# What is Chebyshev polynomials: Definition and 12 Discussions

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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1. ### A Coefficients of Chebyshev polynomials

Not long ago, I derived the formula for Chebyshev polynomials $$T_{n}\left( x\right)= \sum_{k=0}^{\lfloor \frac{n}{2} \rfloor}{n \choose 2k}x^{n-2k}\left( x^2-1\right)^{k}$$ How to extract the coefficients of this polynomial of degree n ? I tried using Newton's binomial but got a double sum...
2. ### I Finding a limit involving Chebyshev polynomials

How could I show that this limit: ##\lim_{N\to\infty}\frac{\sum_{p=1}^N T_{4N} \left(u_0(N)\cdot \cos\frac{p\pi}{2N+1}\right)}{N}## is equal to 0? In the expression above ##T_{4N}## is the Chebyshev polynomials of order ##4N##, ##u_0(N)\geq 1## is a number such that ##T_{4N}(u_0)=b##, with...

Hello everyone. I am studying this article since I am interested in optimization. The article makes use of Clenshaw–Curtis quadrature scheme to discretize the integral part of the cost function to a finite sum using Chebyshev polynomials. The article differentiates between the case of odd...
4. ### I Question about the roots of Chebyshev polynomials

Hello everyone. I am trying to construct an optimization problem using Chebyshev pseudospectral method as described in this article. For that, I need to calculate the zeros of the Chebyshev polynomial of any order. In the article is sugested to do it as tk=cos(πk/N) k=0, ..., N...
5. ### How can I evaluate a Chebishev polynomial in python?

Hello everyone. I need to construct in python a function which returns the evaluation of a Chebishev polynomial of order k evaluated in x. I have tested the function chebval form these documents, but it doesn't provide what I look for, since I have tested the third one, 4t^3-3t and import numpy...
6. ### MHB Identities of Chebyshev polynomials

Hey! :o We are given the polynomial functions $$T_0(x)=1, T_1(x)=x, x \in \mathbb{R} \\ T_{n+1}(x)=2xT_n(x)-T_{n-1}(x), n \in \mathbb{N}, x \in \mathbb{R}$$ (Chebyshev polynomials) Using induction I have to show that: the degree of $T_n$ is $n$ $\forall n \in \mathbb{N}$ : $T_n(1)=1$...
7. ### What is the Solution to the Chebyshev Polynomial Problem?

This is something Chebyshev polynomial problems. I need to show that: ##\sum_{r=0}^{n}T_{2r}(x)=\frac{1}{2}\big ( 1+\frac{U_{2n+1}(x)}{\sqrt{1-x^2}}\big )## by using two type of solution : ##T_n(x)=\cos(n \cos^{-1}x)## and ##U_n(x)=\sin(n \cos^{-1}x)## with ##x=\cos\theta##, I have form the...
8. ### Need help understanding Remez Algorithm and Chebyshev Polynomials

So I've been reading about minimax polynomial approximations and have found them to be pretty impressive. However, i am confused on exactly how to determine the constants. The first step is supposed be solving for the Chebyshev polynomials as an initial guess. I'm reading wikipedia but I'm a...
9. ### Question on Chebyshev polynomials?

Question A Chebyshev polynomial is Tn(x) = cos(arccos^(-1)(x)) My questions are: 1. what are the domain(s) and range(s) of this function? 2. Give equivalent polynomial definitions for Tn(x) when n = 0; 1; 2; 3. That is: show that the definition for Tn above really is a polynomial...
10. ### Anyone have any suggestions on books on chebyshev polynomials?

i find that chebyshev polynomials are quite useful in numerical computations is there any good references?
11. ### Multi-dimensional Chebyshev polynomials?

I was hoping someone could point me in the direction of a suitable extension of Chebyshev polynomials to mutple dimensions? I find Chebyshev polynomials useful in situations when I need to fit some function in a general way, imposing as little pre-concieved ideas about the form as possible...
12. ### A property of Chebyshev polynomials

Hi, I fail finding a proof (even in MathWorld, in my Mathematic dictionary or on the Web) for the following property of Chebyshev polynomials: (T_i o T_j)(x) = (T_j o T_i)(x) = T_ij(x) when x is in ] -inf ; + inf [ Example : T_2(x) = 2x^2-1 T_3(x) = 4x^3-3x T_3(T_2(x)) = T_2(T_3(x)) =...