Chebyshev polynomials

Chebyshev polynomials
The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as and . 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:
That these expressions define polynomials in may not be obvious at first sight but follows by rewriting and using de Moivre's formula or by using the angle sum formulas for and repeatedly. For example, the double angle formulas, which follow directly from the angle sum formulas, may be used to obtain and , which are respectively a polynomial in and a polynomial in multiplied by . Hence and .
An important and convenient property of the is that they are orthogonal with respect to the inner product:
and are orthogonal with respect to another, analogous inner product, given 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 are obtained from the recurrence relation:
The recurrence also allows to represent them explicitly as the determinant of a tridiagonal matrix of size :
The ordinary generating function for is:
There are several other generating functions for the Chebyshev polynomials; the exponential generating function is:
The generating function relevant for 2-dimensional potential theory and multipole expansion is:
The Chebyshev polynomials of the second kind are defined by the recurrence relation:
Notice that the two sets of recurrence relations are identical, except for vs. The ordinary generating function for is:
and the exponential generating function is:
Trigonometric definition
As described in the introduction, the Chebyshev polynomials of the first kind can be defined as the unique polynomials satisfying:
or, in other words, as the unique polynomials satisfying:
for .
The polynomials of the second kind satisfy:
or
which is structurally quite similar to the Dirichlet kernel :
(The Dirichlet kernel, in fact, coincides with what is now known as the Chebyshev polynomial of the fourth kind.)
An equivalent way to state this is via exponentiation of a complex number: given a complex number with absolute value of one:
Chebyshev polynomials can be defined in this form when studying trigonometric polynomials.
That is an th-degree polynomial in can be seen by observing that is the real part of one side of de Moivre's formula:
The real part of the other side is a polynomial in and , in which all powers of are even and thus replaceable through the identity .
By the same reasoning, is the imaginary part of the polynomial, in which all powers of are odd and thus, if one factor of is factored out, the remaining factors can be replaced to create a st-degree polynomial in .
Commuting polynomials definition
Chebyshev polynomials can also be characterized by the following theorem:
If is a family of monic polynomials with coefficients in a field of characteristic such that and for all
and , then, up to a simple change of variables, either for all or
for all .
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:
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 and with parameters and :
It follows that they also satisfy a pair of mutual recurrence equations:
The second of these may be rearranged using the recurrence definition for the Chebyshev polynomials of the second kind to give:
Using this formula iteratively gives the sum formula:
while replacing and using the derivative formula for gives the recurrence relationship for the derivative of :
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
Different approaches to defining Chebyshev polynomials lead to different explicit expressions. The trigonometric definition gives an explicit formula as follows:
From this trigonometric form, the recurrence definition can be recovered by computing directly that the bases cases hold:
and
and that the product-to-sum identity holds:
Using the complex number exponentiation definition of the Chebyshev polynomial, one can derive the following expression:
The two are equivalent because .
An explicit form of the Chebyshev polynomial in terms of monomials follows from de Moivre's formula:
where denotes the real part of a complex number. Expanding the formula, one gets:
The real part of the expression is obtained from summands corresponding to even indices. Noting and , one gets the explicit formula:
which in turn means that:
This can be written as a hypergeometric function:
with inverse:
where the prime at the summation symbol indicates that the contribution of needs to be halved if it appears:
where is a hypergeometric function.
Properties
Symmetry
That is, Chebyshev polynomials of even order have even symmetry and therefore contain only even powers of . Chebyshev polynomials of odd order have odd symmetry and therefore contain only odd powers of .
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 nodes in polynomial interpolation. Using the trigonometric definition and the fact that:
one can show that the roots of are:
Similarly, the roots of are:
The extrema of on the interval are located at:
One unique property of the Chebyshev polynomials of the first kind is that on the interval 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:
The extrema of on the interval where are located at values of . They are , or where and , i.e., and are relatively prime numbers.
Specifically, when is even:
if , or and is even. There are such values of .
if and is odd. There are such values of .
When is odd:
if , or and is even. There are such values of .
if , or and is odd. There are such values of .
This result has been generalized to solutions of , and to and for Chebyshev polynomials of the third and fourth kinds, respectively.
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:
The last two formulas can be numerically troublesome due to the division by zero (indeterminate form, specifically) at and . By L'Hôpital's rule:
More generally,
which is of great use in the numerical solution of eigenvalue problems.
Also, we have:
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 implies that:
and the recurrence relation for the first kind polynomials involving derivatives establishes that for :
The last formula can be further manipulated to express the integral of as a function of Chebyshev polynomials of the first kind only:
Furthermore, we have:
Products of Chebyshev polynomials
The Chebyshev polynomials of the first kind satisfy the relation:
which is easily proved from the product-to-sum formula for the cosine:
For this results in the already known recurrence formula, just arranged differently, and with it forms the recurrence relation for all even or all odd indexed Chebyshev polynomials (depending on the parity of the lowest ) which implies the evenness or oddness of these polynomials. Three more useful formulas for evaluating Chebyshev polynomials can be concluded from this product expansion:
The polynomials of the second kind satisfy the similar relation:
(with the definition by convention ). They also satisfy:
for .
For this recurrence reduces to:
which establishes the evenness or oddness of the even or odd indexed Chebyshev polynomials of the second kind depending on whether starts with 2 or 3.
Composition and divisibility properties
The trigonometric definitions of and imply the composition or nesting properties:
For the order of composition may be reversed, making the family of polynomial functions a commutative semigroup under composition.
Since is divisible by if is odd, it follows that is divisible by if is odd. Furthermore, is divisible by , and in the case that is even, divisible by .
Orthogonality
Both and form a sequence of orthogonal polynomials. The polynomials of the first kind are orthogonal with respect to the weight:
on the interval , i.e. we have:
This can be proven by letting and using the defining identity .
Similarly, the polynomials of the second kind are orthogonal with respect to the weight:
on the interval , i.e. we have:
(The measure 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:
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 also satisfy a discrete orthogonality condition:
where is any integer greater than , and the are the Chebyshev nodes (see above) of :
For the polynomials of the second kind and any integer with the same Chebyshev nodes , there are similar sums:
and without the weight function:
For any integer , based on the zeros of :
one can get the sum:
and again without the weight function:
Minimal -norm
For any given , among the polynomials of degree with leading coefficient 1 (monic polynomials):
is the one of which the maximal absolute value on the interval [−1, 1] is minimal.
This maximal absolute value is:
and reaches this maximum exactly times at:
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:
In particular, when , they are related by and .
Other properties
The curves given by , or equivalently, by the parametric equations are a special case of Lissajous curves with frequency ratio equal to .
Similar to the formula:
we have the analogous formula:
For :
and:
which follows from the fact that this holds by definition for .
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).
Example 1
Consider the Chebyshev expansion of . One can express:
One can find the coefficients either through the application of an inner product or by the discrete orthogonality condition. For the inner product:
which gives:
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:
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
The partial sums of:
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 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:
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 is of the form:
Polynomials in Chebyshev form can be evaluated using the Clenshaw algorithm.
Families of polynomials related to Chebyshev polynomials
Polynomials denoted and closely related to Chebyshev polynomials are sometimes used. They are defined by:
and satisfy:
A. F. Horadam called the polynomials Vieta–Lucas polynomials and denoted them . He called the polynomials Vieta–Fibonacci polynomials and denoted them . Lists of both sets of polynomials are given in Viète's Opera Mathematica, Chapter IX, Theorems VI and VII. The Vieta–Lucas and Vieta–Fibonacci polynomials of real argument are, up to a power of 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 are defined as:
where . In the airfoil literature and are denoted and . The polynomial families , and are orthogonal with respect to the weights:
and are proportional to Jacobi polynomials with:
All four families satisfy the recurrence with , where , or , but they differ according to whether equals , or .
See also
Chebyshev filter
Chebyshev cube root
Dickson polynomials
Legendre polynomials
Laguerre polynomials
Hermite polynomials
Minimal polynomial of 2cos(2pi/n)
Romanovski polynomials
Chebyshev rational functions
Approximation theory
The Chebfun system
Discrete Chebyshev transform
Markov brothers' inequality
Clenshaw algorithm
References
Sources
External links
– includes illustrative Java applet.
Special hypergeometric functions
Orthogonal polynomials
Polynomials
Approximation theory