Chebyshev polynomials

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The Chebyshev polynomials are two sequences of 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, define 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]

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[edit]

Recurrence definition[edit]

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

and the exponential generating function is

Trigonometric definition[edit]

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):

(The Dirichlet kernel, in fact, coincides with what is now known as the Chebyshev polynomial of the fourth 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 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 factor of sin x is factored out, the remaining factors can be replaced to create a (n−1)st-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 integer multiple of an angle solely in terms of the cosine of the base angle.

The first two Chebyshev polynomials of the first kind are computed directly from the definition to be

and

while the rest may be evaluated using a specialization of the product-to-sum identity

as, for example,

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 summation symbol indicates that the contribution of j = 0 needs to be halved if it appears, and .

An immediate corollary is the expression of complex exponentiation in terms of Chebyshev polynomials: given z = a + bi,

Commuting polynomials definition[edit]

Chebyshev polynomials can also be characterized by the following theorem:[3]

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[edit]

The Chebyshev polynomials can also be defined as the solutions to the Pell equation

in a ring R[x].[4] 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[edit]

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:[5]

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[6]

The integral relations are[5]: 187(47)(48) [7]

where integrals are considered as principal value.

Explicit expressions[edit]

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

with inverse[8][9]

where the prime at the summation symbol indicates that the contribution of j = 0 needs to be halved if it appears.

where 2F1 is a hypergeometric function.

Properties[edit]

Symmetry[edit]

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[edit]

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.

Specifically,[10][11] 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 .

This result has been generalized to solutions of ,[11] and to and for Chebyshev polynomials of the third and fourth kinds, respectively.[12]

Differentiation and integration[edit]

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 values – taking the limit as x → 1:

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[edit]

The Chebyshev polynomials of the first kind satisfy the relation

which is easily proved from the product-to-sum formula for the cosine,

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 mn. 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[edit]

The trigonometric definitions of Tn and Un imply the composition or nesting properties[13]

For Tmn the order of composition may be reversed, making the family of polynomial functions Tn a commutative semigroup 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[edit]

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.)


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. (Another way to define the Chebyshev polynomials is as the solutions to those equations.)


The Tn also satisfy a discrete orthogonality condition:

where N is any integer greater than max(i, j),[14] 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[edit]

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[edit]

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[edit]

The Chebyshev polynomials are a special case of the ultraspherical or Gegenbauer polynomials , which themselves are a special case of the Jacobi polynomials :

Chebyshev polynomials are also a special case of Dickson polynomials:

In particular, when , they are related by and .

Other properties[edit]

The curves given by y = Tn(x), or equivalently, by the parametric equations y = Tn(cos θ) = cos , 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 = e.

Examples[edit]

First kind[edit]

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[edit]

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[edit]

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:[15]

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.[15] 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,[15] often in favor of trigonometric series due to generally faster convergence for continuous functions (Gibbs' phenomenon is still a problem).

Example 1[edit]

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[edit]

To provide another example:

Partial sums[edit]

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)st partial sum are usually obtained on the Chebyshev–Gauss–Lobatto[16] 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[edit]

An arbitrary polynomial of degree N can be written in terms of the Chebyshev polynomials of the first kind.[17] 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[edit]

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's Opera 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 polynomials Ln 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

and are proportional to Jacobi polynomials with

[22]

All four families satisfy the recurrence with , where , , , or , but they differ according to whether equals , , , or .[21]

See also[edit]

References[edit]

  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. ^ Ritt, J.F. (1922). "Prime and Composite Polynomials". Trans. Amer. Math. Soc. pp. 51–66. doi:10.1090/S0002-9947-1922-1501189-9.
  4. ^ 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.
  5. ^ a b Erdélyi, Arthur; Magnus, (Hans Heinrich) Wilhelm; Oberhettinger, Fritz; Tricomi, Francesco Giacomo; Bertin, David; Fulks, Watson B.; Harvey, Albert Raymond; Thomsen, Jr., Donald L.; Weber, Maria A.; Whitney, Eoin Laird; Stampfel, Rosemarie (1953). Erdélyi, Arthur (ed.). Higher Transcendental Functions - Volume II - Based, in part, on notes left by Harry Bateman (PDF). Bateman Manuscript Project. Vol. II (1 ed.). New York / Toronto / London: McGraw-Hill Book Company, Inc. p. 184:(3),(4). LCCN 53-5555. Contract No. N6onr-244 Task Order XIV. Project Designation Number: NR 043-045. Order No. 19546. Archived (PDF) from the original on 9 April 2017. Retrieved 23 July 2020. [1][2] (xvii+1 errata page+396 pages, red cloth hardcover) (NB. Copyright was renewed by California Institute of Technology in 1981.); Reprint: Robert E. Krieger Publishing Co., Inc., Melbourne, Florida, USA. 1981. ISBN 0-89874-069-X; Planned Dover reprint: ISBN 0-486-44615-8.
  6. ^ 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, MR 0040487
  7. ^ Mason, J. C.; Handscomb, David C. (17 September 2002). Chebyshev Polynomials. CRC Press. ISBN 978-1-4200-3611-4.
  8. ^ Cody, W. J. (1970). "A survey of practical rational and polynomial approximation of functions". SIAM Review. 12 (3): 400–423. doi:10.1137/1012082.
  9. ^ 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.
  10. ^ 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.
  11. ^ a b Wolfram, 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.
  12. ^ Wolfram, D. A. (2022). "Factoring Chebyshev polynomials with minimal polynomials of ". Bulletin of the Australian Mathematical Society. doi:10.1017/S0004972722000235.
  13. ^ 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
  14. ^ For proof, see: Mason, J.C. & Handscomb, D.C. (2002). Chebyshev Polynomials. Taylor & Francis.
  15. ^ 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.
  16. ^ "Chebyshev Interpolation: An Interactive Tour". Archived from the original on 18 March 2017. Retrieved 2 June 2016.
  17. ^ For more information on the coefficients, see Mason, J. C.; Handscomb, D. C. (2002). Chebyshev Polynomials. Taylor & Francis.
  18. ^ a b Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 22". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 778. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.
  19. ^ Horadam, A. F. (2002), "Vieta polynomials" (PDF), Fibonacci Quarterly, 40 (3): 223–232
  20. ^ Viète, François (1646). Francisci Vietae Opera mathematica : in unum volumen congesta ac recognita / opera atque studio Francisci a Schooten (PDF). Bibliothèque nationale de France.
  21. ^ a b c Mason, 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
  22. ^ a b Desmarais, Robert N.; Bland, Samuel R. (1995), "Tables of properties of airfoil polynomials", NASA Reference Publication 1343, National Aeronautics and Space Administration

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