Chebyshev polynomials

The Chebyshev polynomials are two sequences of polynomials related to the sine and cosine functions, notated as formula_1 and formula_2. They can be defined several ways that have the same end result; in this article the polynomials are defined by starting with trigonometric functions:
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 the formula can be converted into a polynomial with argument , using the double angle formula:
Replacing the terms in the formula with the definitions above, we get
The other are defined similarly, where for the polynomials of the second kind () we must use de Moivre's formula to get as times a polynomial in For instance, 
gives
Once converted to polynomial form, and 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
where the prime at the sum symbol indicates that the contribution of needs to be halved if it appears, and formula_10.
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 product, given below. This follows 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 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.
Chebyshev polynomials are important in approximation theory because 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).
Definition.
The Chebyshev polynomials of the first kind are obtained from the recurrence relation
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 formula_19 vs. 
The ordinary generating function for is
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 which as a technical point is a variant (equivalent transpose) of Schröder's equation. That is, is functionally conjugate to , codified in the nesting property below. 
The polynomials of the second kind satisfy:
or
which is structurally quite similar to the Dirichlet kernel :
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 is factored out, the remaining can be replaced to create a th-degree polynomial in .
The identity is quite useful in conjunction with the recursive generating formula, inasmuch as it enables one to calculate the cosine of any integral multiple of an angle solely in terms of the cosine of the base angle.
Evaluating the first two Chebyshev polynomials,
and
one can straightforwardly determine that
and so forth.
Two immediate corollaries are the "composition identity" (or nesting property specifying a semigroup)
and the expression of complex exponentiation in terms of Chebyshev polynomials: given ,
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:
Products of Chebyshev polynomials.
When working with Chebyshev polynomials quite often products of two of them occur. These products can be reduced to combinations of Chebyshev polynomials with lower or higher degree and concluding statements about the product are easier to make. It shall be assumed that in the following the index m is greater than or equal to the index n and n is not negative. For Chebyshev polynomials of the first kind the product expands to
which is an analogy to the addition theorem
with the identities
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 Chebyshev polynomials (depending on the parity of the lowest ) which allows to design functions with prescribed symmetry properties. Three more useful formulas for evaluating Chebyshev polynomials can be concluded from this product expansion:
For Chebyshev polynomials of the second kind, products may be written as:
for .
By this, like above, with the recurrence formula for Chebyshev polynomials of the second kind reduces for both types of symmetry to
depending on whether starts with 2 or 3.
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 Chebyshev polynomials of the first and second kinds are also connected by the following relations:
The recurrence relationship of the derivative of Chebyshev polynomials can be derived from these relations:
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 such as:
with inverse
where the prime at the sum 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 contain only even powers of . Chebyshev polynomials of odd order have odd symmetry and 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:
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 . It can be shown that:
which, if evaluated as shown above, poses a problem because it is indeterminate at . Since the function is a polynomial, (all of) the derivatives must exist for all real numbers, so the taking to limit on the expression above should yield the desired value:
where only is considered for now. Factoring the denominator:
Since the limit as a whole must exist, the limit of the numerator and denominator must independently exist, and
The denominator (still) limits to zero, which implies that the numerator must be limiting to zero, i.e. which will be useful later on. Since the numerator and denominator are both limiting to zero, L'Hôpital's rule applies:
The proof for is similar, with the fact that being important.
Indeed, the following, more general formula holds:
This latter result 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 implies that 
and the recurrence relation for the first kind polynomials involving derivatives establishes that for 
The latter formula can be further manipulated to express the integral of as a function of Chebyshev polynomials of the first kind only:
Furthermore, we have
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 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
T_n(x)\right| \\
f_n(x) &> 0 \qquad \text{ for }~ x = \cos \frac{2k\pi}{n} ~&&\text{ where } 0 \le 2k \le n \\
f_n(x) &< 0 \qquad \text{ for }~ x = \cos \frac{(2k + 1)\pi}{n} ~&&\text{ where } 0 \le 2k + 1 \le n
\end{align}</math>
From the intermediate value theorem, has at least roots. However, this is impossible, as is a polynomial of degree , so the fundamental theorem of algebra implies it has at most roots.
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 approach 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).
Other properties.
The Chebyshev polynomials are a special case of the ultraspherical or Gegenbauer polynomials, which themselves are a special case of the Jacobi polynomials:
For every nonnegative integer , and are both polynomials of degree . They are even or odd functions of as is even or odd, so when written as polynomials of , it only has even or odd degree terms respectively. In fact,
and
The leading coefficient of is if , but 1 if .
 are a special case of Lissajous curves with frequency ratio equal to .
Several polynomial sequences like Lucas polynomials (), Dickson polynomials (), Fibonacci polynomials () are related to Chebyshev polynomials and .
The Chebyshev polynomials of the first kind satisfy the relation
which is easily proved from the product-to-sum formula for the cosine. The polynomials of the second kind satisfy the similar relation
(with the definition by convention ).
Similar to the formula
we have the analogous formula
For ,
and
which follows from the fact that this holds by definition for .
Define
Then and are commuting polynomials:
as is evident in the Abelian nesting property specified above.
Generalized Chebyshev polynomials.
The generalized Chebyshev polynomials "Ta" are defined by
where is not necessarily an integer, and is the Gaussian hypergeometric function; as an example, formula_96.
The power series expansion 
converges 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 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 th 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.
Shifted Chebyshev polynomials.
Shifted Chebyshev polynomials of the first kind are defined as
When the argument of the Chebyshev polynomial is in the range of the argument of the shifted Chebyshev polynomial is . Similarly, one can define shifted polynomials for generic intervals .