In mathematics the Chebyshev polynomials , named after Pafnuty Chebyshev (Пафнутий Чебышёв ), are special polynomials . They compose a polynomial sequence and can therefore be used to express other polynomials.
One usually distinguishes between Chebyshev polynomials of the first kind which are denoted
T
n
(
x
)
{\displaystyle T_{n}(x)}
Chebyshev polynomials of the second kind which are denoted
U
n
(
x
)
{\displaystyle U_{n}(x)}
Chebyshev polynomials of the first kind are very important in numerical approximation because they are the best approximation of a continuous function under the maximum norm .
Definition
The Chebyshev polynomials of the first kind are defined by the recurrence relation
T
0
(
x
)
=
1
{\displaystyle T_{0}(x)=1\,}
T
1
(
x
)
=
x
{\displaystyle T_{1}(x)=x\,}
T
n
+
1
(
x
)
=
2
x
T
n
(
x
)
−
T
n
−
1
(
x
)
{\displaystyle T_{n+1}(x)=2xT_{n}(x)-T_{n-1}(x)\,}
This image shows the first few Chebyshev polynomials of the first kind in the domain -1¼<x<1¼, -1¼<y<1¼; the flat T0 , and T1 , T2 , T3 , T4 and T5 . One example of a generating function for this recurrence relation is
∑
n
=
0
∞
T
n
(
x
)
t
n
=
1
−
t
x
1
−
2
t
x
+
t
2
.
{\displaystyle \sum _{n=0}^{\infty }T_{n}(x)t^{n}={\frac {1-tx}{1-2tx+t^{2}}}.}
The Chebyshev polynomials of the second kind are defined by the recurrence relation
U
0
(
x
)
=
1
{\displaystyle U_{0}(x)=1\,}
U
1
(
x
)
=
2
x
{\displaystyle U_{1}(x)=2x\,}
U
n
+
1
(
x
)
=
2
x
U
n
(
x
)
−
U
n
−
1
(
x
)
{\displaystyle U_{n+1}(x)=2xU_{n}(x)-U_{n-1}(x)\,}
One example of a generating function for this recurrence relation is
∑
n
=
0
∞
U
n
(
x
)
t
n
=
1
1
−
2
t
x
+
t
2
.
{\displaystyle \sum _{n=0}^{\infty }U_{n}(x)t^{n}={\frac {1}{1-2tx+t^{2}}}.}
Trigonometric definition
The Chebyshev polynomials of the first kind can be defined by the trigonometric identity
T
n
(
cos
(
θ
)
)
=
cos
(
n
θ
)
{\displaystyle T_{n}(\cos(\theta ))=\cos(n\theta )\,}
for n = 0, 1, 2, 3, .... . That cos(nx ) is an n th-degree polynomial in cos(x ) can be seen by observing that cos(nx ) is the real part of one side of De Moivre's formula , and the real part of the other side is a polynomial in cos(x ) and sin(x ), in which all powers of sin(x ) are even and thus replaceable via the identity cos²(x ) + sin²(x ) = 1.
Written explicitly
T
n
(
x
)
=
{
cos
(
n
arccos
(
x
)
)
,
x
∈
[
−
1
,
1
]
cosh
(
n
a
r
c
o
s
h
(
x
)
)
,
x
≥
1
(
−
1
)
n
cosh
(
n
a
r
c
o
s
h
(
−
x
)
)
,
x
≤
−
1
{\displaystyle T_{n}(x)=\left\{{\begin{matrix}&\cos(n\arccos(x))&{\mbox{ , }}\ x\in [-1,1]\\&\cosh(n\,\mathrm {arcosh} (x))&{\mbox{ , }}\ x\geq 1\\&(-1)^{n}\cosh(n\,\mathrm {arcosh} (-x))&{\mbox{ , }}\ x\leq -1\\\end{matrix}}\right.}
Notes
The polynomial T n n simple roots in [−1, 1] called Chebyshev roots .
The Chebyshev polynomials of first kind and second kind are closely related as
d
d
x
T
n
(
x
)
=
n
U
n
−
1
(
x
)
,
n
=
1
,
…
{\displaystyle {\frac {d}{dx}}\,T_{n}(x)=nU_{n-1}(x){\mbox{ , }}n=1,\ldots }
These polynomials are orthogonal with respect to the weight
d
x
1
−
x
2
,
{\displaystyle {\frac {dx}{\sqrt {1-x^{2}}}},}
on the interval [−1,1], i.e., we have
∫
−
1
1
T
n
(
x
)
T
m
(
x
)
d
x
1
−
x
2
=
0
if
n
≠
m
.
{\displaystyle \int _{-1}^{1}T_{n}(x)T_{m}(x)\,{\frac {dx}{\sqrt {1-x^{2}}}}=0\quad {\mbox{if}}\ n\neq m.}
This is because (letting x = cos θ)
∫
0
π
cos
(
n
θ
)
cos
(
m
θ
)
d
θ
=
0
if
n
≠
m
.
{\displaystyle \int _{0}^{\pi }\cos(n\theta )\cos(m\theta )\,d\theta =0\quad {\mbox{if}}\ n\neq m.}
Examples
The first few Chebyshev polynomials of the first kind are
T
0
(
x
)
=
1
{\displaystyle T_{0}(x)=1\,}
T
1
(
x
)
=
x
{\displaystyle T_{1}(x)=x\,}
T
2
(
x
)
=
2
x
2
−
1
{\displaystyle T_{2}(x)=2x^{2}-1\,}
T
3
(
x
)
=
4
x
3
−
3
x
{\displaystyle T_{3}(x)=4x^{3}-3x\,}
T
4
(
x
)
=
8
x
4
−
8
x
2
+
1
{\displaystyle T_{4}(x)=8x^{4}-8x^{2}+1\,}
T
5
(
x
)
=
16
x
5
−
20
x
3
+
5
x
{\displaystyle T_{5}(x)=16x^{5}-20x^{3}+5x\,}
T
6
(
x
)
=
32
x
6
−
48
x
4
+
18
x
2
−
1
{\displaystyle T_{6}(x)=32x^{6}-48x^{4}+18x^{2}-1\,}
T
7
(
x
)
=
64
x
7
−
112
x
5
+
56
x
3
−
7
x
{\displaystyle T_{7}(x)=64x^{7}-112x^{5}+56x^{3}-7x\,}
T
8
(
x
)
=
128
x
8
−
256
x
6
+
160
x
4
−
32
x
2
+
1
{\displaystyle T_{8}(x)=128x^{8}-256x^{6}+160x^{4}-32x^{2}+1\,}
T
9
(
x
)
=
256
x
9
−
576
x
7
+
432
x
5
−
120
x
3
+
9
x
{\displaystyle T_{9}(x)=256x^{9}-576x^{7}+432x^{5}-120x^{3}+9x\,}
The first few Chebyshev polynomials of the second kind are
U
0
(
x
)
=
1
{\displaystyle U_{0}(x)=1\,}
U
1
(
x
)
=
2
x
{\displaystyle U_{1}(x)=2x\,}
U
2
(
x
)
=
4
x
2
−
1
{\displaystyle U_{2}(x)=4x^{2}-1\,}
U
3
(
x
)
=
8
x
3
−
2
x
{\displaystyle U_{3}(x)=8x^{3}-2x\,}
U
4
(
x
)
=
16
x
4
−
12
x
2
+
1
{\displaystyle U_{4}(x)=16x^{4}-12x^{2}+1\,}
U
5
(
x
)
=
32
x
5
−
32
x
3
+
6
x
{\displaystyle U_{5}(x)=32x^{5}-32x^{3}+6x\,}
U
6
(
x
)
=
64
x
6
−
80
x
4
+
24
x
2
−
1
{\displaystyle U_{6}(x)=64x^{6}-80x^{4}+24x^{2}-1\,}
A polynomial of degree N in Chebyshev form is a polynomial p (x ) of the form
p
(
x
)
=
∑
n
=
0
N
a
n
T
n
(
x
)
{\displaystyle p(x)=\sum _{n=0}^{N}a_{n}T_{n}(x)}
where T n n th Chebyshev polynomial.
Clenshaw algorithm
The Clenshaw algorithm can be used to evaluate a polynomial in the Chebyshev form. Given
p
(
x
)
=
∑
n
=
0
N
a
n
T
n
(
x
)
{\displaystyle p(x)=\sum _{n=0}^{N}a_{n}T_{n}(x)}
we define
b
N
{\displaystyle b_{N}\,\!}
:=
a
N
{\displaystyle :=a_{N}\,}
b
N
−
1
{\displaystyle b_{N-1}\,\!}
:=
2
x
b
N
+
a
N
−
1
{\displaystyle :=2xb_{N}+a_{N-1}\,}
b
N
−
n
{\displaystyle b_{N-n}\,\!}
:=
2
x
b
N
−
n
+
1
+
a
N
−
n
+
b
N
−
n
+
2
,
n
=
2
,
…
,
N
−
1
{\displaystyle :=2xb_{N-n+1}+a_{N-n}+b_{N-n+2}\,,\;n=2,\ldots ,N-1\,}
b
0
{\displaystyle b_{0}\,\!}
:=
x
b
1
+
a
0
−
b
2
{\displaystyle :=xb_{1}+a_{0}-b_{2}\,}
then
p
(
x
)
=
∑
n
=
0
N
a
n
T
n
(
x
)
=
b
0
{\displaystyle p(x)=\sum _{n=0}^{N}a_{n}T_{n}(x)=b_{0}}
Chebyshev differential equation
The Chebyshev polynomials are the general solution to the Chebyshev differential equation
(
1
−
x
2
)
y
″
−
x
y
′
+
n
2
y
=
0
{\displaystyle (1-x^{2})\,y''-x\,y'+n^{2}\,y=0}
which is a special case of the Sturm-Liouville differential equation
See also
![{\displaystyle T_{n}(x)=\left\{{\begin{matrix}&\cos(n\arccos(x))&{\mbox{ , }}\ x\in [-1,1]\\&\cosh(n\,\mathrm {arcosh} (x))&{\mbox{ , }}\ x\geq 1\\&(-1)^{n}\cosh(n\,\mathrm {arcosh} (-x))&{\mbox{ , }}\ x\leq -1\\\end{matrix}}\right.}](/media/api/rest_v1/media/math/render/svg/ee73448f475d9752d16fce22be812db8cbfc3e6c)
