Chebyshev polynomials
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In [[mathematics]] the '''Chebyshev polynomials''', named after [[Pafnuty Chebyshev]],<ref>Chebyshev polynomials were first presented in: P. L. Chebyshev (1854) "Théorie des mécanismes connus sous le nom parallelogrammes," ''Mémoires des Savants étrangers présentes à l'Academie de Saint-Pétersbourg'', vol. 7, pages 539-586.</ref> are a [[polynomial sequence|sequence]] of [[orthogonal polynomials]] which are related to [[de Moivre's formula]] and which are easily defined [[recursive]]ly, like [[Fibonacci numbers|Fibonacci]] or [[Lucas number]]s. One usually distinguishes between '''Chebyshev polynomials of the first kind''' which are denoted ''T''<sub>''n''</sub> and '''Chebyshev polynomials of the second kind''' which are denoted ''U''<sub>''n''</sub>. The letter T is used because of the alternative [[transliteration]]s of the name ''Chebyshev'' as ''Tchebyshef'' or ''Tschebyscheff''.
The Chebyshev polynomials ''T''<sub>''n''</sub> or ''U''<sub>''n''</sub> 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 equation]]s they arise as the solution to the
[[Chebyshev equation|'''Chebyshev differential equations''']]
:<math>(1-x^2)\,y'' - x\,y' + n^2\,y = 0 \,\!</math>
and
:<math>(1-x^2)\,y'' - 3x\,y' + n(n+2)\,y = 0 \,\!</math>
for the polynomials of the first and second kind, respectively. These equations are special cases of the [[Sturm-Liouville problem|Sturm-Liouville differential equation]].
==Definition==
The '''Chebyshev polynomials of the first kind''' are defined by the [[recurrence relation]]
:<math>T_0(x) = 1 \,\!</math>
:<math>T_1(x) = x \,\!</math>
:<math>T_{n+1}(x) = 2xT_n(x) - T_{n-1}(x). \,\!</math>
One example of a [[generating function]] for T<sub>n</sub> is
:<math>\sum_{n=0}^{\infty}T_n(x) t^n = \frac{1-tx}{1-2tx+t^2}. \,\!</math>
The '''Chebyshev polynomials of the second kind''' are defined by the [[recurrence relation]]
:<math>U_0(x) = 1 \,\!</math>
:<math>U_1(x) = 2x \,\!</math>
:<math>U_{n+1}(x) = 2xU_n(x) - U_{n-1}(x). \,\!</math>
One example of a [[generating function]] for U<sub>n</sub> is
:<math>\sum_{n=0}^{\infty}U_n(x) t^n = \frac{1}{1-2tx+t^2}. \,\!</math>
===Trigonometric definition===
The Chebyshev polynomials of the first kind can be defined by the [[trigonometric identity]]:
:<math>T_n(x)=\cos(n \arccos x)=\cosh(n\,\mathrm{arccosh}\,x) \,\!</math>
whence:
:<math>T_n(\cos(\theta))=\cos(n\theta) \,\!</math>
for ''n'' = 0, 1, 2, 3, ..., while the polynomials of the second kind satisfy:
:<math> U_n(\cos(\theta)) = \frac{\sin((n+1)\theta)}{\sin\theta} \,\!</math>
which is structurally quite similar to the [[Dirichlet kernel]].
That cos(''nx'') is an ''n''th-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 cos<sup>2</sup>(''x'') + sin<sup>2</sup>(''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:
:<math>T_0(x)=\cos\ 0x\ =1 \,\!</math>
and:
:<math>T_1(\cos(x))=\cos\ (x) \,\!</math>
one can straightforwardly determine that:
:<math>
\cos(2 \theta)=2\cos\theta \cos\theta - \cos(0 \theta) = 2\cos^{2}\,\theta - 1 \,\!
</math>
:<math>
\cos(3 \theta)=2\cos\theta \cos(2\theta) - \cos\theta = 4\cos^3\,\theta - 3\cos\theta \,\!
</math>
and so forth. To trivially check whether the results seem reasonable, sum the coefficients on both sides of the equals sign (that is, setting theta 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")
:<math>T_n(T_m(x)) = T_{n\cdot m}(x).\,\!</math>
===Pell equation definition===
The Chebyshev polynomials can also be defined as the solutions to the [[Pell equation]]
:<math>T_i^2 - (x^2-1) U_{i-1}^2 = 1 \,\!</math>
in a ring R[''x''].<ref>Jeroen Demeyer [http://cage.ugent.be/~jdemeyer/phd.pdf Diophantine Sets over Polynomial Rings and Hilbert's Tenth Problem for Function Fields], Ph.D. theses (2007), p.70.</ref> Thus, they can be generated by the standard technique for Pell equations of taking powers of a fundamental solution:
:<math>T_i + U_{i-1} \sqrt{x^2-1} = (x + \sqrt{x^2-1})^i. \,\!</math>
===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
:<math>\frac{d}{dx} \, T_n(x) = n U_{n-1}(x) \mbox{ , } n=1,\ldots</math>
:<math>T_n(x) = \frac{1}{2} (U_n(x) - \, U_{n-2}(x)). </math>
:<math>T_{n+1}(x) = xT_n(x) - (1 - x^2)U_{n-1}(x)\,</math>
:<math>T_n(x) = U_n(x) - x \, U_{n-1}(x). </math>
The recurrence relationship of the derivative of Chebyshev polynomials can be derived from these relations
:<math>2 T_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</math>
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:
:<math>T_0(x) = 1\,\!</math>
:<math>U_{-1}(x) = 0\,\!</math>
:<math>T_{n+1}(x) = xT_n(x) - (1 - x^2)U_{n-1}(x)\,</math>
:<math>U_n(x) = xU_{n-1}(x) + T_n(x)\,</math>
These can be derived from the trigonometric formulae; for example, if <math>\scriptstyle x = \cos\vartheta</math>, then
:<math>\begin{align}
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{align}</math>
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''<sub>''n''</sub> (the polynomial of degree n) with ''U''<sub>''n''+1</sub> instead.
== Explicit formulas ==
Different approaches to defining Chebyshev polynomials lead to different explicit formulas such as:
:<math>T_n(x) =
\begin{cases}
\cos(n\arccos(x)), & \ x \in [-1,1] \\
\cosh(n \, \mathrm{arccosh}(x)), & \ x \ge 1 \\
(-1)^n \cosh(n \, \mathrm{arccosh}(-x)), & \ x \le -1 \\
\end{cases} \,\!
</math>
:<math>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}</math>
:<math>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}</math>
:<math>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)</math> (due to M. Hovdan)
==Properties==
===Orthogonality===
Both the ''T''<sub>''n''</sub> and the ''U''<sub>''n''</sub> form a sequence of [[orthogonal polynomials]]. The polynomials of the first kind are orthogonal with respect to the weight
:<math>\frac{1}{\sqrt{1-x^2}}, \,\!</math>
on the interval [−1,1], i.e. we have:
:<math>\int_{-1}^1 T_n(x)T_m(x)\,\frac{dx}{\sqrt{1-x^2}}=\left\{
\begin{matrix}
0 &: n\ne m~~~~~\\
\pi &: n=m=0\\
\pi/2 &: n=m\ne 0
\end{matrix}
\right. \,\!
</math>
This can be proven by letting ''x='' cos(θ) and using the identity
''T<sub>n</sub>'' (cos(θ))=cos(nθ). Similarly, the polynomials of the second kind are orthogonal with respect to the weight
:<math>\sqrt{1-x^2} \,\!</math>
on the interval [−1,1], i.e. we have:
:<math>\int_{-1}^1 U_n(x)U_m(x)\sqrt{1-x^2}\,dx =
\begin{cases}
0 &: n\ne m, \\
\pi/2 &: n=m.
\end{cases} \,\!
</math>
(Note that the weight <math>\sqrt{1-x^2} \,\!</math> is, to within a normalizing constant, the density of the [[Wigner semicircle distribution]]).
===Minimal ∞-norm===
For any given <math>n \ge 1</math>, among the polynomials of degree <math>n</math> with leading coefficient 1,
:<math>f(x) = \frac1{2^{n-1}}T_n(x)</math>
is the one of which the maximal absolute value on the interval <math>[-1, 1]</math> is minimal.
This maximal absolute value is
:<math>\frac1{2^{n-1}}</math>
and <math>|f(x)|</math> reaches this maximum exactly <math>n+1</math> times: at <math>x = cos \frac{k\pi}{n}</math> for <math>0 \le k \le n</math>
===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:
:<math>\frac{d T_n}{d x} = n U_{n - 1}\,</math>
:<math>\frac{d U_n}{d x} = \frac{(n + 1)T_{n + 1} - x U_n}{x^2 - 1}\,</math>
:<math>\frac{d^2 T_n}{d x^2} = n \frac{n T_n - x U_{n - 1}}{x^2 - 1} = n \frac{(n + 1)T_n - U_n}{x^2 - 1}.\,</math>
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 (see [[Chebyshev polynomials/Proofs|proof]]) that:
:<math>\frac{d^2 T_n}{d x^2} \Bigg|_{x = 1} \!\! = \frac{n^4 - n^2}{3},</math>
:<math>\frac{d^2 T_n}{d x^2} \Bigg|_{x = -1} \!\! = (-1)^n \frac{n^4 - n^2}{3};</math>
indeed, the following, more general formula holds:
:<math>\frac{d^p T_n}{d x^p} \Bigg|_{x = \pm 1} \!\! = (\pm 1)^{n+p}\prod_{k=0}^{p-1}\frac{n^2-k^2}{2k+1}.</math>
This latter result is of great use in the numerical solution of eigenvalue problems.
Concerning integration, the first derivative of the ''T''<sub>''n''</sub> implies that
:<math>\int U_n\, dx = \frac{T_{n + 1}}{n + 1}\,</math>
and the recurrence relation for the first kind polynomials involving derivatives establishes that
:<math>\int T_n\, dx = \frac{1}{2} \left(\frac{T_{n + 1}}{n + 1} - \frac{T_{n - 1}}{n - 1}\right) = \frac{n T_{n + 1}}{n^2 - 1} - \frac{x T_n}{n - 1}.\,</math>
===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
:<math>\cos\left(\frac{\pi}{2}\,(2k+1)\right)=0</math>
one can easily prove that the roots of ''T''<sub>''n''</sub> are
:<math> x_k = \cos\left(\frac{\pi}{2}\,\frac{2k-1}{n}\right) \mbox{ , } k=1,\ldots,n.</math>
Similarly, the roots of ''U''<sub>''n''</sub> are
:<math> x_k = \cos\left(\frac{k}{n+1}\pi\right) \mbox{ , } k=1,\ldots,n.</math>
One unique property of the Chebyshev polynomials of the first kind is that on the interval −1 ≤ ''x'' ≤ 1 all of the [[Maxima and minima|extrema]] have values that are either −1 or 1. Thus these polynomials have only two finite [[critical value]]s, the defining property of [[Shabat polynomial]]s. Both the first and second kinds of Chebyshev polynomial have extrema at the endpoints, given by:
:<math>T_n(1) = 1\,</math>
:<math>T_n(-1) = (-1)^n\,</math>
:<math>U_n(1) = n + 1\,</math>
:<math>U_n(-1) = (n + 1)(-1)^n\,</math>
===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''<sub>''n''</sub>(''x'') and ''U''<sub>''n''</sub>(''x'') are both polynomials of degree ''n''.
They are [[Even and odd functions|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''<sub>''n''</sub> is 2<sup>''n'' − 1</sup> if 1 ≤ ''n'', but 1 if 0 = ''n''.
''T''<sub>''n''</sub> are a special case of [[Lissajous curve]]s with frequency ratio to equal to ''n''.
==Examples==
[[Image:chebyshev.png|frame|This image shows the first few Chebyshev polynomials of the first kind in the domain −1¼ < ''x'' < 1¼, −1¼ < ''y'' < 1¼; the flat <font color=#333333>''T''<sub>0</sub></font>, and <font color=#b30000>''T''<sub>1</sub></font>, <font color=#0000b3>''T''<sub>2</sub></font>, <font color=#00b300>''T''<sub>3</sub></font>, <font color=#b3b300>''T''<sub>4</sub></font> and <font color=#b3b3b3>''T''<sub>5</sub></font>.]]
The first few Chebyshev polynomials of the first kind are
:<math> T_0(x) = 1 \,</math>
:<math> T_1(x) = x \,</math>
:<math> T_2(x) = 2x^2 - 1 \,</math>
:<math> T_3(x) = 4x^3 - 3x \,</math>
:<math> T_4(x) = 8x^4 - 8x^2 + 1 \,</math>
:<math> T_5(x) = 16x^5 - 20x^3 + 5x \,</math>
:<math> T_6(x) = 32x^6 - 48x^4 + 18x^2 - 1 \,</math>
:<math> T_7(x) = 64x^7 - 112x^5 + 56x^3 - 7x \,</math>
:<math> T_8(x) = 128x^8 - 256x^6 + 160x^4 - 32x^2 + 1 \,</math>
:<math> T_9(x) = 256x^9 - 576x^7 + 432x^5 - 120x^3 + 9x. \,</math>
[[Image:chebyshev2.png|frame|This image shows the first few Chebyshev polynomials of the second kind in the domain −1¼ < ''x'' < 1¼, −1¼ < ''y'' < 1¼; the flat <font color=#333333>''U''<sub>0</sub></font>, and <font color=#b30000>''U''<sub>1</sub></font>, <font color=#0000b3>''U''<sub>2</sub></font>, <font color=#00b300>''U''<sub>3</sub></font>, <font color=#b3b300>''U''<sub>4</sub></font> and <font color=#b3b3b3>''U''<sub>5</sub></font>. Although not visible in the image, ''U''<sub>''n''</sub>(1) = ''n'' + 1 and ''U''<sub>''n''</sub>(−1) = (''n'' + 1)(−1)<sup>''n''</sup>.]]
The first few Chebyshev polynomials of the second kind are
:<math> U_0(x) = 1 \,</math>
:<math> U_1(x) = 2x \,</math>
:<math> U_2(x) = 4x^2 - 1 \,</math>
:<math> U_3(x) = 8x^3 - 4x \,</math>
:<math> U_4(x) = 16x^4 - 12x^2 + 1 \,</math>
:<math> U_5(x) = 32x^5 - 32x^3 + 6x \,</math>
:<math> U_6(x) = 64x^6 - 80x^4 + 24x^2 - 1 \,</math>
:<math> U_7(x) = 128x^7 - 192x^5 + 80x^3 - 8x \,</math>
:<math> U_8(x) = 256x^8 - 448 x^6 + 240 x^4 - 40 x^2 + 1 \,</math>
:<math> U_9(x) = 512x^9 - 1024 x^7 + 672 x^5 - 160 x^3 + 10 x. \,</math>
==As a basis set==
[[Image:ChebyshevExpansion.png|thumb|right|240px|The non-smooth function (top) <math>y = -x^3 H(-x)</math>, where <math>H</math> is the [[Heaviside step function]], and (bottom) the 5<sup>th</sup> partial sum of its Chebyshev expansion. The 7<sup>th</sup> 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:<ref name = boyd>[http://www.math.ucl.ac.be/membres/magnus/num1a/boyd01.pdf Chebyshev and Fourier Spectral Methods] by John P. Boyd.</ref>
:<math>f(x) = \sum_{n = 0}^\infty a_n T_n(x).</math>
Furthermore, as mentioned previously, the Chebyshev polynomials form an [[orthogonal]] basis which (among other things) implies that the coefficients ''a''<sub>''n''</sub> 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.<ref name=boyd/> These attributes include:
* The Chebyshev polynomials form a [[complete]] orthogonal system.
* The Chebyshev series converges to ''f''(''x'') if the function is [[piecewise]] [[Smooth function|smooth]] and [[Continuous function|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]],<ref name=boyd/> 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
:<math>f(x) = \sum_{n = 0}^\infty a_n T_n(x)</math>
are very useful in the [[approximation theory|approximation]] of various functions and in the solution of [[differential equation]]s (see [[spectral method]]). Two common methods for determining the coefficients ''a''<sub>''n''</sub> are through the use of the [[inner product]] as in [[Galerkin's method]] and through the use of [[collocation method|collocation]] which is related to [[interpolation]].
As an interpolant, the ''N'' coefficients of the (''N'' − 1)<sup>th</sup> partial sum are usually obtained on the Chebyshev-Gauss-Lobatto<ref>[http://www.joma.org/images/upload_library/4/vol6/Sarra/Chebyshev.html Chebyshev Interpolation: An Interactive Tour]</ref> 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:
:<math>x_i = -\cos\left(\frac{i \pi}{N - 1}\right) ; \qquad \ i = 0, 1, \dots, N - 1.</math>
===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
:<math>p(x) = \sum_{n=0}^{N} a_n T_n(x)</math>
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==
*[[Chebyshev nodes]]
*[[Chebyshev filter]]
*[[Chebyshev cube root]]
*[[Dickson polynomials]]
*[[Legendre polynomials]]
*[[Hermite polynomials]]
*[[Chebyshev rational functions]]
*[[Clenshaw–Curtis quadrature]]
*[[Approximation theory]]
==References==
* {{Abramowitz_Stegun_ref|22|773}}
<references/>
==External links==
* {{MathWorld|urlname=ChebyshevPolynomialoftheFirstKind|title=Chebyshev Polynomial of the First Kind}}
* [http://math.fullerton.edu/mathews/n2003/ChebyshevPolyMod.html Module for Chebyshev Polynomials by John H. Mathews]
* [http://www.joma.org/images/upload_library/4/vol6/Sarra/Chebyshev.html Chebyshev Interpolation: An Interactive Tour], includes illustrative [[Java applet]].
* [http://www.comlab.ox.ac.uk/projects/chebfun/home.html chebfun project], representing functions by automatic Chebyshev polynomial interpolation in [[MATLAB]].
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