Chebyshev polynomials

The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as formula_1 and formula_2. They can be defined in several equivalent ways, one of which starts with trigonometric functions:
The Chebyshev polynomials of the first kind formula_3 are defined by
Similarly, the Chebyshev polynomials of the second kind formula_5 are defined by
That these expressions define polynomials in formula_7 may not be obvious at first sight, but follows by rewriting formula_8 and formula_9 using de Moivre's formula or by using the angle sum formulas for formula_10 and formula_11 repeatedly. For example, the double angle formulas, which follow directly from the angle sum formulas, may be used to obtain formula_12 and formula_13, which are respectively a polynomial in formula_7 and a polynomial in formula_7 multiplied by formula_16. Hence formula_17 and formula_18.
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.
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).
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 formula_21:
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_27 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 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 factor of is factored out, the remaining factors can be replaced to create a st-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 integer multiple of an angle solely in terms of the cosine of the base angle.
Chebyshev polynomials of the first kind can be computed directly from the Euler's identity
Expanding the latter, one gets
Then, to get the expression for formula_37, one should look on the real part of the expression, which is obtained from summands corresponding to even indices. Noting formula_38 and formula_39, one gets the explicit formula
which in turn means that
Alternatively, the first two Chebyshev polynomials of the first kind are computed directly from the definition to be
and
while the rest may be evaluated using a specialization of the product-to-sum identity
as, for example,
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 summation symbol indicates that the contribution of needs to be halved if it appears, and formula_47.
An immediate corollary is the expression of complex exponentiation in terms of Chebyshev polynomials: given ,
Commuting polynomials definition.
Chebyshev polynomials can also be characterized by the following theorem:
If formula_49 is a family of monic polynomials with coefficients in a field of characteristic formula_50 such that formula_51 and formula_52 for all 
formula_53 and formula_54, then, up to a simple change of variables, either formula_55 for all formula_54 or 
formula_57 for all formula_54.
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 formula_2 and formula_66 using the derivative formula for formula_1 gives the recurrence relationship for the derivative of formula_3:
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 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 formula_1 on the interval formula_85 where formula_86 are located at formula_87 values of formula_88. They are formula_89, or formula_90 where formula_91, formula_92, formula_93 and formula_94, i.e., formula_95 and formula_96 are relatively prime numbers.
Specifically, when formula_97 is even, 
When formula_97 is odd, 
This result has been generalized to solutions of formula_122, and to formula_123 and formula_124 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 . 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 values – taking the limit as :
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.
More general formula states:
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:
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
T_n(x)\right| \\
f_n(x) &amp;&gt; 0 \qquad \text{ for }~ x = \cos \frac{2k\pi}{n} ~&amp;&amp;\text{ where } 0 \le 2k \le n \\
f_n(x) &amp;&lt; 0 \qquad \text{ for }~ x = \cos \frac{(2k + 1)\pi}{n} ~&amp;&amp;\text{ where } 0 \le 2k + 1 \le n
\end{align}&lt;/math&gt;
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 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 formula_160, which themselves are a special case of the Jacobi polynomials formula_161:
Chebyshev polynomials are also a special case of Dickson polynomials:
In particular, when formula_165, they are related by formula_166 and formula_167.
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 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 formula_185 and formula_186 closely related to Chebyshev polynomials are sometimes used. They are defined by
and satisfy
A. F. Horadam called the polynomials formula_185 Vieta–Lucas polynomials and denoted them formula_190. He called the polynomials formula_186 Vieta–Fibonacci polynomials and denoted them formula_192. 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 formula_193 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 formula_197. In the airfoil literature formula_192 and formula_199 are denoted formula_200 and formula_201. The polynomial families formula_1, formula_2, formula_192, and formula_199 are orthogonal with respect to the weights
and are proportional to Jacobi polynomials formula_161 with
All four families satisfy the recurrence formula_209 with formula_210, where formula_211, formula_5, formula_213, or formula_214, but they differ according to whether formula_215 equals formula_88, formula_217, formula_218, or formula_219.