Chebyshev polynomials


The Chebyshev polynomials are two sequences of orthogonal 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 is not obvious at first sight but can be shown using de Moivre's formula.
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 Tchebycheff, Tchebyshev or Tschebyschow.

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

The Chebyshev polynomials of the first and second kind can be defined as the unique polynomials satisfying
and
for.
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 also 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.
For outside the interval , the above definition implies

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:

Generating functions

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 ordinary generating function for is
and the exponential generating function is

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

Using the complex number exponentiation definition of the Chebyshev polynomial, one can derive the following expressions, valid for any real :
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 on the summation symbol indicates that the contribution of needs to be halved if it appears.
A related expression for as a sum of monomials with binomial coefficients and powers of two is
Similarly, can be expressed in terms of hypergeometric functions:

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.

    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 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 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:
. 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.
The also satisfy a discrete orthogonality condition:
where is any integer greater than, and the are the Chebyshev nodes 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

Minimal -norm

For any given, among the polynomials of degree with leading coefficient 1 :
is the one of which the maximal absolute value on the interval is minimal.
This maximal absolute value is:
and reaches this maximum exactly times at: