{2}-1}}{\Big)}^{n}{\bigg)}\qquad&{\text{에}}일)\geq 1\\\end{경우}}\\\\&, =ᆹ\cos(n\arccos))\qquad \quad&{\text{에}}~-1\leq x\leq1\\\\\cosh(n\opera.torname{arcosh}x))qquad \leq x\\\(-1)^{n}\cosh {big (}n\operatorname {paramosh}(-x){\big})\qquad \parames & {\text{}(}~x\leq -1\\\\\end{cases}\\cases}\cases}\coshT_{n}(x)&=_{k=0}^{\left\lfloor {frac {n}{2}\right\loor}{\binom {n}{2k}\left(x^{2}-1\right)^{k}x^{n-2k}\&=x^{n}\sum _{k=0}^{\left\lfloor {frac {n}{2}\right\binom {n}{2k}\left(1-x^{-2}\right)^{k}\&=sum _{k=0}^{\left\lfloor {frac {n}{2}\right\loor }(-1)^{k}{\frac {(n-k-1)!}{k!(n-2k)!}~(2x)^{n-2k}\qquad\qquad\\\&=n\sum_{k=0}^{n}(-2)^{k}{\frac{(n+k-1)!}{(n-k)!(2k)!}}(1-x)^{k}\qquad \qquad ~{\text{ for }}~n>0\\\&={}_{2}}F_{1}\!\leftfracn,n;{\tfrac {1}{2}};{\tfrac {2}}}(1-x)\right)\\end{aligned}}$

역순으로^[8][9]

${{displaystyle x^{n}=2^{1-n}\mathop {{\sum }'}_{j=0 \syslog j,\equiv,nsyspmod {2}}^{n}\!\!{\binom {n}{\tfrac {n-j}{2}}\!\;T_{j}(x),}$

여기서 소수는 j = 0의 기여도가 나타나면 절반으로 줄여야 함을 나타냅니다.

$디스플레이 스타일U_{n}(x)&=paramfrac {left(x+{\paramrt {x^{2}-1}}\right(x-{\paramrt {x^2}-1}\right)^{n+1}}}{2paramrt {x^{2}-1}}}}}\&={\sumforam fl left fright {x}_loor {x}_l left}^{k}x^{n-2k}\&=x^{n}\sum _{k=0}^{\left\lfloor {frac {n}{2}\right\loor}{\binom {n+1}{2k+1}\left(1-x^{-2}\오른쪽)^{k}\&=\sum _{k=0}^{\left\lfloor {frac {n}{\right\loor}{\binom {2k-(n+1)}{k}~(2x)^{n-2k}&{\text{floor}{k=0\frfloor}{\flflflfloor}{\flfloor}{\flflflflfloor}{\floor }}{(n-k)!(2k+1)!}}(1-x)^{k}&{\text{ for }~n>0\&=(n+1)\{}_{2}F_{1}\left(-n,n+2;{\tfrac {3}{2}};{\tfrac {1}{1}{{\end}}}}에 정렬되었습니다.$

여기서₁ F는 초기하 함수입니다.

특성.

대칭

$디스플레이 스타일T_{n}(-x)&=param1)^{n},T_{n}(x)=begin{case}T_{n}(x)\quad &~{\text{even}\\text{n}(x)\quad &~{\text{n}(x)\quad &~{\text{n}\end{case}\\\\\text{cases}\\\\\\\\text{\\\}U_{n}(-x)&=begin1)^{n},U_{n}(x)=begin{case}U_{n}(x)\quad &~{\text{even}\\\text{n}(x)\quad &~{\text{n}(x)\text{n}\end{case}\end{aligned}}\end{n}$

즉, 짝수 차수의 체비셰프 다항식은 짝수 대칭을 가지므로 x의 짝수 거듭제곱만 포함합니다. 홀수 차수의 체비셰프 다항식은 홀수 대칭을 가지므로 x의 홀수 거듭제곱만 포함합니다.

뿌리와 극단

차수가 n인 체비셰프 다항식은 [-1, 1] 구간에 체비셰프 루트라고 불리는 n개의 다른 단순 루트를 가진다.제1종 체비셰프 다항식의 근은 다항식 보간에서 노드로 사용되기 때문에 체비셰프 노드라고 불리기도 한다.삼각법 정의와 다음과 같은 사실을 사용하여

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

T의_n 뿌리가 …라는 것을 알 수 있다.

$\displaystyle x_{k}=\cos\leftfrac\pi(k+1/2)}{n}}\right),\cisco k=0,\ldots,n-1.}$

마찬가지로, U의_n 어근은

${\displaystyle x_{k}=\cos \leftfrac {k}{n+1}\pi \right},\display k=1,\ldots,n.$

간격 -1 µ x µ 1의 T의_n 극점은 다음 위치에 있습니다.

$\displaystyle x_{k}=\cos \leftfrac {k}{n}\pi \right},\display k=0,\ldots,n.}$

첫 번째 종류의 체비셰프 다항식의 한 가지 고유한 특성은 구간 -1 x x 1 1에서 모든 극단값이 -1 또는 1이라는 것이다.따라서 이 다항식들은 샤바트 다항식의 정의 특성인 두 개의 유한 임계값만을 가지고 있다.체비셰프 다항식의 첫 번째와 두 번째 종류는 다음과 같이 끝점에 극단점을 가진다.

${\displaystyle T_{n}(1)=1}$

${\displaystyle T_{n}(-1)=param1)^{n}$

${\displaystyle U_{n}(1)=n+1}$

${\displaystyle U_{n}(-1)=param1)^{n}(n+1)}.$

간격${\displaystyle }$- ${\displaystyle 1x}$ x$$ 1 { $display$ $style$ -$$$1$ \ display style - 1 \ $leq$x \ $leq$1 }의 $극단{\displaystyle }$($x$ $).$ 서 n${\displaystyle }$> { $displaystyle$ n > $1${ $displaystyle$ $n$$$+$$1$$} 값은$$ ${\displaystyle \mathrm{n}}$${\displaystyle {}_{}}$+$$1 입니다.그들은±1{\displaystyle \pm 1}, d>2{\displaystyle d>2}, d2n{\displaystyle d\.\;2n}, 0<>k<>d/2{0<, k<, d/2\displaystyle}과(k, d))1{\displaystyle(k,d)=1}, i.e., 거⁡(2π kd){\displaystyle \cos \left({\frac{2\pi k}{d}}\right)}. k{\displaystyl$ek}$와 d$\displaystyle$d는$$ 비교적 소수입니다.

특히 ^[10][11]$n이$$짝수$일 때

• ${\displaystyle x_{}}$ $=$ ±$$ {$displaystyle$ x=\$pm$1)$$ d> {$displaystyle$ d>$2$및 ${\displaystyle 2}$ n${\displaystyle }$/ ${\displaystyle d}${$displaystyle 2n/d}$인 $경우{\displaystyle }$ T n${\displaystyle {}_{}}$(${\displaystyle x}$ )$=$ ${\displaystyle 1}$ {$displaystyle T_{n$}(x)=1$}$입니다.$x$에는 n${\displaystyle /2\mathrm{}}$+ $(style$$n$/2$+$$1)$의 값이 $있습니다{\displaystyle }$$.$
• ${\displaystyle d_{}}$> { $displaystyle$ d $> 2$ $및$ 2${\displaystyle n}$ /$$ { $display$ style 2$n/d$}인$$$$ 경우 T${\displaystyle {}_{}n}$ (${\displaystyle }$x )$=$ - ${\displaystyle 1}$ { $displaystyle T_{n}(x$)=-$1}.$x$(\displaystyle$ x$)$에는 n${\displaystyle /}$2$(\$$displaystyle$$$n/$2)$의 값이 $있습니다{\displaystyle }$.

n$\displaystyle$n이$$ 이상할 $경우{\displaystyle }$

• ${\displaystyle x_{}}$ ${\displaystyle =}$ { $display style$ x $=$$1$) $>$ $2{\displaystyle \mathrm{}}$ { $displaystyle d >$ 2 ${\displaystyle 및}$2 ${\displaystyle n}$/ d { $displaystyle 2n/d$}이면 T${\displaystyle {}_{}}$n (${\displaystyle x}$ ) ${\displaystyle =}$ { $displaystyle T_{n}(x$)=$$1 입니다.x$({displaystyle$x$\})$에는 (${\displaystyle \mathrm{n}}$ ${\displaystyle +}$1)${\displaystyle }$/ 2$({displaystyle (n+1)/2)$의 값이 있습니다$.{\displaystyle }$
• ${\displaystyle x_{}}$ $=$ -$$({$displaystyle$ x=-$1$ $또는$d$>$ ({$displaystyle d>$2} $및{\displaystyle \mathrm{}\mathrm{2n}}$ / $d($displaystyle $2n/d)$인 경우 T${\displaystyle {}_{}}$n $=$ - $(x$ - 1(x)=-$1)$x$({displaystyle$x$\})$에는 (${\displaystyle \mathrm{n}}$ ${\displaystyle +}$1)${\displaystyle }$/ 2$({displaystyle (n+1)/2)$의 값이 있습니다$.{\displaystyle }$

이 결과는 U ${\displaystyle n{}_{}}$(${\displaystyle }$x${\displaystyle }$) ± ${\displaystyle 1}$ ${\displaystyle =}$ {\$displaystyle U_{n}(x$)\$pm$ 1$=$$0$^[11] 및 V${\displaystyle {}_{}}$n (${\displaystyle x)}$ ± ${\displaystyle 1}$ ${\displaystyle =}$ {$displaystyle V_$${n$$}(x$)\$pm$ 1$=0$} 및$$ ${\displaystyle W_{}}$n (${\displaystyle x)}$ ± 1 ${\displaystyle =}$ 0 {$displaystyle$ W_0($x)$의 $해$로 일반화되었습니다.

차별화 및 통합

다항식의 도함수는 간단하지 않을 수 있다.삼각함수 형태로 다항식을 미분함으로써 다음과 같이 나타낼 수 있습니다.

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

마지막 두 공식은 0으로 나누기 때문에 수치적으로 문제가 있을 수 있습니다. 0/0 부정 형식, 특히 x = 1 및 x = -1에서.다음과 같이 나타낼 수 있습니다.

$좌측에 전시합니다.{\frac {mathrm {d} ^{2}T_{n}{\mathrm {d} x^{2}}\right _{x=1}\!\!&=snfrac {n^{4}-n^{2}}{3}},\\left.{\frac {mathrm {d} ^{2}T_{n}{\mathrm {d} x^{2}}\right _{x=-1}\!!&=param1)^{n}{\frac {n^{4}-n^{2}}{3}}.\end { aligned}}$

More general formula states:

${\displaystyle \left.{\frac {d^{p}T_{n}}{dx^{p}}}\right _{x=\pm 1}\!\!=(\pm 1)^{n+p}\prod _{k=0}^{p-1}{\frac {n^{2}-k^{2}}{2k+1}}~,}$

which is of great use in the numerical solution of eigenvalue problems.

Also, we have

${\displaystyle {\frac {\mathrm {d} ^{p}}{\mathrm {d} x^{p}}}\,T_{n}(x)=2^{p}\,n\mathop {{\sum }'} _{0\leq k\leq n-p \atop k\,\equiv \,n-p{\pmod {2}}}{\binom {{\frac {n+p-k}{2}}-1}{\frac {n-p-k}{2}}}{\frac {\left({\frac {n+p+k}{2}}-1\right)!}{\left({\frac {n-p+k}{2}}\right)!}}\,T_{k}(x),~\qquad p\geq 1,}$

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

Concerning integration, the first derivative of the T_n implies that

${\displaystyle \int U_{n}\,\mathrm {d} x={\frac {T_{n+1}}{n+1}}}$

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

${\displaystyle \int T_{n}\,\mathrm {d} x={\frac {1}{2}}\,\left({\frac {T_{n+1}}{n+1}}-{\frac {T_{n-1}}{n-1}}\right)={\frac {n\,T_{n+1}}{n^{2}-1}}-{\frac {x\,T_{n}}{n-1}}.}$

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

${\displaystyle {\begin{aligned}\int T_{n}\,\mathrm {d} x&={\frac {n}{n^{2}-1}}T_{n+1}-{\frac {1}{n-1}}T_{1}T_{n}\\&={\frac {n}{n^{2}-1}}\,T_{n+1}-{\frac {1}{2(n-1)}}\,(T_{n+1}+T_{n-1})\\&={\frac {1}{2(n+1)}}\,T_{n+1}-{\frac {1}{2(n-1)}}\,T_{n-1}.\end{aligned}}}$

Furthermore, we have

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

Products of Chebyshev polynomials

The Chebyshev polynomials of the first kind satisfy the relation

${\displaystyle T_{m}(x)\,T_{n}(x)={\tfrac {1}{2}}\!\left(T_{m+n}(x)+T_{ m-n }(x)\right)\!,\qquad \forall m,n\geq 0,}$

which is easily proved from the product-to-sum formula for the cosine,

${\displaystyle 2\cos \alpha \,\cos \beta =\cos(\alpha +\beta )+\cos(\alpha -\beta ).}$

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 indexed Chebyshev polynomials (depending on the parity of the lowest m) which implies the evenness or oddness of these polynomials. Three more useful formulas for evaluating Chebyshev polynomials can be concluded from this product expansion:

${\displaystyle {\begin{aligned}T_{2n}(x)&=2\,T_{n}^{2}(x)-T_{0}(x)&&=2T_{n}^{2}(x)-1,\\T_{2n+1}(x)&=2\,T_{n+1}(x)\,T_{n}(x)-T_{1}(x)&&=2\,T_{n+1}(x)\,T_{n}(x)-x,\\T_{2n-1}(x)&=2\,T_{n-1}(x)\,T_{n}(x)-T_{1}(x)&&=2\,T_{n-1}(x)\,T_{n}(x)-x.\end{aligned}}}$

The polynomials of the second kind satisfy the similar relation

${\displaystyle T_{m}(x)\,U_{n}(x)={\begin{cases}{\frac {1}{2}}\left(U_{m+n}(x)+U_{n-m}(x)\right),&~{\text{ if }}~n\geq m-1,\\\\{\frac {1}{2}}\left(U_{m+n}(x)-U_{m-n-2}(x)\right),&~{\text{ if }}~n\leq m-2.\end{cases}}}$

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

${\displaystyle U_{m}(x)\,U_{n}(x)=\sum _{k=0}^{n}\,U_{m-n+2k}(x)=\sum _{\underset {\text{ step 2 }}{p=m-n}}^{m+n}U_{p}(x)~.}$

for m ≥ n. For n = 2 this recurrence reduces to

${\displaystyle U_{m+2}(x)=U_{2}(x)\,U_{m}(x)-U_{m}(x)-U_{m-2}(x)=U_{m}(x)\,{\big (}U_{2}(x)-1{\big )}-U_{m-2}(x)~,}$

which establishes the evenness or oddness of the even or odd indexed Chebyshev polynomials of the second kind depending on whether m starts with 2 or 3.

Composition and divisibility properties

The trigonometric definitions of T_n and U_n imply the composition or nesting properties^[13]

${\displaystyle {\begin{aligned}T_{mn}(x)&=T_{m}(T_{n}(x)),\\U_{mn-1}(x)&=U_{m-1}(T_{n}(x))U_{n-1}(x).\end{aligned}}}$

For T_mn the order of composition may be reversed, making the family of polynomial functions T_n a commutative semigroup under composition.

Since T_m(x) is divisible by x if m is odd, it follows that T_mn(x) is divisible by T_n(x) if m is odd. Furthermore, U_mn−1(x) is divisible by U_n−1(x), and in the case that m is even, divisible by T_n(x)U_n−1(x).

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

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

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

${\displaystyle \int _{-1}^{1}T_{n}(x)\,T_{m}(x)\,{\frac {\mathrm {d} x}{\sqrt {1-x^{2}}}}={\begin{cases}0&~{\text{ if }}~n\neq m,\\\\\pi &~{\text{ if }}~n=m=0,\\\\{\frac {\pi }{2}}&~{\text{ if }}~n=m\neq 0.\end{cases}}}$

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

${\displaystyle {\sqrt {1-x^{2}}}}$

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

${\displaystyle \int _{-1}^{1}U_{n}(x)\,U_{m}(x)\,{\sqrt {1-x^{2}}}\,\mathrm {d} x={\begin{cases}0&~{\text{ if }}~n\neq m,\\{\frac {\pi }{2}}&~{\text{ if }}~n=m.\end{cases}}}$

(The measure √1 − x² dx 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

${\displaystyle (1-x^{2})y''-xy'+n^{2}y=0,}$

${\displaystyle (1-x^{2})y''-3xy'+n(n+2)y=0,}$

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 T_n also satisfy a discrete orthogonality condition:

${\displaystyle \sum _{k=0}^{N-1}{T_{i}(x_{k})\,T_{j}(x_{k})}={\begin{cases}0&~{\text{ if }}~i\neq j,\\N&~{\text{ if }}~i=j=0,\\{\frac {N}{2}}&~{\text{ if }}~i=j\neq 0,\end{cases}}}$

where N is any integer greater than max(i, j),^[14] and the x_k are the N Chebyshev nodes (see above) of T_N (x):

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

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

${\displaystyle \sum _{k=0}^{N-1}{U_{i}(x_{k})\,U_{j}(x_{k})\left(1-x_{k}^{2}\right)}={\begin{cases}0&{\text{ if }}~i\neq j,\\{\frac {N}{2}}&{\text{ if }}~i=j,\end{cases}}}$

and without the weight function:

${\displaystyle \sum _{k=0}^{N-1}{U_{i}(x_{k})\,U_{j}(x_{k})}={\begin{cases}0&~{\text{ if }}~i\not \equiv j{\pmod {2}},\\N\cdot (1+\min\{i,j\})&~{\text{ if }}~i\equiv j{\pmod {2}}.\end{cases}}}$

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

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

one can get the sum:

${\displaystyle \sum _{k=0}^{N-1}{U_{i}(y_{k})\,U_{j}(y_{k})(1-y_{k}^{2})}={\begin{cases}0&~{\text{ if }}i\neq j,\\{\frac {N+1}{2}}&~{\text{ if }}i=j,\end{cases}}}$

and again without the weight function:

${\displaystyle \sum _{k=0}^{N-1}{U_{i}(y_{k})\,U_{j}(y_{k})}={\begin{cases}0&~{\text{ if }}~i\not \equiv j{\pmod {2}},\\{\bigl (}\min\{i,j\}+1{\bigr )}{\bigl (}N-\max\{i,j\}{\bigr )}&~{\text{ if }}~i\equiv j{\pmod {2}}.\end{cases}}}$

Minimal ∞-norm

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

${\displaystyle f(x)={\frac {1}{\,2^{n-1}\,}}\,T_{n}(x)}$

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

This maximal absolute value is

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

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

${\displaystyle x=\cos {\frac {k\pi }{n}}\quad {\text{for }}0\leq k\leq n.}$

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 approximated 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).

Chebyshev polynomials as special cases of more general polynomial families

The Chebyshev polynomials are a special case of the ultraspherical or Gegenbauer polynomials ${\displaystyle C_{n}^{(\lambda )}(x)}$, which themselves are a special case of the Jacobi polynomials ${\displaystyle P_{n}^{(\alpha ,\beta )}(x)}$:

${\displaystyle {\begin{aligned}T_{n}(x)&={\frac {n}{2}}\lim _{q\to 0}{\frac {1}{q}}\,C_{n}^{(q)}(x)\qquad ~{\text{ if }}~n\geq 1,\\&={\frac {1}{\binom {n-{\frac {1}{2}}}{n}}}P_{n}^{\left(-{\frac {1}{2}},-{\frac {1}{2}}\right)}(x)={\frac {2^{2n}}{\binom {2n}{n}}}P_{n}^{\left(-{\frac {1}{2}},-{\frac {1}{2}}\right)}(x)~,\\U_{n}(x)&=C_{n}^{(1)}(x)\\&={\frac {n+1}{\binom {n+{\frac {1}{2}}}{n}}}P_{n}^{\left({\frac {1}{2}},{\frac {1}{2}}\right)}(x)={\frac {2^{2n+1}}{\binom {2n+2}{n+1}}}P_{n}^{\left({\frac {1}{2}},{\frac {1}{2}}\right)}(x)~.\end{aligned}}}$

Chebyshev polynomials are also a special case of Dickson polynomials:

${\displaystyle D_{n}(2x\alpha ,\alpha ^{2})=2\alpha ^{n}T_{n}(x)\,}$

${\displaystyle E_{n}(2x\alpha ,\alpha ^{2})=\alpha ^{n}U_{n}(x).\,}$

In particular, when ${\displaystyle \alpha ={\tfrac {1}{2}}}$, they are related by ${\displaystyle D_{n}(x,{\tfrac {1}{4}})=2^{1-n}T_{n}(x)}$ and ${\displaystyle E_{n}(x,{\tfrac {1}{4}})=2^{-n}U_{n}(x)}$.

Other properties

The curves given by y = T_n(x), or equivalently, by the parametric equations y = T_n(cos θ) = cos nθ, x = cos θ, are a special case of Lissajous curves with frequency ratio equal to n.

Similar to the formula

${\displaystyle T_{n}(\cos \theta )=\cos(n\theta ),}$

we have the analogous formula

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

For x ≠ 0,

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

and

${\displaystyle x^{n}=T_{n}\!\left({\frac {x+x^{-1}}{2}}\right)+{\frac {x-x^{-1}}{2}}\ U_{n-1}\!\left({\frac {x+x^{-1}}{2}}\right),}$

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

Examples

First kind

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

${\displaystyle {\begin{aligned}T_{0}(x)&=1\\T_{1}(x)&=x\\T_{2}(x)&=2x^{2}-1\\T_{3}(x)&=4x^{3}-3x\\T_{4}(x)&=8x^{4}-8x^{2}+1\\T_{5}(x)&=16x^{5}-20x^{3}+5x\\T_{6}(x)&=32x^{6}-48x^{4}+18x^{2}-1\\T_{7}(x)&=64x^{7}-112x^{5}+56x^{3}-7x\\T_{8}(x)&=128x^{8}-256x^{6}+160x^{4}-32x^{2}+1\\T_{9}(x)&=256x^{9}-576x^{7}+432x^{5}-120x^{3}+9x\\T_{10}(x)&=512x^{10}-1280x^{8}+1120x^{6}-400x^{4}+50x^{2}-1\\T_{11}(x)&=1024x^{11}-2816x^{9}+2816x^{7}-1232x^{5}+220x^{3}-11x\end{aligned}}}$

Second kind

The first few Chebyshev polynomials of the second kind are OEIS: A053117

${\displaystyle {\begin{aligned}U_{0}(x)&=1\\U_{1}(x)&=2x\\U_{2}(x)&=4x^{2}-1\\U_{3}(x)&=8x^{3}-4x\\U_{4}(x)&=16x^{4}-12x^{2}+1\\U_{5}(x)&=32x^{5}-32x^{3}+6x\\U_{6}(x)&=64x^{6}-80x^{4}+24x^{2}-1\\U_{7}(x)&=128x^{7}-192x^{5}+80x^{3}-8x\\U_{8}(x)&=256x^{8}-448x^{6}+240x^{4}-40x^{2}+1\\U_{9}(x)&=512x^{9}-1024x^{7}+672x^{5}-160x^{3}+10x\end{aligned}}}$

As a basis set

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:^[15]

${\displaystyle f(x)=\sum _{n=0}^{\infty }a_{n}T_{n}(x).}$

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.^[15] 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,^[15] 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

${\displaystyle \log(1+x)=\sum _{n=0}^{\infty }a_{n}T_{n}(x)~.}$

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,

${\displaystyle \int _{-1}^{+1}\,{\frac {T_{m}(x)\,\log(1+x)}{\sqrt {1-x^{2}}}}\,\mathrm {d} x=\sum _{n=0}^{\infty }a_{n}\int _{-1}^{+1}{\frac {T_{m}(x)\,T_{n}(x)}{\sqrt {1-x^{2}}}}\,\mathrm {d} x,}$

which gives

${\displaystyle a_{n}={\begin{cases}-\log 2&{\text{ for }}~n=0,\\{\frac {-2(-1)^{n}}{n}}&{\text{ for }}~n>0.\end{cases}}}$

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

${\displaystyle a_{n}\approx {\frac {\,2-\delta _{0n}\,}{N}}\,\sum _{k=0}^{N-1}T_{n}(x_{k})\,\log(1+x_{k}),}$

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

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

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

${\displaystyle a_{n}\approx {\frac {2-\delta _{0n}}{N}}\sum _{k=0}^{N-1}\cos \left({\frac {n\pi \left(\,k+{\tfrac {1}{2}}\right)}{N}}\right)\log(1+x_{k}).}$

Example 2

To provide another example:

${\displaystyle {\begin{aligned}(1-x^{2})^{\alpha }&=-{\frac {1}{\sqrt {\pi }}}\,{\frac {\Gamma \left({\tfrac {1}{2}}+\alpha \right)}{\Gamma (\alpha +1)}}+2^{1-2\alpha }\,\sum _{n=0}(-1)^{n}\,{2\alpha \choose \alpha -n}\,T_{2n}(x)\\&=2^{-2\alpha }\,\sum _{n=0}(-1)^{n}\,{2\alpha +1 \choose \alpha -n}\,U_{2n}(x).\end{aligned}}}$

Partial sums

The partial sums of

${\displaystyle f(x)=\sum _{n=0}^{\infty }a_{n}T_{n}(x)}$

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)st partial sum are usually obtained on the Chebyshev–Gauss–Lobatto^[16] 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:

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

Polynomial in Chebyshev form

An arbitrary polynomial of degree N can be written in terms of the Chebyshev polynomials of the first kind.^[17] Such a polynomial p(x) is of the form

${\displaystyle p(x)=\sum _{n=0}^{N}a_{n}T_{n}(x).}$

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

Families of polynomials related to Chebyshev polynomials

Polynomials denoted ${\displaystyle C_{n}(x)}$ and ${\displaystyle S_{n}(x)}$ closely related to Chebyshev polynomials are sometimes used. They are defined by^[18]

${\displaystyle C_{n}(x)=2T_{n}\left({\frac {x}{2}}\right),\qquad S_{n}(x)=U_{n}\left({\frac {x}{2}}\right)}$

and satisfy

${\displaystyle C_{n}(x)=S_{n}(x)-S_{n-2}(x).}$

A. F. Horadam called the polynomials ${\displaystyle C_{n}(x)}$ Vieta–Lucas polynomials and denoted them ${\displaystyle v_{n}(x)}$. He called the polynomials ${\displaystyle S_{n}(x)}$ Vieta–Fibonacci polynomials and denoted them ${\displaystyle V_{n}(x)}$.^[19] Lists of both sets of polynomials are given in Viète's Opera Mathematica, Chapter IX, Theorems VI and VII.^[20] The Vieta–Lucas and Vieta–Fibonacci polynomials of real argument are, up to a power of ${\displaystyle i}$ and a shift of index in the case of the latter, equal to Lucas and Fibonacci polynomials L_n and F_n of imaginary argument.

Shifted Chebyshev polynomials of the first and second kinds are related to the Chebyshev polynomials by^[18]

${\displaystyle T_{n}^{*}(x)=T_{n}(2x-1),\qquad U_{n}^{*}(x)=U_{n}(2x-1).}$

When the argument of the Chebyshev polynomial satisfies 2x − 1 ∈ [−1, 1] the argument of the shifted Chebyshev polynomial satisfies x ∈ [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 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."^[21] The Chebyshev polynomials of the third kind are defined as

${\displaystyle V_{n}(x)={\frac {\cos \left(\left(n+{\frac {1}{2}}\right)\theta \right)}{\cos \left({\frac {\theta }{2}}\right)}}={\sqrt {\frac {2}{1+x}}}T_{2n+1}\left({\sqrt {\frac {x+1}{2}}}\right)}$

and the Chebyshev polynomials of the fourth kind are defined as

${\displaystyle W_{n}(x)={\frac {\sin \left(\left(n+{\frac {1}{2}}\right)\theta \right)}{\sin \left({\frac {\theta }{2}}\right)}}=U_{2n}\left({\sqrt {\frac {x+1}{2}}}\right),}$

where ${\displaystyle \theta =\arccos x}$.^[21][22] In the airfoil literature ${\displaystyle V_{n}(x)}$ and ${\displaystyle W_{n}(x)}$ are denoted ${\displaystyle t_{n}(x)}$ and ${\displaystyle u_{n}(x)}$. The polynomial families ${\displaystyle T_{n}(x)}$, ${\displaystyle U_{n}(x)}$, ${\displaystyle V_{n}(x)}$, and ${\displaystyle W_{n}(x)}$ are orthogonal with respect to the weights

${\displaystyle \left(1-x^{2}\right)^{-1/2},\quad \left(1-x^{2}\right)^{1/2},\quad (1-x)^{-1/2}(1+x)^{1/2},\quad (1+x)^{-1/2}(1-x)^{1/2}}$

and are proportional to Jacobi polynomials ${\displaystyle P_{n}^{(\alpha ,\beta )}(x)}$ with

${\displaystyle (\alpha ,\beta )=\left(-{\frac {1}{2}},-{\frac {1}{2}}\right),\quad (\alpha ,\beta )=\left({\frac {1}{2}},{\frac {1}{2}}\right),\quad (\alpha ,\beta )=\left(-{\frac {1}{2}},{\frac {1}{2}}\right),\quad (\alpha ,\beta )=\left({\frac {1}{2}},-{\frac {1}{2}}\right).}$^[22]

All four families satisfy the recurrence ${\displaystyle p_{n}(x)=2xp_{n-1}(x)-p_{n-2}(x)}$ with ${\displaystyle p_{0}(x)=1}$, where ${\displaystyle p_{n}=T_{n}}$, ${\displaystyle U_{n}}$, ${\displaystyle V_{n}}$, or ${\displaystyle W_{n}}$, but they differ according to whether ${\displaystyle p_{1}(x)}$ equals ${\displaystyle x}$, ${\displaystyle 2x}$, ${\displaystyle 2x-1}$, or ${\displaystyle 2x+1}$.^[21]

See also

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

References

Sources

• Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 22". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 773. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.
• 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.
• 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". 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. Vol. 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).
• Salzer, Herbert E. (1976). "Converting interpolation series into Chebyshev series by recurrence formulas". Math. Comp. 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". Math. Comput. 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 to Chebyshev 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 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.