The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as and . They can be defined several ways that have the same end result; in this article the polynomials are defined by starting with trigonometric functions:
The Chebyshev polynomials of the first kind are given by
Similarly, define the Chebyshev polynomials of the second kind as
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 n = 2 the T2 formula can be converted into a polynomial with argument x = cos(θ), using the double angle formula:
Replacing the terms in the formula with the definitions above, we get
The other Tn(x) are defined similarly, where for the polynomials of the second kind (Un) we must use de Moivre's formula to get sin(n θ) as sin(θ) times a polynomial in cos(θ) . For instance,
Once converted to polynomial form, Tn(x) and Un(x) 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 j = 0 needs to be halved if it appears, and .
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.
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).
Plot of the first five Tn Chebyshev polynomials of the first kind
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 cos2x + sin2x = 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
in a ring R[x]. 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 sequencesṼn(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:
That is, Chebyshev polynomials of even order have even symmetry and contain only even powers of x. Chebyshev polynomials of odd order have odd symmetry and contain only odd powers of x.
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 can be shown that:
The last two formulas can be numerically troublesome due to the division by zero (0/0indeterminate form, specifically) at x = 1 and x = −1. It can be shown that:
which, if evaluated as shown above, poses a problem because it is indeterminate at x = ±1. 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 x = 1 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. Un − 1(1) = nTn(1) = n which will be useful later on. Since the numerator and denominator are both limiting to zero, L'Hôpital's rule applies:
The proof for x = −1 is similar, with the fact that Tn(−1) = (−1)n 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 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
The last formula can be further manipulated to express the integral of Tn as a function of Chebyshev polynomials of the first kind only:
Furthermore, we have
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
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, products may be written as:
for m ≥ n.
By this, like above, with n = 2 the recurrence formula for Chebyshev polynomials of the second kind reduces for both types of symmetry to
depending on whether m starts with 2 or 3.
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
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.
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 U−1 ≡ 0 by convention ).
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 .
The first few Chebyshev polynomials of the first kind in the domain −1 < x < 1: The flat T0, T1, T2, T3, T4 and T5.
The first few Chebyshev polynomials of the first kind are OEIS: A028297
The first few Chebyshev polynomials of the second kind in the domain −1 < x < 1: The flat U0, U1, U2, U3, U4 and U5. Although not visible in the image, Un(1) = n + 1 and Un(−1) = (n + 1)(−1)n.
The first few Chebyshev polynomials of the second kind are OEIS: A053117
As a basis set
The non-smooth function (top) y = −x3H(−x), where H is the Heaviside step function, and (bottom) the 5th partial sum of its Chebyshev expansion. The 7th sum is indistinguishable from the original function at the resolution of the graph.
In the appropriate Sobolev space, the set of Chebyshev polynomials form an orthonormal basis, so that a function in the same space can, on −1 ≤ x ≤ 1 be expressed via the expansion:
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 piecewisesmooth 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 the inner product of the function being approximated cannot be evaluated, 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.
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
An arbitrary polynomial of degree N can be written in terms of the Chebyshev polynomials of the first kind. Such a polynomial p(x) is of the form
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].
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