Chebyshev polynomials

Summary

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,

gives

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 .

An important and convenient property of the Tn(x) is that they are orthogonal with respect to the inner product

and Un(x) are orthogonal with respect to another, analogous inner 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 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.[1]

Chebyshev polynomials are important in approximation theory because the roots of Tn(x), 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.[2] The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebycheff, Tchebyshev (French) or Tschebyschow (German).

Definitions

Recurrence definition

Plot of the first five Tn Chebyshev polynomials of the first kind

The Chebyshev polynomials of the first kind are obtained from the recurrence relation

The ordinary generating function for Tn 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

Plot of the first five Un Chebyshev polynomials of the second kind

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 vs. . The ordinary generating function for Un 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 n = 0, 1, 2, 3, … which as a technical point is a variant (equivalent transpose) of Schröder's equation. That is, Tn(x) is functionally conjugate to n x, codified in the nesting property below.

The polynomials of the second kind satisfy:

or

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,

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 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].[3] 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 n(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.

Explicit expressions

Different approaches to defining Chebyshev polynomials lead to different explicit expressions such as:

with inverse[4][5]

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.

Properties

Symmetry

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 (0/0 indeterminate form, specifically) at x = 1 and x = −1. It can be shown that:

Proof —

The second derivative of the Chebyshev polynomial of the first kind is

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

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, products may be written as:

for mn.

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.

Orthogonality

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 .

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 1 − x2 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 max(i,j),[6] 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:

Minimal ∞-norm

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

Proof —

Let's assume that wn(x) is a polynomial of degree n with leading coefficient 1 with maximal absolute value on the interval [−1,1] less than 1 / 2n − 1.

Define

Because at extreme points of Tn we have

From the intermediate value theorem, fn(x) has at least n roots. However, this is impossible, as fn(x) is a polynomial of degree n − 1, so the fundamental theorem of algebra implies it has at most n − 1 roots.

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).

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 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,

and

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.

Several polynomial sequences like Lucas polynomials (Ln), Dickson polynomials (Dn), Fibonacci polynomials (Fn) are related to Chebyshev polynomials Tn and Un.

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,

and

which follows from the fact that this holds by definition for x = e.

Define

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 .

Examples

First kind

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 OEISA028297

Second kind

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 OEISA053117

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:[7]

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.[7] 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,[7] 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 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,

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 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

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 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[8] 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.[9] Such a polynomial p(x) 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 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].

See also

References

  1. ^ Rivlin, Theodore J. (1974). "Chapter  2, Extremal properties". The Chebyshev Polynomials. Pure and Applied Mathematics (1st ed.). New York-London-Sydney: Wiley-Interscience [John Wiley & Sons]. pp. 56–123. ISBN 978-047172470-4.
  2. ^ Chebyshev polynomials were first presented in Chebyshev, P. L. (1854). "Théorie des mécanismes connus sous le nom de parallélogrammes". Mémoires des Savants étrangers présentés à l'Académie de Saint-Pétersbourg (in French). 7: 539–586.
  3. ^ Demeyer, Jeroen (2007). Diophantine Sets over Polynomial Rings and Hilbert's Tenth Problem for Function Fields (PDF) (Ph.D. thesis). p. 70. Archived from the original (PDF) on 2 July 2007.
  4. ^ Cody, W. J. (1970). "A survey of practical rational and polynomial approximation of functions". SIAM Review. 12 (3): 400–423. doi:10.1137/1012082.
  5. ^ Mathar, R. J. (2006). "Chebyshev series expansion of inverse polynomials". J. Comput. Appl. Math. 196 (2): 596–607. arXiv:math/0403344. Bibcode:2006JCoAM..196.596M. doi:10.1016/j.cam.2005.10.013. S2CID 16476052.
  6. ^ For proof, see: Mason, J.C. & Handscomb, D.C. (2002). Chebyshev Polynomials. Taylor & Francis.
  7. ^ a b c Boyd, John P. (2001). Chebyshev and Fourier Spectral Methods (PDF) (second ed.). Dover. ISBN 0-486-41183-4. Archived from the original (PDF) on 31 March 2010. Retrieved 19 March 2009.
  8. ^ "Chebyshev Interpolation: An Interactive Tour". Archived from the original on 18 March 2017. Retrieved 2 June 2016.
  9. ^ For more information on the coefficients, see: Mason, J.C. & Handscomb, D.C. (2002). Chebyshev Polynomials. Taylor & Francis.

Sources

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  • Dette, Holger (1995). "A note on some peculiar nonlinear extremal phenomena of the Chebyshev polynomials". Proceedings of the Edinburgh Mathematical Society. 38 (2): 343–355. arXiv:math/9406222. doi:10.1017/S001309150001912X. S2CID 16703489.
  • Elliott, David (1964). "The evaluation and estimation of the coefficients in the Chebyshev Series expansion of a function". Math. Comp. 18 (86): 274–284. doi:10.1090/S0025-5718-1964-0166903-7. MR 0166903.
  • Eremenko, A.; Lempert, L. (1994). "An Extremal Problem For Polynomials" (PDF). Proceedings of the American Mathematical Society. 122 (1): 191–193. doi:10.1090/S0002-9939-1994-1207536-1. MR 1207536.
  • Hernandez, M. A. (2001). "Chebyshev's approximation algorithms and applications". Comp. Math. Applic. 41 (3–4): 433–445. doi:10.1016/s0898-1221(00)00286-8.
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  • Mason, J. C.; Handscomb, D.C. (2002). Chebyshev Polynomials. Taylor & Francis.
  • Mathar, R. J. (2006). "Chebyshev series expansion of inverse polynomials". J. Comput. Appl. Math. 196 (2): 596–607. arXiv:math/0403344. Bibcode:2006JCoAM.196..596M. doi:10.1016/j.cam.2005.10.013. S2CID 16476052.
  • Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), "Orthogonal Polynomials", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248
  • Remes, Eugene. "On an Extremal Property of Chebyshev Polynomials" (PDF).
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External links

  • Media related to Chebyshev polynomials at Wikimedia Commons
  • Weisstein, Eric W. "Chebyshev polynomial[s] of the first kind". MathWorld.
  • Mathews, John H. (2003). "Module for Chebyshev polynomials". Department of Mathematics. Course notes for Math 340 Numerical Analysis & Math 440 Advanced Numerical Analysis. Fullerton, CA: California State University. Archived from the original on 29 May 2007. Retrieved 17 August 2020.
  • "Chebyshev interpolation: An interactive tour". Mathematical Association of America (MAA) – includes illustrative Java applet.
  • "Numerical computing with functions". The Chebfun Project.
  • "Is there an intuitive explanation for an extremal property of Chebyshev polynomials?". Math Overflow. Question 25534.
  • "Chebyshev polynomial evaluation and the Chebyshev transform". Boost. Math.