Chebyshev polynomials

The Chebyshev polynomials are two sequences of polynomials related to the sine and cosine functions, notated as




T

n


(
x
)


{\displaystyle T_{n}(x)}
and




U

n


(
x
)


{\displaystyle U_{n}(x)}
. They can be defined several ways that have the same end result; in this article the polynomials are defined by starting with trigonometric functions:

The Chebyshev polynomials of the first kind




T

n




{\displaystyle T_{n}}
are given by





T

n



(

cos
⁡

θ


)

=
cos
⁡

(
n
θ
)

.


{\displaystyle T_{n}\left(\cos {\theta }\right)=\cos {(n\theta )}.}
Similarly, define the Chebyshev polynomials of the second kind




U

n




{\displaystyle U_{n}}
as





U

n



(

cos
⁡

θ


)

sin
⁡

θ

=
sin
⁡



(




(


n
+
1
)
θ


)



.


{\displaystyle U_{n}\left(\cos {\theta }\right)\sin {\theta }=\sin {{\big (}{\big (}n+1)\theta {\big )}}.}
These definitions do not appear to be polynomials, but by using various trigonometric identities they can be converted to an explicitly polynomial form. For example, for n = 2 the T2 formula can be converted into a polynomial with argument x = cos(θ), using the double angle formula:




cos
⁡
(
2
θ
)
=
2

cos

2


⁡
(
θ
)
−
1.


{\displaystyle \cos(2\theta )=2\cos ^{2}(\theta )-1.}
Replacing the terms in the formula with the definitions above, we get





T

2


(
x
)
=
2

x

2


−
1.


{\displaystyle T_{2}(x)=2x^{2}-1.}
The other Tn(x) are defined similarly, where for the polynomials of the second kind (Un) we must use de Moivre's formula to get sin(n θ) as sin(θ) times a polynomial in cos(θ) . For instance,




sin
⁡
(
3
θ
)
=
(
4

cos

2


⁡
(
θ
)
−
1
)

sin
⁡
(
θ
)


{\displaystyle \sin(3\theta )=(4\cos ^{2}(\theta )-1)\,\sin(\theta )}
gives





U

2


(
x
)
=
4

x

2


−
1.


{\displaystyle U_{2}(x)=4x^{2}-1.}
Once converted to polynomial form, Tn(x) and Un(x) are called Chebyshev polynomials of the first and second kind, respectively.
Conversely, an arbitrary integer power of trigonometric functions may be expressed as a linear combination of trigonometric functions using Chebyshev polynomials





cos

n



θ
=

2

1
−
n









∑

′



j
=
0


n




n
−
j


e
v
e
n








(


n



n
−
j

2




)





T

j


(
cos
⁡
θ
)
,


{\displaystyle \cos ^{n}\!\theta =2^{1-n}\!\mathop {\mathop {{\sum }'} _{j=0}^{n}} _{n-j\,\mathrm {even} }\!\!{\binom {n}{\tfrac {n-j}{2}}}\,T_{j}(\cos \theta ),}
where the prime at the sum symbol indicates that the contribution of j = 0 needs to be halved if it appears, and




T

j


(
cos
⁡
θ
)
=
cos
⁡
j
θ


{\displaystyle T_{j}(\cos \theta )=\cos j\theta }
.
An important and convenient property of the Tn(x) is that they are orthogonal with respect to the inner product






⟨



f
(
x
)
,

g
(
x
)



⟩



=


∫

−
1


1



f
(
x
)

g
(
x
)





d

x





1
−

x

2










,


{\displaystyle {\bigl \langle }\,f(x),\,g(x)\,{\bigr \rangle }~=~\int _{-1}^{1}\,f(x)\,g(x)\,{\frac {\mathrm {d} x}{\,{\sqrt {1-x^{2}\,}}\,}}~,}
and Un(x) are orthogonal with respect to another, analogous inner product product, given below. This follows from the fact that the Chebyshev polynomials solve the Chebyshev differential equations




(
1
−

x

2


)


y
″

−
x


y
′

+

n

2



y
=
0

,


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





(
1
−

x

2


)


y
″

−
3

x


y
′

+
n

(
n
+
2
)

y
=
0

,


{\displaystyle (1-x^{2})\,y''-3\,x\,y'+n\,(n+2)\,y=0~,}
which are Sturm–Liouville differential equations. It is a general feature of such differential equations that there is a distinguished orthonormal set of solutions. (Another way to define the Chebyshev polynomials is as the solutions to those equations.)
The Chebyshev polynomials Tn are polynomials with the largest possible leading coefficient, whose absolute value on the interval [−1, 1] is bounded by 1. They are also the "extremal" polynomials for many other properties.Chebyshev polynomials are important in approximation theory because the roots of Tn(x), which are also called Chebyshev nodes, are used as matching-points for optimizing polynomial interpolation. The resulting interpolation polynomial minimizes the problem of Runge's phenomenon, and provides an approximation that is close to the best polynomial approximation to a continuous function under the maximum norm, also called the "minimax" criterion. This approximation leads directly to the method of Clenshaw–Curtis quadrature.
These polynomials were named after Pafnuty Chebyshev. The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebycheff, Tchebyshev (French) or Tschebyschow (German).

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