Chebyshev polynomials

In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of orthogonal polynomials which are related to de Moivre's formula and which are easily defined recursively, like Fibonacci or Lucas numbers. One usually distinguishes between Chebyshev polynomials of the first kind which are denoted T_n and Chebyshev polynomials of the second kind which are denoted U_n. The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebyshef or Tschebyscheff.

The Chebyshev polynomials T_n or U_n are polynomials of degree n and the sequence of Chebyshev polynomials of either kind composes a polynomial sequence.

Chebyshev polynomials are important in approximation theory because the roots of the Chebyshev polynomials of the first kind, 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.

In the study of differential equations they arise as the solution to the Chebyshev differential equations

$(1-x^{2})\,y''-x\,y'+n^{2}\,y=0\,\!$

and

$(1-x^{2})\,y''-3x\,y'+n(n+2)\,y=0\,\!$

for the polynomials of the first and second kind, respectively. These equations are special cases of the Sturm-Liouville differential equation.

Definition

The Chebyshev polynomials of the first kind are defined by the recurrence relation

$T_{0}(x)=1\,\!$

$T_{1}(x)=x\,\!$

$T_{n+1}(x)=2xT_{n}(x)-T_{n-1}(x).\,\!$

One example of a generating function for T_n is

$\sum _{n=0}^{\infty }T_{n}(x)t^{n}={\frac {1-tx}{1-2tx+t^{2}}}.\,\!$

The Chebyshev polynomials of the second kind are defined by the recurrence relation

$U_{0}(x)=1\,\!$

$U_{1}(x)=2x\,\!$

$U_{n+1}(x)=2xU_{n}(x)-U_{n-1}(x).\,\!$

One example of a generating function for U_n is

$\sum _{n=0}^{\infty }U_{n}(x)t^{n}={\frac {1}{1-2tx+t^{2}}}.\,\!$

Trigonometric definition

The Chebyshev polynomials of the first kind can be defined by the trigonometric identity:

$T_{n}(x)=\cos(n\arccos x)=\cosh(n\,\mathrm {arccosh} \,x)\,\!$

whence:

$T_{n}(\cos(\vartheta ))=\cos(n\vartheta )\,\!$

for n = 0, 1, 2, 3, ..., while the polynomials of the second kind satisfy:

$U_{n}(\cos(\vartheta ))={\frac {\sin((n+1)\vartheta )}{\sin \vartheta }}\,\!$

which is structurally quite similar to the Dirichlet kernel.

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, and 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 via the identity cos²(x) + sin²(x) = 1.

This identity is extremely 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:

$T_{0}(x)=\cos \ 0x\ =1\,\!$

and:

$T_{1}(\cos(x))=\cos \ (x)\,\!$

one can straightforwardly determine that:

$\cos(2\vartheta )=2\cos \vartheta \cos \vartheta -\cos(0\vartheta )=2\cos ^{2}\,\vartheta -1\,\!$

$\cos(3\vartheta )=2\cos \vartheta \cos(2\vartheta )-\cos \vartheta =4\cos ^{3}\,\vartheta -3\cos \vartheta \,\!$

and so forth. To trivially check whether the results seem reasonable, sum the coefficients on both sides of the equals sign (that is, setting $\vartheta$ equal to zero, for which the cosine is unity), and one sees that 1 = 2 − 1 in the former expression and 1 = 4 − 3 in the latter.

An immediate corollary is the composition identity (or the "nesting property")

$T_{n}(T_{m}(x))=T_{n\cdot m}(x).\,\!$

Pell equation definition

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

$T_{i}^{2}-(x^{2}-1)U_{i-1}^{2}=1\,\!$

in a ring R[x]. Thus, they can be generated by the standard technique for Pell equations of taking powers of a fundamental solution:

$T_{i}+U_{i-1}{\sqrt {x^{2}-1}}=(x+{\sqrt {x^{2}-1}})^{i}.\,\!$

Relation between Chebyshev polynomials of the first and second kind

The Chebyshev polynomials of the first and second kind are closely related by the following equations

${\frac {d}{dx}}\,T_{n}(x)=nU_{n-1}(x){\mbox{ , }}n=1,\ldots$

$T_{n}(x)={\frac {1}{2}}(U_{n}(x)-\,U_{n-2}(x)).$

$T_{n+1}(x)=xT_{n}(x)-(1-x^{2})U_{n-1}(x)\,$

$T_{n}(x)=U_{n}(x)-x\,U_{n-1}(x).$

The recurrence relationship of the derivative of Chebyshev polynomials can be derived from these relations

$2T_{n}(x)={\frac {1}{n+1}}\;{\frac {d}{dx}}T_{n+1}(x)-{\frac {1}{n-1}}\;{\frac {d}{dx}}T_{n-1}(x){\mbox{ , }}\quad n=1,\ldots$

This relationship is used in the Chebyshev spectral method of solving differential equations.

Equivalently, the two sequences can also be defined from a pair of mutual recurrence equations:

$T_{0}(x)=1\,\!$

$U_{-1}(x)=0\,\!$

$T_{n+1}(x)=xT_{n}(x)-(1-x^{2})U_{n-1}(x)\,$

$U_{n}(x)=xU_{n-1}(x)+T_{n}(x)\,$

These can be derived from the trigonometric formulae; for example, if $x=\cos \vartheta$, then

${\begin{aligned}T_{n+1}(x)&=T_{n+1}(\cos(\vartheta ))\\&=\cos((n+1)\vartheta )\\&=\cos(n\vartheta )\cos(\vartheta )-\sin(n\vartheta )\sin(\vartheta )\\&=T_{n}(\cos(\vartheta ))\cos(\vartheta )-U_{n-1}(\cos(\vartheta ))\sin ^{2}(\vartheta )\\&=xT_{n}(x)-(1-x^{2})U_{n-1}(x).\\\end{aligned}}$

Note that both these equations and the trigonometric equations take a simpler form if we, like some works, follow the alternate convention of denoting our U_n (the polynomial of degree n) with U_n+1 instead.

Explicit formulas

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

$T_{n}(x)={\begin{cases}\cos(n\arccos(x)),&\ x\in [-1,1]\\\cosh(n\,\mathrm {arccosh} (x)),&\ x\geq 1\\(-1)^{n}\cosh(n\,\mathrm {arccosh} (-x)),&\ x\leq -1\\\end{cases}}\,\!$

$T_{n}(x)={\frac {(x+{\sqrt {x^{2}-1}})^{n}+(x-{\sqrt {x^{2}-1}})^{n}}{2}}=\sum _{k=0}^{\lfloor n/2\rfloor }{\binom {n}{2k}}(x^{2}-1)^{k}x^{n-2k}$

$U_{n}(x)={\frac {(x+{\sqrt {x^{2}-1}})^{n+1}-(x-{\sqrt {x^{2}-1}})^{n+1}}{2{\sqrt {x^{2}-1}}}}=\sum _{k=0}^{\lfloor n/2\rfloor }{\binom {n+1}{2k+1}}(x^{2}-1)^{k}x^{n-2k}$

$T_{n}(x)=1+n^{2}{(x-1)}\prod _{k=1}^{n-1}\left({1+{{x-1} \over 2\sin ^{2}\left({k\pi \over n}\right)}}\right)$ (due to M. Hovdan)

Properties

Orthogonality

Both the T_n and the U_n form a sequence of orthogonal polynomials. The polynomials of the first kind are orthogonal with respect to the weight

${\frac {1}{\sqrt {1-x^{2}}}},\,\!$

on the interval [−1,1], i.e. we have:

$\int _{-1}^{1}T_{n}(x)T_{m}(x)\,{\frac {dx}{\sqrt {1-x^{2}}}}=\left\{{\begin{matrix}0&:n\neq m~~~~~\\\pi &:n=m=0\\\pi /2&:n=m\neq 0\end{matrix}}\right.\,\!$

This can be proven by letting $x=\cos(\vartheta )$ and using the identity $T_{n}(\cos(\vartheta ))=\cos(n\vartheta )$. Similarly, the polynomials of the second kind are orthogonal with respect to the weight

${\sqrt {1-x^{2}}}\,\!$

on the interval [−1,1], i.e. we have:

$\int _{-1}^{1}U_{n}(x)U_{m}(x){\sqrt {1-x^{2}}}\,dx={\begin{cases}0&:n\neq m,\\\pi /2&:n=m.\end{cases}}\,\!$

(Note that the weight ${\sqrt {1-x^{2}}}\,\!$ is, to within a normalizing constant, the density of the Wigner semicircle distribution).

Minimal ∞-norm

For any given n ≥ 1, among the polynomials of degree n with leading coefficient 1,

$f(x)={\frac {1}{2^{n-1}}}T_{n}(x)$

is the one of which the maximal absolute value on the interval [−1, 1] is minimal.

This maximal absolute value is

${\frac {1}{2^{n-1}}}$

and |ƒ(x)| reaches this maximum exactly n + 1 times: at

$x=\cos {\frac {k\pi }{n}}{\text{ for }}0\leq k\leq n.$

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:

${\frac {dT_{n}}{dx}}=nU_{n-1}\,$

${\frac {dU_{n}}{dx}}={\frac {(n+1)T_{n+1}-xU_{n}}{x^{2}-1}}\,$

${\frac {d^{2}T_{n}}{dx^{2}}}=n{\frac {nT_{n}-xU_{n-1}}{x^{2}-1}}=n{\frac {(n+1)T_{n}-U_{n}}{x^{2}-1}}.\,$

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:

${\frac {d^{2}T_{n}}{dx^{2}}}{\Bigg |}_{x=1}\!\!={\frac {n^{4}-n^{2}}{3}},$

${\frac {d^{2}T_{n}}{dx^{2}}}{\Bigg |}_{x=-1}\!\!=(-1)^{n}{\frac {n^{4}-n^{2}}{3}}.$

Proof

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

$T''_{n}=n{\frac {nT_{n}-xU_{n-1}}{x^{2}-1}}$

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:

$T''_{n}(1)=\lim _{x\to 1}n{\frac {nT_{n}-xU_{n-1}}{x^{2}-1}}$

where only $x=1$ is considered for now. Factoring the denominator:

$T''_{n}(1)=\lim _{x\to 1}n{\frac {nT_{n}-xU_{n-1}}{(x+1)(x-1)}}=\lim _{x\to 1}n{\frac {\frac {nT_{n}-xU_{n-1}}{x-1}}{x+1}}.$

Since the limit as a whole must exist, the limit of the numerator and denominator must independently exist, and

$T''_{n}(1)=n{\frac {\lim _{x\to 1}{\frac {nT_{n}-xU_{n-1}}{x-1}}}{\lim _{x\to 1}(x+1)}}={\frac {n}{2}}\lim _{x\to 1}{\frac {nT_{n}-xU_{n-1}}{x-1}}.$

The denominator (still) limits to zero, which implies that the numerator must be limiting to zero, ie $U_{n-1}(1)=nT_{n}(1)=n$ which will be useful later on. Since the numerator and denominator are both limiting to zero, L'Hôpital's rule applies:

${\begin{aligned}T''_{n}(1)&={\frac {n}{2}}\lim _{x\to 1}{\frac {{\frac {d}{dx}}(nT_{n}-xU_{n-1})}{{\frac {d}{dx}}(x-1)}}\\&={\frac {n}{2}}\lim _{x\to 1}{\frac {d}{dx}}(nT_{n}-xU_{n-1})\\&={\frac {n}{2}}\lim _{x\to 1}\left(n^{2}U_{n-1}-U_{n-1}-x{\frac {d}{dx}}(U_{n-1})\right)\\&={\frac {n}{2}}\left(n^{2}U_{n-1}(1)-U_{n-1}(1)-\lim _{x\to 1}x{\frac {d}{dx}}(U_{n-1})\right)\\&={\frac {n^{4}}{2}}-{\frac {n^{2}}{2}}-{\frac {1}{2}}\lim _{x\to 1}{\frac {d}{dx}}(nU_{n-1})\\&={\frac {n^{4}}{2}}-{\frac {n^{2}}{2}}-{\frac {T''_{n}(1)}{2}}\\T''_{n}(1)&={\frac {n^{4}-n^{2}}{3}}.\\\end{aligned}}$

The proof for $x=-1$ is similar, with the fact that $T_{n}(-1)=(-1)^{n}$ being important.

Indeed, the following, more general formula holds:

${\frac {d^{p}T_{n}}{dx^{p}}}{\Bigg |}_{x=\pm 1}\!\!=(\pm 1)^{n+p}\prod _{k=0}^{p-1}{\frac {n^{2}-k^{2}}{2k+1}}.$

This latter result is of great use in the numerical solution of eigenvalue problems.

Concerning integration, the first derivative of the T_n implies that

$\int U_{n}\,dx={\frac {T_{n+1}}{n+1}}\,$

and the recurrence relation for the first kind polynomials involving derivatives establishes that

$\int T_{n}\,dx={\frac {1}{2}}\left({\frac {T_{n+1}}{n+1}}-{\frac {T_{n-1}}{n-1}}\right)={\frac {nT_{n+1}}{n^{2}-1}}-{\frac {xT_{n}}{n-1}}.\,$

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 are sometimes called Chebyshev nodes because they are used as nodes in polynomial interpolation. Using the trigonometric definition and the fact that

$\cos \left({\frac {\pi }{2}}\,(2k+1)\right)=0$

one can easily prove that the roots of T_n are

$x_{k}=\cos \left({\frac {\pi }{2}}\,{\frac {2k-1}{n}}\right),\quad k=1,\ldots ,n.$

Similarly, the roots of U_n are

$x_{k}=\cos \left({\frac {k}{n+1}}\pi \right),\quad k=1,\ldots ,n.$

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:

$T_{n}(1)=1\,$

$T_{n}(-1)=(-1)^{n}\,$

$U_{n}(1)=n+1\,$

$U_{n}(-1)=(n+1)(-1)^{n}.\,$

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, T_n(x) and U_n(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.

The leading coefficient of T_n is 2^n − 1 if 1 ≤ n, but 1 if 0 = n.

T_n are a special case of Lissajous curves with frequency ratio to equal to n. Several polynomil sequences like Lucas polynomials , Dickson polynomials, Fibonacci polynomials and Boubaker polynomials are related to Chebyshev polynomials T_n and U_n .i e.:

$B_{n}(x)={\frac {4x}{n}}{\frac {d}{dx}}T_{n}(x)-2T_{n}(x)\,$

$B_{n}(x)=-2T_{n}(x)+4xU_{n-1}(x)\,$

Examples

This image shows the first few Chebyshev polynomials of the first kind in the domain −1¼ < x < 1¼, −1¼ < y < 1¼; the flat T₀, and T₁, T₂, T₃, T₄ and T₅.

The first few Chebyshev polynomials of the first kind are

$T_{0}(x)=1\,$

$T_{1}(x)=x\,$

$T_{2}(x)=2x^{2}-1\,$

$T_{3}(x)=4x^{3}-3x\,$

$T_{4}(x)=8x^{4}-8x^{2}+1\,$

$T_{5}(x)=16x^{5}-20x^{3}+5x\,$

$T_{6}(x)=32x^{6}-48x^{4}+18x^{2}-1\,$

$T_{7}(x)=64x^{7}-112x^{5}+56x^{3}-7x\,$

$T_{8}(x)=128x^{8}-256x^{6}+160x^{4}-32x^{2}+1\,$

$T_{9}(x)=256x^{9}-576x^{7}+432x^{5}-120x^{3}+9x.\,$

This image shows the first few Chebyshev polynomials of the second kind in the domain −1¼ < x < 1¼, −1¼ < y < 1¼; the flat U₀, and U₁, U₂, U₃, U₄ and U₅. Although not visible in the image, U_n(1) = n + 1 and U_n(−1) = (n + 1)(−1)ⁿ.

The first few Chebyshev polynomials of the second kind are

$U_{0}(x)=1\,$

$U_{1}(x)=2x\,$

$U_{2}(x)=4x^{2}-1\,$

$U_{3}(x)=8x^{3}-4x\,$

$U_{4}(x)=16x^{4}-12x^{2}+1\,$

$U_{5}(x)=32x^{5}-32x^{3}+6x\,$

$U_{6}(x)=64x^{6}-80x^{4}+24x^{2}-1\,$

$U_{7}(x)=128x^{7}-192x^{5}+80x^{3}-8x\,$

$U_{8}(x)=256x^{8}-448x^{6}+240x^{4}-40x^{2}+1\,$

$U_{9}(x)=512x^{9}-1024x^{7}+672x^{5}-160x^{3}+10x.\,$

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 a complete basis set, so that a function in the same space can, on −1 ≤ x ≤ 1 be expressed via the expansion:

$f(x)=\sum _{n=0}^{\infty }a_{n}T_{n}(x).$

Furthermore, as mentioned previously, the Chebyshev polynomials form an orthogonal basis which (among other things) implies that the coefficients a_n 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 ƒ(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 ƒ(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).

Partial sums

The partial sums of

$f(x)=\sum _{n=0}^{\infty }a_{n}T_{n}(x)$

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 a_n 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:

$x_{i}=-\cos \left({\frac {i\pi }{N-1}}\right);\qquad \ i=0,1,\dots ,N-1.$

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

$p(x)=\sum _{n=0}^{N}a_{n}T_{n}(x).$

Polynomials in Chebyshev form can be evaluated using the Clenshaw algorithm.

Spread polynomials

The spread polynomials are in a sense equivalent to the Chebyshev polynomials of the first kind, but enable one to avoid square roots and conventional trigonometric functions in certain contexts, notably in rational trigonometry.

See also

• Boubaker polynomials
• Chebyshev nodes
• Chebyshev filter
• Chebyshev cube root
• Dickson polynomials
• Legendre polynomials
• Hermite polynomials
• Chebyshev rational functions
• Clenshaw–Curtis quadrature
• Approximation theory

References

• 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. 773. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.

External links

• Weisstein, Eric W. "Chebyshev Polynomial of the First Kind". MathWorld.
• Module for Chebyshev Polynomials by John H. Mathews
• Chebyshev Interpolation: An Interactive Tour, includes illustrative Java applet.
• chebfun project, representing functions by automatic Chebyshev polynomial interpolation in MATLAB.