The Chebyshev polynomials are two sequences of polynomials, denoted Tn(x) and Un(x). They are defined as follows. By the double angle formula,
is a polynomial in cos(θ), so define T2(x)=2x2-1. The other Tn(x) are defined similarly, using cos(nθ)=Tn(cos(θ)). Similarly, define the other sequence by sin(nθ)=Un-1(cos(θ))sin(θ), where we have used de Moivre's formula to note that sin(nθ) is sin(θ) times a polynomial in cos(θ). For instance,
gives U2(θ)=4x2-1. The Tn(x) and Un(x) are called Chebyshev polynomials of the first and second kind, respectively.
The Tn(x) are orthogonal with respect to the inner product
and Un(x) are orthogonal with respect to a different product. 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.
The Chebyshev polynomials Tn are polynomials with the largest possible leading coefficient whose absolute value on the interval [−1,1] 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 Tn(x), which are also called Chebyshev nodes, are used as nodes in polynomial interpolation. The resulting interpolation polynomial minimizes the problem of Runge's phenomenon and provides an approximation that is close to the polynomial of best approximation to a continuous function under the maximum norm. This approximation leads directly to the method of Clenshaw–Curtis quadrature.
These polynomials were named after Pafnuty Chebyshev,. The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebycheff, Tchebyshev (French) or Tschebyschow (German).
The Chebyshev polynomials of the first kind are defined by the recurrence relation
The ordinary generating function for Tn is
The Chebyshev polynomials of the second kind are defined by the recurrence relation
The ordinary generating function for Un is
the exponential generating function is
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 n = 0, 1, 2, 3, ... which is a variant (equivalent transpose) of Schröder's equation, viz. Tn(x) is functionally conjugate to nx, codified in the nesting property below. Further compare to the spread polynomials, in the section below.
The polynomials of the second kind satisfy:
which is structurally quite similar to the Dirichlet kernel Dn(x):
That cos nx is an nth-degree polynomial in cos x can be seen by observing that cos nx is the real part of one side of de Moivre's formula. The real part of the other side is a polynomial in cos x and sin x, in which all powers of sin x are even and thus replaceable through the identity cos2 x + sin2 x = 1. By the same reasoning, sin nx is the imaginary part of the polynomial, in which all powers of sin x are odd and thus, if one is factored out, the remaining can be replaced to create a (n-1)th-degree polynomial in cos x.
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,
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 z = a + bi,
Pell equation definition
The Chebyshev polynomials can also be defined as the solutions to the Pell equation
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 n = 1 this results in the already known recurrence formula, just arranged differently, and with n = 2 it forms the recurrence relation for all even or all odd Chebyshev polynomials (depending on the parity of the lowest m) 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 with products may be written as:
for m ≥ n.
By this, like above, with n = 2 the recurrence formula of Chebyshev polynomials of the second kind forms for both types of symmetry to
depending on whether m starts with 2 or 3.
Relations between Chebyshev polynomials of the first and second kinds
The Chebyshev polynomials of the first and second kinds correspond to a complementary pair of Lucas sequences Ṽn(P,Q) and Ũn(P,Q) with parameters P = 2x and Q = 1:
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.
Different approaches to defining Chebyshev polynomials lead to different explicit expressions such as:
where the prime at the sum symbol indicates that the contribution of j = 0 needs to be halved if it appears.
where 2F1 is a hypergeometric function.
Roots and extrema
A Chebyshev polynomial of either kind with degree n has n different simple roots, called Chebyshev roots, in the interval [−1,1]. 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 Tn are
Similarly, the roots of Un are
The extrema of Tn on the interval −1 ≤ x ≤ 1 are located at
One unique property of the Chebyshev polynomials of the first kind is that on the interval −1 ≤ x ≤ 1 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's easy to show that:
The last two formulas can be numerically troublesome due to the division by zero (0/ indeterminate form, specifically) at x = 1 and x = −1. It can be shown that:
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 k = 0 is to be halved, if it appears.
Concerning integration, the first derivative of the Tn implies that
and the recurrence relation for the first kind polynomials involving derivatives establishes that for n ≥ 2
Both Tn and Un form a sequence of orthogonal polynomials. The polynomials of the first kind Tn are orthogonal with respect to the weight
on the interval [−1,1], i.e. we have:
This can be proven by letting x = cos θ and using the defining identity Tn(cos θ) = cos nθ.
Similarly, the polynomials of the second kind Un are orthogonal with respect to the weight
on the interval [−1,1], i.e. we have:
(The measure √ dx is, to within a normalizing constant, the Wigner semicircle distribution.)
The Tn also satisfy a discrete orthogonality condition:
where N is any integer greater than i+j, and the xk are the N Chebyshev nodes (see above) of TN(x):
For the polynomials of the second kind and any integer N>i+j with the same Chebyshev nodes xk, there are similar sums:
and without the weight function:
For any integer N>i+j, based on the N zeros of UN(x):
one can get the sum:
and again without the weight function:
For any given n ≥ 1, among the polynomials of degree n 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 |f(x)| reaches this maximum exactly n + 1 times at
Remark: By the Equioscillation theorem, among all the polynomials of degree ≤ n, the polynomial f minimizes ||f||∞ on [−1,1] if and only if there are n + 2 points −1 ≤ x0 < x1 < ... < xn + 1 ≤ 1 such that |f(xi)| = ||f||∞.
Of course, the null polynomial on the interval [−1,1] can be approach by itself and minimizes the ∞-norm.
Above, however, |f| reaches its maximum only n + 1 times because we are searching for the best polynomial of degree n ≥ 1 (therefore the theorem evoked previously cannot be used).
For every nonnegative integer n, Tn(x) and Un(x) are both polynomials of degree n. They are even or odd functions of x as n is even or odd, so when written as polynomials of x, it only has even or odd degree terms respectively. In fact,
The leading coefficient of Tn is 2n − 1 if 1 ≤ n, but 1 if 0 = n.
Tn are a special case of Lissajous curves with frequency ratio equal to n.
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 convention U−1 ≡ 0).
Similar to the formula
we have the analogous formula
For x ≠ 0,
which follows from the fact that this holds by definition for x = eiθ.
Then Cn(x) and Cm(x) 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 a is not necessarily an integer, and 2F1(a, b; c; z) is the Gaussian hypergeometric function; as an example . The power series expansion
converges for .
As a basis set
Furthermore, as mentioned previously, the Chebyshev polynomials form an orthogonal basis which (among other things) implies that the coefficients an 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 f(x) 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 f(x) 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).
Consider the Chebyshev expansion of log(1 + x). One can express
One can find the coefficients an either through the application of an inner product or by the discrete orthogonality condition. For the inner product,
Alternatively, when you cannot evaluate the inner product of the function you are trying to approximate, the discrete orthogonality condition gives an often useful result for approximate coefficients,
where δij is the Kronecker delta function and the xk are the N Gauss–Chebyshev zeros of TN(x):
For any N, these approximate coefficients provide an exact approximation to the function at xk with a controlled error between those points. The exact coefficients are obtained with N = ∞, thus representing the function exactly at all points in [−1,1]. The rate of convergence depends on the function and its smoothness.
This allows us to compute the approximate coefficients an very efficiently through the discrete cosine transform
To provide another example:
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 an 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 N coefficients of the (N − 1)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
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 2x − 1 ∈ [−1,1] the argument of the shifted Chebyshev polynomial is x ∈ [0,1]. Similarly, one can define shifted polynomials for generic intervals [a,b].
The spread polynomials are a rescaling of the shifted Chebyshev polynomials of the first kind so that the range is also [0,1]. That is,
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