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Wikipedia / Chebyshev polynomials

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

The Chebyshev polynomials of the first kind are defined by

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

is not obvious at first sight but can be shown using de Moivre's formula (see below).

The Chebyshev polynomials are polynomials with the largest possible leading coefficient whose absolute value on the interval is bounded by 1. They are also the "extremal" polynomials for many other properties.

In 1952, Cornelius Lanczos showed that the Chebyshev polynomials are important in approximation theory for the solution of linear systems; the roots of , 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 is used because of the alternative transliterations of the name Chebyshev as , (French) or (German).

Definitions

Recurrence definition

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

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

which differs from the above only by the rule for n=1.

Trigonometric definition

for .

An equivalent way to state this is via exponentiation of a complex number: given a complex number

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

th-degree is the real part of one side of de Moivre's formula

are even is the imaginary part are odd.

Commuting polynomials definition

Chebyshev polynomials can also be characterized by the following theorem:

Pell equation definition

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

in a ring . Thus, they can be generated by the standard technique for Pell equations of taking powers of a fundamental solution

Generating functions

The ordinary generating function

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

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

The ordinary generating function for

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:

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

The second of these may be rearranged using the recurrence definition

using the derivative formula.

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

Turán's inequalities for the Chebyshev polynomials are:

The integral relations are

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 :

denote the real part, the definition of , de Moivre's formula, and the binomial theoremThis can be written as a hypergeometric functionwith inverse needs to be halved if it appears.

Properties

Symmetry

That is, Chebyshev polynomials of even order have even symmetry. Chebyshev polynomials of odd order have odd symmetry.

Roots and extrema

A Chebyshev polynomial of either kind with degree has different simple roots, called Chebyshev roots, in the interval . The roots of the Chebyshev polynomial of the first kind are sometimes called Chebyshev nodes because they are used as nodesThe extrema all of the extrema have values that are either −1 or 1. Thus these polynomials have only two finite critical valuess, the defining property of Shabat polynomialsThe extrema are relatively prime.

Specifically (Minimal polynomial of 2cos(2pi/n) is even:

is odd:

Differentiation and integration

The last two formulas can be numerically troublesome due to the division by zero ( indeterminate form. By L'Hôpital's rulewhich is of great use in the numerical solution of eigenvalue problems.

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

Concerning integration, the first derivative of the

Products of Chebyshev polynomials

for all non-negative values of and , which is easily proved from the product-to-sum formula it forms the recurrence relation for all even or all odd indexed Chebyshev polynomials (depending on the parity of the lowest starts with 2 or 3.

Composition and divisibility properties

imply the composition or nesting properties: a commutative semigroup under composition.

Orthogonality

form a sequence of orthogonal polynomialson the interval .

Similarly, the polynomials of the second kind on the interval 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 equationswhich 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.)

, Chebyshev nodesand without the weight function

Minimal ∞-norm

with leading coefficient 1 (monic is the one of which the maximal absolute value on the interval is minimal.

Remark

By the equioscillation theorem, among all the polynomials of degree , the polynomial minimizes on if and only if there are points such that .

Of course, the null polynomial on the interval can be approximated by itself and minimizes the -norm.

Above, however, reaches its maximum only times because we are searching for the best polynomial of degree (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, which themselves are a special case of the Jacobi polynomials

Chebyshev polynomials are also a special case of Dickson polynomials.

Other properties

The curves given by , or equivalently, by the parametric equations , , are a special case of Lissajous curves with frequency ratio equal to .

For which follows from the fact that this holds by definition for .

There are relations between Legendre polynomials 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:

Examples

First kind

The first few Chebyshev polynomials of the first kind are

Second kind

The first few Chebyshev polynomials of the second kind are

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 , be expressed via the expansion: Furthermore, as mentioned previously, the Chebyshev polynomials form an orthogonal basis which (among other things) implies that the coefficients 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. These attributes include:

The Chebyshev polynomials form a complete orthogonal system.
The Chebyshev series converges to 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 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, 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

One can find the coefficients

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

where is the Kronecker delta function and the are the Gauss–Chebyshev zeros of

For any , these approximate coefficients provide an exact approximation to the function at with a controlled error between those points. The exact coefficients are obtained with , thus representing the function exactly at all points in . The rate of convergence depends on the function and its smoothness.

This allows us to compute the approximate coefficients very efficiently through the discrete cosine transform:

Example 2

To provide another example:

Partial sums

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 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 coefficients of the st partial sum are usually obtained on the Chebyshev–Gauss–Lobatto points (or Lobatto grid), which results in minimum error and avoids Runge's phenomenon

Polynomial in Chebyshev form

An arbitrary polynomial of degree can be written in terms of the Chebyshev polynomials of the first kind. Such a polynomial

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

Families of polynomials related to Chebyshev polynomials

closely related to Chebyshev polynomials are sometimes used. They are defined by:

and satisfy:

Vieta–Lucas polynomials Vieta–Fibonacci polynomials and denoted them 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. and a shift of index in the case of the latter, equal to Lucas and Fibonacci polynomials and of imaginary argument.

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

When the argument of the Chebyshev polynomial satisfies the argument of the shifted Chebyshev polynomial satisfies . Similarly, one can define shifted polynomials for generic intervals .

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." The Chebyshev polynomials of the third kind are defined as:

and the Chebyshev polynomials of the fourth kind

. They coincide with the Dirichlet kernel.

In the airfoil

with:

, or

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.

is a primitive root of unity is the th cyclotomic polynomial.

, corresponding to the cyclotomic polynomial is the well known Euler's totient function.

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

is that it maps the product of two or more polynomials to the product of the images of the individual polynomials.

are also irreducible.

For more details, see .

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. 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.

where

is an N-th order Chebyshev polynomial
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.