Chebyshev polynomials

Short description: Polynomial sequence

Not to be confused with discrete Chebyshev polynomials.

Plot of the first five Tn Chebyshev polynomials (first kind) Plot of the first five Un Chebyshev polynomials (second kind)

The Chebyshev polynomials are two sequences of orthogonal polynomials related to the cosine and sine functions, notated as Tn(x) and Un(x). They can be defined in several equivalent ways, one of which starts with trigonometric functions:

The Chebyshev polynomials of the first kind Tn are defined by

Tn(cos⁡θ)=cos⁡(nθ).

Similarly, the Chebyshev polynomials of the second kind Un are defined by

Un(cos⁡θ)sin⁡θ=sin⁡((n+1)θ).

That these expressions define polynomials in cos⁡θ is not obvious at first sight but can be shown using de Moivre's formula (see below).

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.[1]

In 1952, Cornelius Lanczos showed that the Chebyshev polynomials are important in approximation theory for the solution of linear systems;[2] 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.[3] The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebycheff, Tchebyshev (French) or Tschebyschow (German).

## Contents

* 1 Definitions
* 1.1 Recurrence definition
* 1.2 Trigonometric definition
* 1.3 Commuting polynomials definition
* 1.4 Pell equation definition
* 1.5 Generating functions
* 2 Relations between the two kinds of Chebyshev polynomials
* 3 Explicit expressions
* 4 Properties
* 4.1 Symmetry
* 4.2 Roots and extrema
* 4.3 Differentiation and integration
* 4.4 Products of Chebyshev polynomials
* 4.5 Composition and divisibility properties
* 4.6 Orthogonality
* 4.7 Minimal ∞-norm
* 4.7.1 Remark
* 4.8 Chebyshev polynomials as special cases of more general polynomial families
* 4.9 Other properties
* 4.9.1 Chebyshev polynomials as determinants
* 5 Examples
* 5.1 First kind
* 5.2 Second kind
* 6 As a basis set
* 6.1 Example 1
* 6.2 Example 2
* 6.3 Partial sums
* 6.4 Polynomial in Chebyshev form
* 7 Families of polynomials related to Chebyshev polynomials
* 7.1 Irreducible Factorization of Chebyshev Polynomials
* 7.2 Even order modified Chebyshev polynomials
* 8 See also
* 9 References
* 10 Sources
* 11 Further reading
* 12 External links

## Definitions

### Recurrence definition

The Chebyshev polynomials of the first kind can be defined by the recurrence relation

T0(x)=1,T1(x)=x,Tn+1(x)=2xTn(x)−Tn−1(x).

The Chebyshev polynomials of the second kind can be defined by the recurrence relation

U0(x)=1,U1(x)=2x,Un+1(x)=2xUn(x)−Un−1(x), which differs from the above only by the rule for n=1.

### Trigonometric definition

The Chebyshev polynomials of the first and second kind can be defined as the unique polynomials satisfying

Tn(cos⁡θ)=cos⁡(nθ)

and

Un(cos⁡θ)=sin⁡((n+1)θ)sin⁡θ,

for n = 0, 1, 2, 3, ….

An equivalent way to state this is via exponentiation of a complex number: given a complex number z = a \+ bi with absolute value of one,

zn=Tn(a)+ibUn−1(a).

Chebyshev polynomials can also be defined in this form when studying trigonometric polynomials.[4]

That cos⁡(nx) is an nth-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:

cos⁡nθ+isin⁡nθ=(cos⁡θ+isin⁡θ)n.

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 through the identity cos2(x)+sin2(x)=1. By the same reasoning, sin⁡(nx) is the imaginary part of the polynomial, in which all powers of sin⁡(x) are odd and thus, if one factor of sin⁡(x) is factored out, the remaining factors can be replaced to create a n−1st-degree polynomial in cos⁡(x).

For x outside the interval [-1,1], the above definition implies

Tn(x)={cos⁡(narccos⁡x) if |x|≤1,cosh⁡(narcosh⁡x) if x≥1,(−1)ncosh⁡(narcosh⁡(−x)) if x≤−1.

### Commuting polynomials definition

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

If Fn(x) is a family of monic polynomials with coefficients in a field of characteristic 0 such that deg⁡Fn(x)=n and Fm(Fn(x))=Fn(Fm(x)) for all m and n, then, up to a simple change of variables, either Fn(x)=xn for all n or Fn(x)=2⋅Tn(x/2) for all n.

### Pell equation definition

The Chebyshev polynomials can also be defined as the solutions to the Pell equation:

[ Tn(x) ]2 − ( x2−1 )[ Un−1(x) ]2=1

in a ring R[x].[6] Thus, they can be generated by the standard technique for Pell equations of taking powers of a fundamental solution:

Tn(x) + Un−1(x) x2−1 = (x+x2−1 )n.

### Generating functions

The ordinary generating function for Tn is

∑n=0∞Tn(x)tn=1−tx1−2tx+t2.

There are several other generating functions for the Chebyshev polynomials; the exponential generating function is

∑n=0∞Tn(x)tnn!=12(exp(t(x−x2−1))+exp(t(x+x2−1)))=etxcosh⁡(tx2−1).

The generating function relevant for 2-dimensional potential theory and multipole expansion is

∑n=1∞Tn(x)tnn=ln⁡(11−2tx+t2).

The ordinary generating function for Un is

∑n=0∞Un(x)tn=11−2tx+t2,

and the exponential generating function is

∑n=0∞Un(x)tnn!=etx(cosh⁡(tx2−1)+xx2−1sinh⁡(tx2−1)).

## Relations between the two kinds of Chebyshev polynomials

The Chebyshev polynomials of the first and second kinds correspond to a complementary pair of Lucas sequences V~n(P,Q) and U~n(P,Q) with parameters P=2x and Q=1:

U~n(2x,1)=Un−1(x),V~n(2x,1)=2Tn(x).

It follows that they also satisfy a pair of mutual recurrence equations:[7]

Tn+1(x)=xTn(x)−(1−x2)Un−1(x),Un+1(x)=xUn(x)+Tn+1(x).

The second of these may be rearranged using the recurrence definition for the Chebyshev polynomials of the second kind to give:

Tn(x)=12(Un(x)−Un−2(x)).

Using this formula iteratively gives the sum formula:

Un(x)={2∑ odd j>0nTj(x) for odd n.2∑ even j≥0nTj(x)−1 for even n,

while replacing Un(x) and Un−2(x) using the derivative formula for Tn(x) gives the recurrence relationship for the derivative of Tn:

2Tn(x)=1n+1ddxTn+1(x)−1n−1ddxTn−1(x),n=2,3,…

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

Turán's inequalities for the Chebyshev polynomials are:[8]

Tn(x)2−Tn−1(x)Tn+1(x)=1−x2>0 for −1<x<1 and Un(x)2−Un−1(x)Un+1(x)=1>0.

The integral relations are[9][10]

∫−11Tn(y)y−xdy1−y2=πUn−1(x),∫−11Un−1(y)y−x1−y2dy=−πTn(x)

where integrals are considered as principal value.

## Explicit expressions

Using the complex number exponentiation definition of the Chebyshev polynomial, one can derive the following expressions, valid for any real x: Tn(x) = 12( ( x−x2−1 )n + ( x+x2−1 )n ) = 12( ( x−x2−1 )n + ( x−x2−1 )−n ).

The two are equivalent because ( x+x2−1 )±1=( x−x2−1 )∓1.

An explicit form of the Chebyshev polynomial in terms of monomials  xk follows from de Moivre's formula:

Tn( cos⁡(θ) ) = Rℯ⁡( cos⁡nθ + i sin⁡nθ ) = Rℯ⁡( ( cos⁡θ + i sin⁡θ )n ) ,

where  Rℯ⁡ denotes the real part of a complex number. Expanding the formula, one gets

(cos⁡θ + i sin⁡θ)n = ∑j=0n (nj) ij sinjθcosn−jθ.

The real part of the expression is obtained from summands corresponding to even indices. Noting  i2j=(−1)j and  sin2jθ=( 1−cos2θ )j , one gets the explicit formula:

cos⁡nθ = ∑j=0⌊n2⌋ (n2j) ( cos2θ − 1 )jcosn−2jθ ,

which in turn means that

Tn(x) = ∑j=0⌊n2⌋(n2j)( x2−1 )j xn−2j.

This can be written as a  2F1 hypergeometric function:

Tn(x) = ∑k=0⌊n2⌋ (n2k) ( x2−1 )k xn−2k = xn ∑k=0⌊n2⌋ (n2k) ( 1−x−2 )k = n2 ∑k=0⌊n2⌋ (−1)k (n−k−1)!k! (n−2k)! ( 2 x )n−2kforn>0 = n ∑k=0n(−2)k (n+k−1)!(n−k)! (2k)! ( 1−x )kforn>0 = 2F1( −n, n ; 12 ; 12( 1−x ) )

with inverse[11][12]

xn = 12n−1 ∑′j=0j≡n(mod2)n (nn−j2) Tj(x) ,

where the prime on the summation symbol indicates that the contribution of  j=0 needs to be halved if it appears.

A related expression for  Tn as a sum of monomials with binomial coefficients and powers of two is

Tn(x) = ∑m=0⌊n2⌋ (−1)m ( (n−mm)+(n−m−1n−2m) ) ⋅ 2n−2m−1 ⋅ xn−2m.

Similarly,  Un can be expressed in terms of hypergeometric functions:

Un(x) = ( x+x2−1 )n+1−( x−x2−1 )n+12 x2−1 = ∑k=0⌊n2⌋ (n+12k+1) ( x2−1 )kxn−2k = xn∑k=0⌊n2⌋ (n+12k+1) ( 1−x−2 )k = ∑k=0⌊n2⌋ (2k−(n+1)k) ( 2 x )n−2kforn>0 = ∑k=0⌊n/2⌋(−1)k(n−kk) ( 2 x )n−2kforn>0 = ∑k=0n (−2)k (n+k+1)!(n−k)! (2k+1)! ( 1−x )kforn>0 = (n+1)⋅ 2F1( −n, n+2 ; 32 ; 12( 1−x ) ).

## Properties

### Symmetry

Tn(−x)=(−1)nTn(x),Un(−x)=(−1)nUn(x).

That is, Chebyshev polynomials of even order have even symmetry and therefore contain only even powers of x. Chebyshev polynomials of odd order have odd symmetry and therefore contain only odd powers of x.

### Roots and extrema

A Chebyshev polynomial of either kind with degree n has n different simple roots, called Chebyshev roots, in the interval [−1, 1]. The roots of the Chebyshev polynomial of the first kind are sometimes called Chebyshev nodes because they are used as nodes in polynomial interpolation. Using the trigonometric definition and the fact that:

cos⁡((2k+1)π2)=0

one can show that the roots of Tn are:

xk=cos⁡(2k+12nπ),k=0,…,n−1.

Similarly, the roots of Un are:

xk=cos⁡(kn+1π),k=1,…,n.

The extrema of Tn on the interval −1≤x≤1 are located at:

xk=cos⁡(knπ),k=0,…,n.

One unique property of the Chebyshev polynomials of the first kind is that on the interval −1≤x≤1 all of the extrema have values that are either −1 or 1. Thus these polynomials have only two finite critical values, the defining property of Shabat polynomials. Both the first and second kinds of Chebyshev polynomial have extrema at the endpoints, given by:

Tn(1)=1Tn(−1)=(−1)nUn(1)=n+1Un(−1)=(−1)n(n+1).

The extrema of Tn(x) on the interval −1≤x≤1 where n>0 are located at n+1 values of x. They are ±1, or cos⁡(2πkd) where d>2, d|2n, 0<k<d/2 and (k,d)=1, i.e., k and d are relatively prime numbers.

Specifically (Minimal polynomial of 2cos(2pi/n)[13][14]) when n is even:

* Tn(x)=1 if x=±1, or d>2 and 2n/d is even. There are n/2+1 such values of x.
* Tn(x)=−1 if d>2 and 2n/d is odd. There are n/2 such values of x.

When n is odd:

* Tn(x)=1 if x=1, or d>2 and 2n/d is even. There are (n+1)/2 such values of x.
* Tn(x)=−1 if x=−1, or d>2 and 2n/d is odd. There are (n+1)/2 such values of 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:

dTndx=nUn−1dUndx=(n+1)Tn+1−xUnx2−1d2Tndx2=nnTn−xUn−1x2−1=n(n+1)Tn−Unx2−1.

The last two formulas can be numerically troublesome due to the division by zero (⁠0/0⁠ indeterminate form, specifically) at x=1 and x=−1. By L'Hôpital's rule:

d2Tndx2|x=1=n4−n23,d2Tndx2|x=−1=(−1)nn4−n23.

More generally,

dpTndxp|x=±1=(±1)n+p∏k=0p−1n2−k22k+1,

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

Also, we have:

dpdxpTn(x)=2pn∑′0≤k≤n−pk≡n−p(mod2)(n+p−k2−1n−p−k2)(n+p+k2−1)!(n−p+k2)!Tk(x),p≥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 Tn implies that:

∫Undx=Tn+1n+1

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

∫Tndx=12(Tn+1n+1−Tn−1n−1)=nTn+1n2−1−xTnn−1.

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

∫Tndx=nn2−1Tn+1−1n−1T1Tn=nn2−1Tn+1−12(n−1)(Tn+1+Tn−1)=12(n+1)Tn+1−12(n−1)Tn−1.

Furthermore, we have:

∫−11Tn(x)dx={(−1)n+11−n2 if n≠10 if n=1.

### Products of Chebyshev polynomials

The Chebyshev polynomials of the first kind satisfy the relation:

Tm(x)Tn(x)=12(Tm+n(x)+T|m−n|(x)),∀m,n≥0,

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

2cos⁡αcos⁡β=cos⁡(α+β)+cos⁡(α−β).

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:

T2n(x)=2Tn2(x)−T0(x)=2Tn2(x)−1,T2n+1(x)=2Tn+1(x)Tn(x)−T1(x)=2Tn+1(x)Tn(x)−x,T2n−1(x)=2Tn−1(x)Tn(x)−T1(x)=2Tn−1(x)Tn(x)−x.

The polynomials of the second kind satisfy the similar relation:

Tm(x)Un(x)={12(Um+n(x)+Un−m(x)), if n≥m−1,12(Um+n(x)−Um−n−2(x)), if n≤m−2.

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

Um(x)Un(x)=∑k=0nUm−n+2k(x)=∑p=m−n step 2 m+nUp(x).

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

Um+2(x)=U2(x)Um(x)−Um(x)−Um−2(x)=Um(x)(U2(x)−1)−Um−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 Tn and Un imply the composition or nesting properties:[15]

Tmn(x)=Tm(Tn(x)),Umn−1(x)=Um−1(Tn(x))Un−1(x).

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

Since Tm(x) is divisible by x if m is odd, it follows that Tmn(x) is divisible by Tn(x) if m is odd. Furthermore, Umn−1(x) is divisible by Un−1(x), and in the case that m is even, divisible by Tn(x)Un−1(x).

### Orthogonality

Both Tn and Un form a sequence of orthogonal polynomials. The polynomials of the first kind Tn are orthogonal with respect to the weight:

11−x2,

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

∫−11Tn(x)Tm(x)dx1−x2={0 if n≠m,π if n=m=0,π2 if n=m≠0.

This can be proven by letting x=cos⁡(θ) and using the defining identity Tn(cos⁡(θ))=cos⁡(nθ).

Similarly, the polynomials of the second kind Un are orthogonal with respect to the weight:

1−x2 on the interval [−1, 1], i.e. we have:

∫−11Un(x)Um(x)1−x2dx={0 if n≠m,π2 if n=m.

(The measure 1−x2dx 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:

(1−x2)Tn″−xTn′+n2Tn=0,(1−x2)Un″−3xUn′+n(n+2)Un=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 Tn also satisfy a discrete orthogonality condition:

∑k=0N−1Ti(xk)Tj(xk)={0 if i≠j,N if i=j=0,N2 if i=j≠0,

where N is any integer greater than max⁡(i,j),[10] and the xk are the N Chebyshev nodes (see above) of TN(x):

xk=cos⁡(π2k+12N) for k=0,1,…,N−1.

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

∑k=0N−1Ui(xk)Uj(xk)(1−xk2)={0 if i≠j,N2 if i=j,

and without the weight function:

∑k=0N−1Ui(xk)Uj(xk)={0 if i≢j(mod2),N⋅(1+min⁡{i,j}) if i≡j(mod2).

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

yk=cos⁡(πk+1N+1) for k=0,1,…,N−1,

one can get the sum:

∑k=0N−1Ui(yk)Uj(yk)(1−yk2)={0 if i≠j,N+12 if i=j,

and again without the weight function:

∑k=0N−1Ui(yk)Uj(yk)={0 if i≢j(mod2),(min⁡{i,j}+1)(N−max⁡{i,j}) if i≡j(mod2).

### Minimal ∞-norm

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

f(x)=12n−1Tn(x)

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

This maximal absolute value is:

12n−1

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

x=cos⁡kπnfor 0≤k≤n.

Proof

Let's assume that wn(x) is a polynomial of degree n with leading coefficient 1 with maximal absolute value on the interval [−1, 1] less than 1 / 2n − 1.

Define

fn(x)=12n−1Tn(x)−wn(x)

Because at extreme points of Tn we have

|wn(x)|<|12n−1Tn(x)|fn(x)>0 for x=cos⁡2kπn where 0≤2k≤nfn(x)<0 for x=cos⁡(2k+1)πn where 0≤2k+1≤n

From the intermediate value theorem, fn(x) has at least n roots. However, this is impossible, as fn(x) is a polynomial of degree n − 1, so the fundamental theorem of algebra implies it has at most n − 1 roots.

#### 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 < ⋯ < xn \+ 1 ≤ 1 such that | f(xi)| = || 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 Cn(λ)(x), which themselves are a special case of the Jacobi polynomials Pn(α,β)(x):

Tn(x)=n2limq->01qCn(q)(x) if n≥1,=1(n−12n)Pn(−12,−12)(x)=22n(2nn)Pn(−12,−12)(x),Un(x)=Cn(1)(x)=n+1(n+12n)Pn(12,12)(x)=22n+1(2n+2n+1)Pn(12,12)(x).

Chebyshev polynomials are also a special case of Dickson polynomials:

Dn(2xα,α2)=2αnTn(x)

En(2xα,α2)=αnUn(x).

In particular, when α=12, they are related by Dn(x,14)=21−nTn(x) and En(x,14)=2−nUn(x).

### Other properties

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

Similar to the formula:

Tn(cos⁡θ)=cos⁡(nθ),

we have the analogous formula:

T2n+1(sin⁡θ)=(−1)nsin⁡((2n+1)θ).

For x ≠ 0:

Tn(x+x−12)=xn+x−n2

and:

xn=Tn(x+x−12)+x−x−12 Un−1(x+x−12), which follows from the fact that this holds by definition for x = eiθ.

There are relations between Legendre polynomials and Chebyshev polynomials

∑k=0nPk(x)Tn−k(x)=(n+1)Pn(x)

∑k=0nPk(x)Pn−k(x)=Un(x)

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 as determinants of special tridiagonal matrices of size k×k:

Tk(x)=det⁡[x10⋯012x1⋱⋮012x⋱0⋮⋱⋱⋱10⋯012x], and similarly for Uk.

## Examples

### First kind

The first few Chebyshev polynomials of the first kind in the domain −1 < x < 1: The flat T0, T1, T2, T3, T4 and T5.

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

T0(x)=1T1(x)=xT2(x)=2x2−1T3(x)=4x3−3xT4(x)=8x4−8x2+1T5(x)=16x5−20x3+5xT6(x)=32x6−48x4+18x2−1T7(x)=64x7−112x5+56x3−7xT8(x)=128x8−256x6+160x4−32x2+1T9(x)=256x9−576x7+432x5−120x3+9xT10(x)=512x10−1280x8+1120x6−400x4+50x2−1

### Second kind

The first few Chebyshev polynomials of the second kind in the domain −1 < x < 1: The flat U0, U1, U2, U3, U4 and U5. Although not visible in the image, Un(1) = n + 1 and Un(−1) = (n + 1)(−1)n.

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

U0(x)=1U1(x)=2xU2(x)=4x2−1U3(x)=8x3−4xU4(x)=16x4−12x2+1U5(x)=32x5−32x3+6xU6(x)=64x6−80x4+24x2−1U7(x)=128x7−192x5+80x3−8xU8(x)=256x8−448x6+240x4−40x2+1U9(x)=512x9−1024x7+672x5−160x3+10xU10(x)=1024x10−2304x8+1792x6−560x4+60x2−1

## As a basis set

The non-smooth function (top) y = −x3H(−x), where H is the Heaviside 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 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:[16]

f(x)=∑n=0∞anTn(x).

Furthermore, as mentioned previously, the Chebyshev polynomials form an orthogonal basis which (among other things) implies that the coefficients an 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.[16] 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,[16] often in favor of trigonometric series due to generally faster convergence for continuous functions (Gibbs' phenomenon is still a problem).

The Chebfun software package supports function manipulation based on their expansion in the Chebyshev basis.

### Example 1

Consider the Chebyshev expansion of log(1 + x). One can express:

log⁡(1+x)=∑n=0∞anTn(x).

One can find the coefficients an either through the application of an inner product or by the discrete orthogonality condition. For the inner product:

∫−1+1Tm(x)log⁡(1+x)1−x2dx=∑n=0∞an∫−1+1Tm(x)Tn(x)1−x2dx, which gives: an={−log⁡2 for n=0,−2(−1)nn for n>0.

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:

an≈2−δ0nN∑k=0N−1Tn(xk)log⁡(1+xk),

where δij is the Kronecker delta function and the xk are the N Gauss–Chebyshev zeros of TN (x):

xk=cos⁡(π(k+12)N).

For any N, these approximate coefficients provide an exact approximation to the function at xk 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 an very efficiently through the discrete cosine transform:

an≈2−δ0nN∑k=0N−1cos⁡(nπ(k+12)N)log⁡(1+xk).

### Example 2

To provide another example:

(1−x2)α=−1πΓ(12+α)Γ(α+1)+21−2α∑n=0(−1)n(2αα−n)T2n(x)=2−2α∑n=0(−1)n(2α+1α−n)U2n(x).

### Partial sums

The partial sums of:

f(x)=∑n=0∞anTn(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 an 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[17] 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:

xk=−cos⁡(kπN−1);k=0,1,…,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.[10] Such a polynomial p(x) is of the form:

p(x)=∑n=0NanTn(x).

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

## Families of polynomials related to Chebyshev polynomials

Polynomials denoted Cn(x) and Sn(x) closely related to Chebyshev polynomials are sometimes used. They are defined by:[18]

Cn(x)=2Tn(x2),Sn(x)=Un(x2)

and satisfy:

Cn(x)=Sn(x)−Sn−2(x).

A. F. Horadam called the polynomials Cn(x) Vieta–Lucas polynomials and denoted them vn(x). He called the polynomials Sn(x) Vieta–Fibonacci polynomials and denoted them Vn(x).[19] All of these polynomials have 1 as their leading coefficient. 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 i and a shift of index in the case of the latter, equal to Lucas and Fibonacci polynomials Ln and Fn of imaginary argument.

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

Tn*(x)=Tn(2x−1),Un*(x)=Un(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:

Vn(x)=cos⁡((n+12)θ)cos⁡(θ2)=21+xT2n+1(x+12) and the Chebyshev polynomials of the fourth kind are defined as: Wn(x)=sin⁡((n+12)θ)sin⁡(θ2)=U2n(x+12),

where θ=arccos⁡x.[21][22] They coincide with the Dirichlet kernel.

In the airfoil literature Vn(x) and Wn(x) are denoted tn(x) and un(x). The polynomial families Tn(x), Un(x), Vn(x), and Wn(x) are orthogonal with respect to the weights:

(1−x2)−1/2,(1−x2)1/2,(1−x)−1/2(1+x)1/2,(1+x)−1/2(1−x)1/2

and are proportional to Jacobi polynomials Pn(α,β)(x) with:[22]

(α,β)=(−12,−12),(α,β)=(12,12),(α,β)=(−12,12),(α,β)=(12,−12).

All four families satisfy the recurrence pn(x)=2xpn−1(x)−pn−2(x) with p0(x)=1, where pn=Tn, Un, Vn, or Wn, but they differ according to whether p1(x) equals x, 2x, 2x−1, or 2x+1.[21]

### Irreducible Factorization of Chebyshev Polynomials

It is easier to discuss this detail by first examining the factorization of the Vieta-Lucas and Vieta-Fibonacci polynomials.

Given the roots of the Chebyshev polynomials, it is easy to see—by comparing their root sets—that xnCn(x+1x)=x2n+1 and xnSn(x+1x)=∑k=0nx2k.

By expressing the right-hand side expressions in form x2n+1=x4n−1x2n−1, and ∑k=0nx2k=x2n+2−1x2−1, the numerators and denominators of these fractions—and consequently the fractions themselves—can be written as products of expressions likex−gi where each gi is a primitive root of unity. Thus, we obtain: xnCn(x+1x)=∏d≥3,d∣4n,d∤2nΦd(x) and xnSn(x+1x)=∏d≥3,d∣2n+2Φd(x), where Φd(x) is the dth cyclotomic polynomial.

It can be shown that, for every n≥3, corresponding to the cyclotomic polynomial Φn(x) of degree φ(n) there exists a unique polynomial Ψn(x) of degree φ(n)/2 such that xφ(n)/2Ψn(x+1x)=Φn(x), where φ(n) is the well known Euler's totient function.

The polynomials Ψn(x) may be referred to as cyclotomic pre-polynomials, since the cyclotomic polynomials can be obtained from them via a well-defined mapping.

An obvious property of the mapping Pn(x)->xnPn(x+1x) applicable to any polynomial Pn(x) of degree n is that it maps the product of two or more polynomials to the product of the images of the individual polynomials.

From all of the above, it follows that Cn(x)=∏d≥3,d∣4n,d∤2nΨd(x) and Sn(x)=∏d≥3,d∣2n+2Ψd(x).

Now, it follows directly that the Chebyshev polynomials Tn(x) and Un(x) can be factorized as follows: Tn(x)=12∏d≥3,d∣4n,d∤2nΨd(2x) and Un(x)=∏d≥3,d∣2n+2Ψd(2x).

From the irreducibility of the polynomials Φn(x) it follows that the polynomials Ψn(x) are also irreducible.

For more details, see .[23]

### 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 order Chebyshev filter designs using equally terminated passive networks are an example of this.[24] 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 the Chebyshev nodes in the same manner as standard Chebyshev polynomials.

PN=∏i=1N(x−Ci)

where

* PN is an N-th order Chebyshev polynomial
* Ci is the i-th Chebyshev node

In the case of even order modified Chebyshev polynomials, the even order modified Chebyshev nodes are used to construct the even order modified Chebyshev polynomials.

PeN=∏i=1N(x−Cei)

where

* PeN is an N-th order even order modified Chebyshev polynomial
* Cei is the i-th even order modified Chebyshev node

For example, the 4th order Chebyshev polynomial from the example above is X4−X2+.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 X4−.828427X2, which by inspection contains two roots at zero, and may be used in applications requiring roots at zero.

## See also

* Chebyshev rational functions
* Function approximation
* Discrete Chebyshev transform
* Markov brothers' inequality

## References

1. ↑ 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.
2. ↑ 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.
3. ↑ 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" (in fr). Mémoires des Savants étrangers présentés à l'Académie de Saint-Pétersbourg 7: 539–586. https://archive.org/details/mmoiresprsentsla07impe/page/537/. Also published separately as Chebyshev, 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.
4. ↑ 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. https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-47/issue-8/Inequalities-of-A-Markoff-and-S-Bernstein-for-polynomials-and/bams/1183503783.full.
5. ↑ Ritt, J. F. (1922). "Prime and Composite Polynomials". Trans. Amer. Math. Soc. 23: 51–66. doi:10.1090/S0002-9947-1922-1501189-9. https://www.ams.org/journals/tran/1922-023-01/S0002-9947-1922-1501189-9.
6. ↑ Demeyer, Jeroen (2007). Diophantine sets over polynomial rings and Hilbert's tenth problem for function fields (PDF) (Ph.D. thesis). p. 70\. Archived from the original (PDF) on 2 July 2007.
7. ↑ Bateman & Bateman Manuscript Project 1953, Template:Pgs, eqs. 3–4.
8. ↑ 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
9. ↑ Bateman & Bateman Manuscript Project 1953, Template:Pgs, eqs. 47–48.
10. ↑ 10.0 10.1 10.2 Mason & Handscomb 2002.
11. ↑ Cody, W.J. (1970). "A survey of practical rational and polynomial approximation of functions". SIAM Review 12 (3): 400–423. doi:10.1137/1012082.
12. ↑ Mathar, Richard J. (2006). "Chebyshev series expansion of inverse polynomials". Journal of Computational and Applied Mathematics 196 (2): 596–607. doi:10.1016/j.cam.2005.10.013.
13. ↑ Gürtaş, Y. Z. (2017). "Chebyshev Polynomials and the minimal polynomial of cos⁡(2π/n)". American Mathematical Monthly 124 (1): 74–78. doi:10.4169/amer.math.monthly.124.1.74.
14. ↑ Wolfram, D. A. (2022). "Factoring Chebyshev polynomials of the first and second kinds with minimal polynomials of cos⁡(2π/d)". American Mathematical Monthly 129 (2): 172–176. doi:10.1080/00029890.2022.2005391.
15. ↑ 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
16. ↑ 16.0 16.1 16.2 Boyd, John P. (2001). Chebyshev and Fourier Spectral Methods (second ed.). Dover. ISBN 0-486-41183-4. http://www-personal.umich.edu/~jpboyd/aaabook_9500may00.pdf. Retrieved 2009-03-19.
17. ↑ "Chebyshev Interpolation: An Interactive Tour". http://www.scottsarra.org/chebyApprox/chebyshevApprox.html.
18. ↑ 18.0 18.1 Hochstrasser 1972, p. 778.
19. ↑ Horadam, A. F. (2002), "Vieta polynomials", Fibonacci Quarterly 40 (3): 223–232, https://www.fq.math.ca/Scanned/40-3/horadam2.pdf
20. ↑ Viète, François (1646). Francisci Vietae Opera mathematica : in unum volumen congesta ac recognita / opera atque studio Francisci a Schooten. Bibliothèque nationale de France. https://gallica.bnf.fr/ark:/12148/bpt6k107597d.pdf.
21. ↑ 21.0 21.1 21.2 Mason, 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
22. ↑ 22.0 22.1 Desmarais, Robert N.; Bland, Samuel R. (1995), "Tables of properties of airfoil polynomials", NASA Reference Publication 1343 (National Aeronautics and Space Administration), https://ntrs.nasa.gov/citations/19960001864
23. ↑ Kéri, Gerzson (2021): Compressed Chebyshev Polynomials and Multiple-Angle Formulas, Omniscriptum Publishing Company, ISBN 978-620-0-62498-7.
24. ↑ Saal, Rudolf (January 1979) (in English, German). Handbook of Filter Design (1st ed.). Munich, Germany: Allgemeine Elektricitais-Gesellschaft. pp. 25, 26, 56–61, 116, 117. ISBN 3-87087-070-2. https://archive.org/details/handbuchzumfilte0000saal.

## Sources

* Hochstrasser, Urs W. (1972). "Orthogonal Polynomials". Handbook of Mathematical Functions (10th printing, with corrections; first ed.). Washington D.C.: National Bureau of Standards. Ch. 22, Template:Pgs. https://personal.math.ubc.ca/~cbm/aands/page_771.htm. Reprint: 1983. New York: Dover. ISBN 978-0-486-61272-0.
* Bateman, Harry; Bateman Manuscript Project (1953). "Tchebichef polynomials". in Erdélyi, Arthur. Higher Transcendental Functions. 2. Research associates: W. Magnus, F. Oberhettinger (de), F. Tricomi (1st ed.). New York: McGraw-Hill. § 10.11, Template:Pgs. Caltech eprint 43491. https://archive.org/details/highertranscende02bate. 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. https://books.google.com/books?id=8FHf0P3to0UC.

## 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. 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.
* Eremenko, A.; Lempert, L. (1994). "An Extremal Problem For Polynomials". Proceedings of the American Mathematical Society 122 (1): 191–193. doi:10.1090/S0002-9939-1994-1207536-1. https://www.math.purdue.edu/~eremenko/dvi/lempert.pdf.
* 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. 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", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F. et al., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, http://dlmf.nist.gov/18
* Remes, Eugene. "On an Extremal Property of Chebyshev Polynomials". https://www.math.technion.ac.il/hat/fpapers/remeztrans.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.
* 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.
* Smith, Lyle B. (1966). "Computation of Chebyshev series coefficients". Comm. ACM 9 (2): 86–87. doi:10.1145/365170.365195. Algorithm 277.
* Hazewinkel, Michiel, ed. (2001), "Chebyshev polynomials", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page

## External links

* Weisstein, Eric W.. "Chebyshev polynomial[s of the first kind"]. http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html.
* Mathews, John H. (2003). "Module for Chebyshev polynomials". California State University. http://math.fullerton.edu/mathews/n2003/ChebyshevPolyMod.html.
* "Numerical computing with functions". https://www.chebfun.org.
* "Is there an intuitive explanation for an extremal property of Chebyshev polynomials?". https://mathoverflow.net/q/25534.
* "Chebyshev polynomial evaluation and the Chebyshev transform". https://www.boost.org/doc/libs/release/libs/math/doc/html/math_toolkit/sf_poly/chebyshev.html.

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