This is an old revision of this page, as edited by Uscitizenjason(talk | contribs) at 16:51, 17 September 2023(Mention Lanczos historic contribution). The present address (URL) is a permanent link to this revision, which may differ significantly from the current revision.
Revision as of 16:51, 17 September 2023 by Uscitizenjason(talk | contribs)(Mention Lanczos historic contribution)
Plot of the Chebyshev polynomial of the first kind T n(x) with n=5 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D
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 may not be obvious at first sight, but follows by rewriting and using de Moivre's formula or by using the angle sum formulas for and repeatedly. For example, the double angle formulas, which follow directly from the angle sum formulas, may be used to obtain and , which are respectively a polynomial in and a polynomial in multiplied by . Hence 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.
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]
These polynomials were named after Pafnuty Chebyshev.[3] 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 (first kind)
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 :
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 factor of sin x is factored out, the remaining factors can be replaced to create a (n−1)st-degree polynomial in cos x.
Commuting polynomials definition
Chebyshev polynomials can also be characterized by the following theorem:[5]
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 ringR[x].[6] 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:[7]
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 :
where integrals are considered as principal value.
Explicit expressions
Different approaches to defining Chebyshev polynomials lead to different explicit expressions. The trigonometric definition gives an explicit formula as follows.
From this trigonometric form, the recurrence definition can be recovered by computing directly that the bases cases hold,
That is, Chebyshev polynomials of even order have even symmetry and therefore contain only even powers of x. Chebyshev polynomials of odd order have odd symmetry and therefore 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:
The extrema of on the interval where are located at values of . They are , or where , , and , i.e., and are relatively prime numbers.
This result has been generalized to solutions of ,[13] and to and for Chebyshev polynomials of the third and fourth kinds, respectively.[14]
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. 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 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
The Chebyshev polynomials of the first kind satisfy the relation
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 indexed Chebyshev polynomials (depending on the parity of the lowest m) 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 U−1 ≡ 0 by convention ).
They also satisfy
for m ≥ n.
For n = 2 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 m starts with 2 or 3.
Composition and divisibility properties
The trigonometric definitions of Tn and Un imply the composition or nesting properties[15]
For Tmn the order of composition may be reversed, making the family of polynomial functions Tn a commutativesemigroup under composition.
Since Tm(x) is divisible by x if m is odd, it follows that Tmn(x) is divisible by Tn(x) if m is odd. Furthermore, Umn−1(x) is divisible by Un−1(x), and in the case that m is even, divisible by Tn(x)Un−1(x).
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(nθ).
Similarly, the polynomials of the second kind Un are orthogonal with respect to the weight
The Tn also satisfy a discrete orthogonality condition:
where N is any integer greater than max(i, j),[9] and the xk are the NChebyshev 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.
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 approximated 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).
Chebyshev polynomials as special cases of more general polynomial families
The curves given by y = Tn(x), or equivalently, by the parametric equations y = Tn(cos θ) = cos nθ, x = cos θ, are a special case of Lissajous curves with frequency ratio equal to n.
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 = eiθ.
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 OEIS: A028297
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 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:[16]
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.[16] 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,[16] 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
As an interpolant, the N coefficients of the (N − 1)st partial sum are usually obtained on the Chebyshev–Gauss–Lobatto[17] 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.
Families of polynomials related to Chebyshev polynomials
Polynomials denoted and closely related to Chebyshev polynomials are sometimes used. They are defined by[18]
and satisfy
A. F. Horadam called the polynomials Vieta–Lucas polynomials and denoted them . He called the polynomials Vieta–Fibonacci polynomials and denoted them .[19] Lists of both sets of polynomials are given in Viète'sOpera Mathematica, Chapter IX, Theorems VI and VII.[20] The Vieta–Lucas and Vieta–Fibonacci polynomials of real argument are, up to a power of and a shift of index in the case of the latter, equal to Lucas and Fibonacci polynomialsLn and Fn of imaginary argument.
Shifted Chebyshev polynomials of the first and second kinds are related to the Chebyshev polynomials by[18]
When the argument of the Chebyshev polynomial satisfies 2x − 1 ∈ [−1, 1] the argument of the shifted Chebyshev polynomial satisfies x ∈ [0, 1]. Similarly, one can define shifted polynomials for generic intervals [a, b].
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."[21] The Chebyshev polynomials of the third kind are defined as
and the Chebyshev polynomials of the fourth kind are defined as
where .[21][22] In the airfoil literature and are denoted and . The polynomial families , , , and are orthogonal with respect to the weights
^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. ISBN978-047172470-4.
^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.
^Beckenbach, E. F.; Seidel, W.; Szász, Otto (1951), "Recurrent determinants of Legendre and of ultraspherical polynomials", Duke Math. J., 18: 1–10, doi:10.1215/S0012-7094-51-01801-7, MR0040487
^Gürtaş, Y. Z. (2017). "Chebyshev Polynomials and the minimal polynomial of ". American Mathematical Monthly. 124 (1): 74--78. doi:10.4169/amer.math.monthly.124.1.74.
^ abWolfram, D. A. (2022). "Factoring Chebyshev polynomials of the first and second kinds with minimal polynomials of ". American Mathematical Monthly. 129 (2): 172--176. doi:10.1080/00029890.2022.2005391.
^Wolfram, D. A. (2022). "Factoring Chebyshev polynomials with minimal polynomials of ". Bulletin of the Australian Mathematical Society. arXiv:2106.14585. doi:10.1017/S0004972722000235.
^Rayes, M. O.; Trevisan, V.; Wang, P. S. (2005), "Factorization properties of chebyshev polynomials", Computers & Mathematics with Applications, 50 (8–9): 1231–1240, doi:10.1016/j.camwa.2005.07.003
^ abcMason, J. C.; Elliott, G. H. (1993), "Near-minimax complex approximation by four kinds of Chebyshev polynomial expansion", J. Comput. Appl. Math., 46: 291–300, doi:10.1016/0377-0427(93)90303-S
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. S2CID16703489.
Mason, J. C. (1984). "Some properties and applications of Chebyshev polynomial and rational approximation". Rational Approximation and Interpolation. Lecture Notes in Mathematics. Vol. 1105. pp. 27–48. doi:10.1007/BFb0072398. ISBN978-3-540-13899-0.
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.