Template:Short description Template:Distinguish Template:Use American English Template:Use dmy dates
The Chebyshev polynomials are two sequences of orthogonal polynomials related to the cosine and sine functions, notated as <math>T_n(x)</math> and <math>U_n(x)</math>. They can be defined in several equivalent ways, one of which starts with trigonometric functions:
The Chebyshev polynomials of the first kind <math>T_n</math> are defined by <math display="block">T_n(\cos \theta) = \cos(n\theta).</math>
Similarly, the Chebyshev polynomials of the second kind <math>U_n</math> are defined by <math display="block">U_n(\cos \theta) \sin \theta = \sin\big((n + 1)\theta\big).</math>
That these expressions define polynomials in <math>\cos\theta</math> is not obvious at first sight but can be shown using de Moivre's formula (see below).
The Chebyshev polynomials Template:Math are polynomials with the largest possible leading coefficient whose absolute value on the interval Template:Closed-closed is bounded by 1. They are also the "extremal" polynomials for many other properties.<ref>Template:Cite book</ref>
In 1952, Cornelius Lanczos showed that the Chebyshev polynomials are important in approximation theory for the solution of linear systems;<ref>Template:Cite journal</ref> the roots of Template:Math, 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.<ref>Chebyshev first presented his eponymous polynomials in a paper read before the St. Petersburg Academy in 1853: Template:Pb Template:Cite journal Also published separately as Template:Cite book</ref> The letter Template:Mvar is used because of the alternative transliterations of the name Chebyshev as {{#invoke:Lang|lang}}, {{#invoke:Lang|lang}} (French) or {{#invoke:Lang|lang}} (German).
DefinitionsEdit
Recurrence definitionEdit
The Chebyshev polynomials of the first kind can be defined by the recurrence relation <math display="block">\begin{align}
T_0(x) & = 1, \\
T_1(x) & = x, \\
T_{n+1}(x) & = 2 x\,T_n(x) - T_{n-1}(x).
\end{align}</math>
The Chebyshev polynomials of the second kind can be defined by the recurrence relation <math display="block">\begin{align}
U_0(x) & = 1, \\
U_1(x) & = 2 x, \\
U_{n+1}(x) & = 2 x\,U_n(x) - U_{n-1}(x),
\end{align}</math> which differs from the above only by the rule for n=1.
Trigonometric definitionEdit
The Chebyshev polynomials of the first and second kind can be defined as the unique polynomials satisfying <math display="block">T_n(\cos\theta) = \cos(n\theta)</math> and <math display="block">U_n(\cos\theta) = \frac{\sin\big((n + 1)\theta\big)}{\sin\theta},</math> for Template:Math.
An equivalent way to state this is via exponentiation of a complex number: given a complex number Template:Math with absolute value of one, <math display="block">z^n = T_n(a) + ib U_{n-1}(a).</math> Chebyshev polynomials can be defined in this form when studying trigonometric polynomials.<ref>Template:Cite journal</ref>
That Template:Math is an Template:Mvarth-degree polynomial in Template:Math can be seen by observing that Template:Math is the real part of one side of de Moivre's formula: <math display="block">\cos n \theta + i \sin n \theta = (\cos \theta + i \sin \theta)^n.</math> The real part of the other side is a polynomial in Template:Math and Template:Math, in which all powers of Template:Math are even and thus replaceable through the identity Template:Math. By the same reasoning, Template:Math is the imaginary part of the polynomial, in which all powers of Template:Math are odd and thus, if one factor of Template:Math is factored out, the remaining factors can be replaced to create a Template:Mathst-degree polynomial in Template:Math.
For x outside the interval [-1,1], the above definition implies <math display="block">T_n(x) = \begin{cases}
\cos(n \arccos x) & \text{ if }~ |x| \le 1, \\
\cosh(n \operatorname{arcosh} x) & \text{ if }~ x \ge 1, \\
(-1)^n \cosh(n \operatorname{arcosh}(-x) ) & \text{ if }~ x \le -1.
\end{cases}</math>
Commuting polynomials definitionEdit
Chebyshev polynomials can also be characterized by the following theorem:<ref>Template:Cite journal</ref>
If <math> F_n(x)</math> is a family of monic polynomials with coefficients in a field of characteristic <math>0</math> such that <math> \deg F_n(x) = n</math> and <math> F_m(F_n(x)) = F_n(F_m(x))</math> for all <math>m</math> and <math> n</math>, then, up to a simple change of variables, either <math> F_n(x) = x^n</math> for all <math> n</math> or <math>F_n(x) = 2\cdot T_n(x/2)</math> for all <math> n</math>.
Pell equation definitionEdit
The Chebyshev polynomials can also be defined as the solutions to the Pell equation: <math display="block">T_n(x)^2 - \left(x^2 - 1\right) U_{n-1}(x)^2 = 1</math> in a ring Template:Math.<ref>Template:Cite thesis</ref> Thus, they can be generated by the standard technique for Pell equations of taking powers of a fundamental solution: <math display="block">T_n(x) + U_{n-1}(x)\,\sqrt{x^2-1} = \left(x + \sqrt{x^2-1}\right)^n~. </math>
Generating functionsEdit
The ordinary generating function for Template:Mvar is <math display="block">\sum_{n=0}^\infty T_n(x)\,t^n = \frac{1 - tx}{1 - 2tx + t^2}.</math>
There are several other generating functions for the Chebyshev polynomials; the exponential generating function is <math display="block">\begin{align} \sum_{n=0}^\infty T_n(x) \frac{t^n}{n!}
&= {\tfrac{1}{2}} \Bigl({\exp}\Bigl({\textstyle t\bigl(x - \sqrt{x^2 - 1}~\!\bigr)}\Bigr)
+ {\exp}\Bigl({\textstyle t\bigl(x + \sqrt{x^2 - 1}~\!\bigr)}\Bigr)\Bigr) \\
&= e^{tx} \cosh\left({\textstyle t\sqrt{x^2 - 1} }~\! \right).
\end{align}</math>
The generating function relevant for 2-dimensional potential theory and multipole expansion is <math display="block">\sum\limits_{n=1}^\infty T_{n}(x)\,\frac{t^n}{n} = \ln\left(\frac{1}{\sqrt{1 - 2tx + t^2 }}\right).</math>
The ordinary generating function for Template:Mvar is <math display="block">\sum_{n=0}^\infty U_n(x)\,t^n = \frac{1}{1 - 2tx + t^2},</math> and the exponential generating function is <math display="block">
\sum_{n=0}^\infty U_n(x) \frac{t^n}{n!} = e^{tx} \biggl(\!\cosh\left(t\sqrt{x^2 - 1}\right) + \frac{x}{\sqrt{x^2 - 1}} \sinh\left(t\sqrt{x^2 - 1}\right)\biggr).
</math>
Relations between the two kinds of Chebyshev polynomialsEdit
The Chebyshev polynomials of the first and second kinds correspond to a complementary pair of Lucas sequences Template:Math and Template:Math with parameters Template:Math and Template:Math: <math display="block">\begin{align} {\tilde U}_n(2x,1) &= U_{n-1}(x), \\ {\tilde V}_n(2x,1) &= 2\, T_n(x). \end{align}</math> It follows that they also satisfy a pair of mutual recurrence equations:Template:Sfn <math display="block">\begin{align} T_{n+1}(x) &= x\,T_n(x) - (1 - x^2)\,U_{n-1}(x), \\ U_{n+1}(x) &= x\,U_n(x) + T_{n+1}(x). \end{align}</math>
The second of these may be rearranged using the recurrence definition for the Chebyshev polynomials of the second kind to give: <math display="block">T_n(x) = \frac{1}{2} \big(U_n(x) - U_{n-2}(x)\big).</math>
Using this formula iteratively gives the sum formula: <math display="block"> U_n(x) = \begin{cases} 2\sum_{\text{ odd }j>0}^n T_j(x) & \text{ for odd }n.\\ 2\sum_{\text{ even }j\ge 0}^n T_j(x) - 1 & \text{ for even }n, \end{cases} </math> while replacing <math>U_n(x)</math> and <math>U_{n-2}(x)</math> using the derivative formula for <math>T_n(x)</math> gives the recurrence relationship for the derivative of <math>T_n</math>:
<math display="block">2\,T_n(x) = \frac{1}{n+1}\, \frac{\mathrm{d}}{\mathrm{d}x}\, T_{n+1}(x) - \frac{1}{n-1}\,\frac{\mathrm{d}}{\mathrm{d}x}\, T_{n-1}(x), \qquad n=2,3,\ldots</math>
This relationship is used in the Chebyshev spectral method of solving differential equations.
Turán's inequalities for the Chebyshev polynomials are:<ref>Template:Citation</ref> <math display="block">\begin{align} T_n(x)^2 - T_{n-1}(x)\,T_{n+1}(x)&= 1-x^2 > 0 &&\text{ for } -1<x<1 &&\text{ and }\\ U_n(x)^2 - U_{n-1}(x)\,U_{n+1}(x)&= 1 > 0~. \end{align}</math>
The integral relations areTemplate:SfnTemplate:Sfn <math display="block">\begin{align} \int_{-1}^1 \frac{T_n(y)}{y-x} \, \frac{\mathrm{d}y}{\sqrt{1 - y^2}} &= \pi\,U_{n-1}(x)~, \\[1.5ex] \int_{-1}^1\frac{U_{n-1}(y)}{y-x}\, \sqrt{1 - y^2}\mathrm{d}y &= -\pi\,T_n(x) \end{align}</math> where integrals are considered as principal value.
Explicit expressionsEdit
Using the complex number exponentiation definition of the Chebyshev polynomial, one can derive the following expression: <math display="block"> T_n(x) = \dfrac{1}{2} \bigg( \Big(x-\sqrt{x^2-1} \Big)^n + \Big(x+\sqrt{x^2-1} \Big)^n \bigg) \quad \text{ for } x \in \mathbb {R},</math> <math display="block"> T_n(x) = \dfrac{1}{2} \bigg( \Big(x-\sqrt{x^2-1} \Big)^n + \Big(x-\sqrt{x^2-1} \Big)^{-n} \bigg) \quad \text{ for } x \in \mathbb {R}.</math> The two are equivalent because <math>(x + \sqrt{x^2 - 1})(x - \sqrt{x^2 - 1}) = 1</math>.
An explicit form of the Chebyshev polynomial in terms of monomials Template:Math follows from de Moivre's formula: <math display="block">T_n(\cos(\theta)) = \operatorname{Re}(\cos n \theta + i \sin n \theta) = \operatorname{Re}((\cos \theta + i \sin \theta)^n),</math> where Template:Math denotes the real part of a complex number. Expanding the formula, one gets <math display="block">(\cos \theta + i \sin \theta)^n = \sum\limits_{j=0}^n \binom{n}{j} i^j \sin^j \theta \cos^{n-j} \theta.</math> The real part of the expression is obtained from summands corresponding to even indices. Noting <math>i^{2j} = (-1)^j</math> and <math>\sin^{2j} \theta = (1-\cos^2 \theta)^j</math>, one gets the explicit formula: <math display="block">\cos n \theta = \sum\limits_{j=0}^{\lfloor n / 2 \rfloor} \binom{n}{2j} (\cos^2 \theta - 1)^j \cos^{n-2j} \theta,</math> which in turn means that <math display="block">T_n(x) = \sum\limits_{j=0}^{\lfloor n / 2 \rfloor} \binom{n}{2j} (x^2-1)^j x^{n-2j}.</math> This can be written as a Template:Math hypergeometric function: <math display="block">\begin{align} T_n(x) & = \sum_{k=0}^{\left \lfloor \frac{n}{2} \right \rfloor} \binom{n}{2k} \left (x^2-1 \right )^k x^{n-2k} \\ & = x^n \sum_{k=0}^{\left \lfloor \frac{n}{2} \right \rfloor} \binom{n}{2k} \left (1 - x^{-2} \right )^k \\ & = \frac{n}{2} \sum_{k=0}^{\left \lfloor \frac{n}{2} \right \rfloor}(-1)^k \frac{(n-k-1)!}{k!(n-2k)!}~(2x)^{n-2k} \quad \text{ for } n > 0 \\ \\ & = n \sum_{k=0}^{n}(-2)^{k} \frac{(n+k-1)!} {(n-k)!(2k)!}(1 - x)^k \quad \text{ for } n > 0 \\ \\ & = {}_2F_1\!\left(-n,n;\tfrac 1 2; \tfrac{1}{2}(1-x)\right) \\ \end{align}</math> with inverse<ref name=Cody>Template:Cite journal</ref><ref name=Mathar>Template:Cite journal</ref> <math display="block">x^n = 2^{1-n}\mathop{{\sum}'}^n_{j=0\atop j \equiv n \pmod 2} \!\!\binom{n}{\tfrac{n-j}{2}}\!\;T_j(x),</math> where the prime at the summation symbol indicates that the contribution of Template:Math needs to be halved if it appears.
A related expression for Template:Math as a sum of monomials with binomial coefficients and powers of two is <math display="block">
T_n(x) = \sum\limits_{m=0}^{\left\lfloor \frac{n}{2} \right\rfloor} (-1)^m \left(\binom{n - m}{m} + \binom{n - m - 1}{n - 2m}\right) \cdot 2^{n-2m-1} \cdot x^{n-2m}.</math>
Similarly, Template:Math can be expressed in terms of hypergeometric functions: <math display="block">\begin{align} U_n(x) &= \frac{\left (x+\sqrt{x^2-1} \right )^{n+1} - \left (x-\sqrt{x^2-1} \right )^{n+1}}{2\sqrt{x^2-1}} \\
&= \sum_{k=0}^{\left \lfloor {n}/{2} \right \rfloor} \binom{n+1}{2k+1} \left (x^2-1 \right )^k x^{n-2k} \\
&= x^n \sum_{k=0}^{\left \lfloor {n}/{2} \right \rfloor} \binom{n+1}{2k+1} \left (1 - x^{-2} \right )^k \\
&= \sum_{k=0}^{\left \lfloor {n}/{2} \right \rfloor} \binom{2k-(n+1)}{k}~(2x)^{n-2k} & \text{ for } n > 0 \\
&= \sum_{k=0}^{\left \lfloor {n}/{2} \right \rfloor} (-1)^k \binom{n-k}{k}~(2x)^{n-2k} & \text{ for } n > 0 \\
&= \sum_{k=0}^{n}(-2)^{k} \frac{(n+k+1)!} {(n-k)!(2k+1)!}(1 - x)^k & \text{ for } n > 0 \\
&= (n + 1)\, {}_2F_1\big(-n, n + 2; \tfrac{3}{2}; \tfrac{1}{2}(1 - x)\big).
\end{align}</math>
PropertiesEdit
SymmetryEdit
<math display="block">\begin{align}
T_n(-x) &= (-1)^n\, T_n(x),\\[1ex] U_n(-x) &= (-1)^n\, U_n(x).
\end{align}</math>
That is, Chebyshev polynomials of even order have even symmetry and therefore contain only even powers of Template:Mvar. Chebyshev polynomials of odd order have odd symmetry and therefore contain only odd powers of Template:Mvar.
Roots and extremaEdit
A Chebyshev polynomial of either kind with degree Template:Mvar has Template:Mvar different simple roots, called Chebyshev roots, in the interval Template:Closed-closed. 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: <math display="block">\cos\left((2k+1)\frac{\pi}{2}\right)=0</math> one can show that the roots of Template:Mvar are: <math display="block"> x_k = \cos\left(\frac{\pi(k+1/2)}{n}\right),\quad k=0,\ldots,n-1.</math> Similarly, the roots of Template:Mvar are: <math display="block"> x_k = \cos\left(\frac{k}{n+1}\pi\right),\quad k=1,\ldots,n.</math> The extrema of Template:Mvar on the interval Template:Math are located at: <math display="block"> x_k = \cos\left(\frac{k}{n}\pi\right),\quad k=0,\ldots,n.</math>
One unique property of the Chebyshev polynomials of the first kind is that on the interval Template:Math 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: <math display="block">\begin{align} T_n(1) &= 1 \\ T_n(-1) &= (-1)^n \\ U_n(1) &= n+1 \\ U_n(-1) &= (-1)^n (n+1). \end{align}</math>
The extrema of <math>T_n(x)</math> on the interval <math>-1 \leq x \leq 1</math> where <math>n>0</math> are located at <math>n+1</math> values of <math>x</math>. They are <math> \pm 1</math>, or <math> \cos\left(\frac{2\pi k}{d}\right)</math> where <math>d > 2</math>, <math>d \;|\; 2n</math>, <math>0 < k < d/2</math> and <math>(k, d) = 1</math>, i.e., <math>k</math> and <math>d</math> are relatively prime numbers.
Specifically (Minimal polynomial of 2cos(2pi/n)<ref name=Gurtas>Template:Cite journal</ref><ref name=Wolfram0>Template:Cite journal</ref>) when <math>n</math> is even:
- <math>T_n(x) = 1</math> if <math>x = \pm 1</math>, or <math>d > 2</math> and <math>2n/d</math> is even. There are <math>n/2 + 1</math> such values of <math>x</math>.
- <math>T_n(x) = -1</math> if <math>d > 2</math> and <math>2n/d</math> is odd. There are <math>n/2</math> such values of <math>x</math>.
When <math>n</math> is odd:
- <math>T_n(x) = 1</math> if <math>x = 1</math>, or <math>d > 2</math> and <math>2n/d</math> is even. There are <math>(n+1)/2</math> such values of <math>x</math>.
- <math>T_n(x) = -1</math> if <math>x = -1</math>, or <math>d > 2</math> and <math>2n/d</math> is odd. There are <math>(n+1)/2</math> such values of <math>x</math>.
Differentiation and integrationEdit
The derivatives of the polynomials can be less than straightforward. By differentiating the polynomials in their trigonometric forms, it can be shown that: <math display="block">\begin{align} \frac{\mathrm{d}T_n}{\mathrm{d}x} &= n U_{n - 1} \\ \frac{\mathrm{d}U_n}{\mathrm{d}x} &= \frac{(n + 1)T_{n + 1} - x U_n}{x^2 - 1} \\ \frac{\mathrm{d}^2 T_n}{\mathrm{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}. \end{align}</math>
The last two formulas can be numerically troublesome due to the division by zero (Template:Sfrac indeterminate form, specifically) at Template:Math and Template:Math. By L'Hôpital's rule: <math display="block">\begin{align} \left. \frac{\mathrm{d}^2 T_n}{\mathrm{d}x^2} \right|_{x = 1} \!\! &= \frac{n^4 - n^2}{3}, \\ \left. \frac{\mathrm{d}^2 T_n}{\mathrm{d}x^2} \right|_{x = -1} \!\! &= (-1)^n \frac{n^4 - n^2}{3}. \end{align}</math>
More generally, <math display="block">\left.\frac{\mathrm{d}^p T_n}{\mathrm{d}x^p} \right|_{x = \pm 1} \!\! = (\pm 1)^{n+p}\prod_{k=0}^{p-1}\frac{n^2-k^2}{2k+1}~,</math> which is of great use in the numerical solution of eigenvalue problems.
Also, we have: <math display="block">\frac{\mathrm{d}^p}{\mathrm{d}x^p}\,T_n(x) = 2^p\,n\mathop{{\sum}'}_{0\leq k\leq n-p\atop k \,\equiv\, n-p \pmod 2} \binom{\frac{n+p-k}{2}-1}{\frac{n-p-k}{2}}\frac{\left(\frac{n+p+k}{2}-1\right)!}{\left(\frac{n-p+k}{2}\right)!}\,T_k(x),~\qquad p \ge 1,</math> where the prime at the summation symbols means that the term contributed by Template:Math is to be halved, if it appears.
Concerning integration, the first derivative of the Template:Mvar implies that: <math display="block">\int U_n\, \mathrm{d}x = \frac{T_{n + 1}}{n + 1}</math> and the recurrence relation for the first kind polynomials involving derivatives establishes that for Template:Math: <math display="block">\int T_n\, \mathrm{d}x = \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>
The last formula can be further manipulated to express the integral of Template:Mvar as a function of Chebyshev polynomials of the first kind only: <math display="block">\begin{align} \int T_n\, \mathrm{d}x &= \frac{n}{n^2 - 1} T_{n + 1} - \frac{1}{n - 1} T_1 T_n \\ &= \frac{n}{n^2 - 1}\,T_{n + 1} - \frac{1}{2(n - 1)}\,(T_{n + 1} + T_{n - 1}) \\ &= \frac{1}{2(n + 1)}\,T_{n + 1} - \frac{1}{2(n - 1)}\,T_{n - 1}. \end{align}</math>
Furthermore, we have: <math display="block">\int_{-1}^1 T_n(x)\, \mathrm{d}x = \begin{cases} \frac{(-1)^n + 1}{1 - n^2} & \text{ if }~ n \ne 1 \\ 0 & \text{ if }~ n = 1. \end{cases}</math>
Products of Chebyshev polynomialsEdit
The Chebyshev polynomials of the first kind satisfy the relation: <math display="block">T_m(x)\,T_n(x) = \tfrac{1}{2}\!\left(T_{m+n}(x) + T_{|m-n|}(x)\right)\!,\qquad \forall m,n \ge 0,</math> which is easily proved from the product-to-sum formula for the cosine: <math display="block">2 \cos \alpha \, \cos \beta = \cos (\alpha + \beta) + \cos (\alpha - \beta).</math> For Template:Math this results in the already known recurrence formula, just arranged differently, and with Template:Math it forms the recurrence relation for all even or all odd indexed Chebyshev polynomials (depending on the parity of the lowest Template:Mvar) which implies the evenness or oddness of these polynomials. Three more useful formulas for evaluating Chebyshev polynomials can be concluded from this product expansion: <math display="block">\begin{align}
T_{2n}(x) &= 2\,T_n^2(x) - T_0(x) &&= 2 T_n^2(x) - 1, \\
T_{2n+1}(x) &= 2\,T_{n+1}(x)\,T_n(x) - T_1(x) &&= 2\,T_{n+1}(x)\,T_n(x) - x, \\ T_{2n-1}(x) &= 2\,T_{n-1}(x)\,T_n(x) - T_1(x) &&= 2\,T_{n-1}(x)\,T_n(x) - x . \end{align}</math>
The polynomials of the second kind satisfy the similar relation: <math display="block"> T_m(x)\,U_n(x) = \begin{cases} \frac{1}{2}\left(U_{m+n}(x) + U_{n-m}(x)\right), & ~\text{ if }~ n \ge m-1,\\ \\ \frac{1}{2}\left(U_{m+n}(x) - U_{m-n-2}(x)\right), & ~\text{ if }~ n \le m-2. \end{cases} </math> (with the definition Template:Math by convention ). They also satisfy: <math display="block"> U_m(x)\,U_n(x) = \sum_{k=0}^n\,U_{m-n+2k}(x) = \sum_\underset{\text{ step 2 }}{p=m-n}^{m+n} U_p(x)~.</math> for Template:Math. For Template:Math this recurrence reduces to: <math display="block"> U_{m+2}(x) = U_2(x)\,U_m(x) - U_m(x) - U_{m-2}(x) = U_m(x)\,\big(U_2(x) - 1\big) - U_{m-2}(x)~,</math> which establishes the evenness or oddness of the even or odd indexed Chebyshev polynomials of the second kind depending on whether Template:Mvar starts with 2 or 3.
Composition and divisibility propertiesEdit
The trigonometric definitions of Template:Math and Template:Math imply the composition or nesting properties:<ref>Template:Citation</ref> <math display="block">\begin{align} T_{mn}(x) &= T_m(T_n(x)),\\ U_{mn-1}(x) &= U_{m-1}(T_n(x))U_{n-1}(x). \end{align} </math> For Template:Math the order of composition may be reversed, making the family of polynomial functions Template:Math a commutative semigroup under composition.
Since Template:Math is divisible by Template:Mvar if Template:Mvar is odd, it follows that Template:Math is divisible by Template:Math if Template:Mvar is odd. Furthermore, Template:Math is divisible by Template:Math, and in the case that Template:Mvar is even, divisible by Template:Math.
OrthogonalityEdit
Both Template:Mvar and Template:Mvar form a sequence of orthogonal polynomials. The polynomials of the first kind Template:Mvar are orthogonal with respect to the weight: <math display="block">\frac{1}{\sqrt{1 - x^2}},</math> on the interval Template:Closed-closed, i.e. we have: <math display="block">\int_{-1}^1 T_n(x)\,T_m(x)\,\frac{\mathrm{d}x}{\sqrt{1-x^2}} = \begin{cases} 0 & ~\text{ if }~ n \ne m, \\[5mu] \pi & ~\text{ if }~ n=m=0, \\[5mu] \frac{\pi}{2} & ~\text{ if }~ n=m \ne 0. \end{cases}</math>
This can be proven by letting Template:Math and using the defining identity Template:Math.
Similarly, the polynomials of the second kind Template:Mvar are orthogonal with respect to the weight: <math display="block">\sqrt{1-x^2}</math> on the interval Template:Closed-closed, i.e. we have: <math display="block">\int_{-1}^1 U_n(x)\,U_m(x)\,\sqrt{1-x^2} \,\mathrm{d}x = \begin{cases} 0 & ~\text{ if }~ n \ne m, \\[5mu] \frac{\pi}{2} & ~\text{ if }~ n = m. \end{cases}</math>
(The measure Template:Math 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: <math display="block">\begin{align} (1 - x^2)T_n - xT_n' + n^2 T_n &= 0, \\[1ex] (1 - x^2)U_n - 3xU_n' + n(n + 2) U_n &= 0, \end{align}</math>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 Template:Mvar also satisfy a discrete orthogonality condition: <math display="block">\sum_{k=0}^{N-1}{T_i(x_k)\,T_j(x_k)} = \begin{cases} 0 & ~\text{ if }~ i \ne j, \\[5mu] N & ~\text{ if }~ i = j = 0, \\[5mu] \frac{N}{2} & ~\text{ if }~ i = j \ne 0, \end{cases} </math> where Template:Mvar is any integer greater than Template:Math,Template:Sfn and the Template:Math are the Template:Mvar Chebyshev nodes (see above) of Template:Math: <math display="block">x_k = \cos\left(\pi\,\frac{2k+1}{2N}\right) \quad ~\text{ for }~ k = 0, 1, \dots, N-1.</math>
For the polynomials of the second kind and any integer Template:Math with the same Chebyshev nodes Template:Math, there are similar sums: <math display="block">\sum_{k=0}^{N-1}{U_i(x_k)\,U_j(x_k)\left(1-x_k^2\right)} = \begin{cases} 0 & \text{ if }~ i \ne j, \\[5mu] \frac{N}{2} & \text{ if }~ i = j, \end{cases}</math> and without the weight function: <math display="block">\sum_{k=0}^{N-1}{ U_i(x_k) \, U_j(x_k) } = \begin{cases} 0 & ~\text{ if }~ i \not\equiv j \pmod{2}, \\[5mu] N \cdot (1 + \min\{i,j\}) & ~\text{ if }~ i \equiv j\pmod{2}. \end{cases} </math>
For any integer Template:Math, based on the Template:Mvar zeros of Template:Math: <math display="block">y_k = \cos\left(\pi\,\frac{k+1}{N+1}\right) \quad ~\text{ for }~ k=0, 1, \dots, N-1,</math> one can get the sum: <math display="block">\sum_{k=0}^{N-1}{U_i(y_k)\,U_j(y_k)(1-y_k^2)} = \begin{cases} 0 & ~\text{ if } i \ne j, \\[5mu] \frac{N+1}{2} & ~\text{ if } i = j, \end{cases}</math> and again without the weight function: <math display="block">\sum_{k=0}^{N-1}{U_i(y_k)\,U_j(y_k)} = \begin{cases} 0 & ~\text{ if }~ i \not\equiv j \pmod{2}, \\[5mu] \bigl(\min\{i,j\} + 1\bigr)\bigl(N-\max\{i,j\}\bigr) & ~\text{ if }~ i \equiv j\pmod{2}. \end{cases}</math>
Minimal Template:Math-normEdit
For any given Template:Math, among the polynomials of degree Template:Mvar with leading coefficient 1 (monic polynomials): <math display="block">f(x) = \frac{1}{\,2^{n-1}\,}\,T_n(x)</math> is the one of which the maximal absolute value on the interval Template:Closed-closed is minimal.
This maximal absolute value is: <math display="block">\frac1{2^{n-1}}</math> and Template:Math reaches this maximum exactly Template:Math times at: <math display="block">x = \cos \frac{k\pi}{n}\quad\text{for }0 \le k \le n.</math>
Template:Math proofT_n(x)\right| \\ f_n(x) &> 0 \qquad \text{ for }~ x = \cos \frac{2k\pi}{n} ~&&\text{ where } 0 \le 2k \le n \\ f_n(x) &< 0 \qquad \text{ for }~ x = \cos \frac{(2k + 1)\pi}{n} ~&&\text{ where } 0 \le 2k + 1 \le n \end{align}</math>
From the intermediate value theorem, Template:Math has at least Template:Mvar roots. However, this is impossible, as Template:Math is a polynomial of degree Template:Math, so the fundamental theorem of algebra implies it has at most Template:Math roots. }}
RemarkEdit
By the equioscillation theorem, among all the polynomials of degree Template:Math, the polynomial Template:Mvar minimizes Template:Math on Template:Closed-closed if and only if there are Template:Math points Template:Math such that Template:Math.
Of course, the null polynomial on the interval Template:Closed-closed can be approximated by itself and minimizes the Template:Math-norm.
Above, however, Template:Math reaches its maximum only Template:Math times because we are searching for the best polynomial of degree Template:Math (therefore the theorem evoked previously cannot be used).
Chebyshev polynomials as special cases of more general polynomial familiesEdit
The Chebyshev polynomials are a special case of the ultraspherical or Gegenbauer polynomials <math>C_n^{(\lambda)}(x)</math>, which themselves are a special case of the Jacobi polynomials <math>P_n^{(\alpha,\beta)}(x)</math>: <math display="block">\begin{align} T_n(x) &= \frac{n}{2} \lim_{q \to 0} \frac{1}{q}\,C_n^{(q)}(x) \qquad ~\text{ if }~ n \ge 1, \\
&= \frac{1}{\binom{n-\frac{1}{2}}{n}} P_n^{\left(-\frac{1}{2}, -\frac{1}{2}\right)}(x) = \frac{2^{2n}}{\binom{2n}{n}} P_n^{\left(-\frac{1}{2}, -\frac{1}{2}\right)}(x)~,
\\[2ex] U_n(x) & = C_n^{(1)}(x)\\
&= \frac{n+1}{\binom{n+\frac{1}{2}}{n}} P_n^{\left(\frac{1}{2}, \frac{1}{2}\right)}(x) = \frac{2^{2n+1}}{\binom{2n+2}{n+1}} P_n^{\left(\frac{1}{2}, \frac{1}{2}\right)}(x)~.
\end{align}</math>
Chebyshev polynomials are also a special case of Dickson polynomials: <math display="block">D_n(2x\alpha,\alpha^2)= 2\alpha^{n}T_n(x) \, </math> <math display="block">E_n(2x\alpha,\alpha^2)= \alpha^{n}U_n(x). \, </math> In particular, when <math>\alpha=\tfrac{1}{2}</math>, they are related by <math>D_n(x,\tfrac{1}{4}) = 2^{1-n}T_n(x)</math> and <math>E_n(x,\tfrac{1}{4}) = 2^{-n}U_n(x)</math>.
Other propertiesEdit
The curves given by Template:Math, or equivalently, by the parametric equations Template:Math, Template:Math, are a special case of Lissajous curves with frequency ratio equal to Template:Mvar.
Similar to the formula: <math display="block">T_n(\cos\theta) = \cos(n\theta),</math> we have the analogous formula: <math display="block">T_{2n+1}(\sin\theta) = (-1)^n \sin\left(\left(2n+1\right)\theta\right).</math>
For Template:Math: <math display="block">T_n\!\left(\frac{x + x^{-1}}{2}\right) = \frac{x^n+x^{-n}}{2}</math> and: <math display="block">x^n = T_n\! \left(\frac{x+x^{-1}}{2}\right) + \frac{x-x^{-1}}{2}\ U_{n-1}\!\left(\frac{x+x^{-1}}{2}\right),</math> which follows from the fact that this holds by definition for Template:Math.
There are relations between Legendre polynomials and Chebyshev polynomials
<math>\sum_{k=0}^{n}P_{k}\left(x\right)T_{n-k}\left(x\right) = \left(n+1\right)P_{n}\left(x\right)</math>
<math>\sum_{k=0}^{n}P_{k}\left(x\right)P_{n-k}\left(x\right) = U_{n}\left(x\right)</math>
These identities can be proven using generating functions and discrete convolution
Chebyshev polynomials as determinantsEdit
From their definition by recurrence it follows that the Chebyshev polynomials can be obtained as determinants of special tridiagonal matrices of size <math>k \times k</math>: <math display="block">T_k(x) = \det \begin{bmatrix}
x & 1 & 0 & \cdots & 0 \\
1 & 2x & 1 & \ddots & \vdots \\
0 & 1 & 2x & \ddots & 0 \\
\vdots & \ddots & \ddots & \ddots & 1 \\
0 & \cdots & 0 & 1 & 2x
\end{bmatrix},</math> and similarly for <math>U_k</math>.
ExamplesEdit
First kindEdit
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The first few Chebyshev polynomials of the first kind are Template:OEIS2C <math display="block"> \begin{align} 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 \\ T_{10}(x) &= 512x^{10} - 1280x^8 + 1120x^6 - 400x^4 + 50x^2-1 \end{align}</math>
Second kindEdit
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The first few Chebyshev polynomials of the second kind are Template:OEIS2C <math display="block">\begin{align} 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 - 448 x^6 + 240 x^4 - 40 x^2 + 1 \\ U_9(x) &= 512x^9 - 1024 x^7 + 672 x^5 - 160 x^3 + 10 x \\ U_{10}(x) &= 1024x^{10} - 2304 x^8 + 1792 x^6 - 560 x^4 + 60 x^2-1 \end{align}</math>
As a basis setEdit
In the appropriate Sobolev space, the set of Chebyshev polynomials form an orthonormal basis, so that a function in the same space can, on Template:Math, be expressed via the expansion:<ref name=boyd>Template:Cite book</ref> <math display="block">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 Template:Math 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 Template:Math if the function is piecewise smooth and continuous. The smoothness requirement can be relaxed in most casesTemplate:Snd as long as there are a finite number of discontinuities in Template:Math 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).
The Chebfun software package supports function manipulation based on their expansion in the Chebysev basis.
Example 1Edit
Consider the Chebyshev expansion of Template:Math. One can express: <math display="block"> \log(1+x) = \sum_{n = 0}^\infty a_n T_n(x)~. </math>
One can find the coefficients Template:Math either through the application of an inner product or by the discrete orthogonality condition. For the inner product: <math display="block">\int_{-1}^{+1}\,\frac{T_m(x)\,\log(1 + x)}{\sqrt{1-x^2}}\,\mathrm{d}x = \sum_{n=0}^{\infty}a_n\int_{-1}^{+1}\frac{T_m(x)\,T_n(x)}{\sqrt{1-x^2}}\,\mathrm{d}x,</math> which gives: <math display="block">a_n = \begin{cases} -\log 2 & \text{ for }~ n = 0, \\ \frac{-2(-1)^n}{n} & \text{ for }~ n > 0. \end{cases}</math>
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: <math display="block">a_n \approx \frac{\,2-\delta_{0n}\,}{N}\,\sum_{k=0}^{N-1}T_n(x_k)\,\log(1+x_k),</math> where Template:Mvar is the Kronecker delta function and the Template:Mvar are the Template:Mvar Gauss–Chebyshev zeros of Template:Math: <math display="block"> x_k = \cos\left(\frac{\pi\left(k+\tfrac{1}{2}\right)}{N}\right) .</math> For any Template:Mvar, these approximate coefficients provide an exact approximation to the function at Template:Mvar with a controlled error between those points. The exact coefficients are obtained with Template:Math, thus representing the function exactly at all points in Template:Closed-closed. The rate of convergence depends on the function and its smoothness.
This allows us to compute the approximate coefficients Template:Mvar very efficiently through the discrete cosine transform: <math display="block">a_n \approx \frac{2-\delta_{0n}}{N}\sum_{k=0}^{N-1}\cos\left(\frac{n\pi\left(\,k+\tfrac{1}{2}\right)}{N}\right)\log(1+x_k).</math>
Example 2Edit
To provide another example: <math display="block">\begin{align} \left(1-x^2\right)^\alpha &= -\frac{1}{\sqrt{\pi}} \, \frac{\Gamma\left(\tfrac{1}{2} + \alpha\right)}{\Gamma(\alpha+1)} + 2^{1-2\alpha}\,\sum_{n=0} \left(-1\right)^n \, {2 \alpha \choose \alpha-n}\,T_{2n}(x) \\[1ex]
&= 2^{-2\alpha}\,\sum_{n=0} \left(-1\right)^n \, {2\alpha+1 \choose \alpha-n}\,U_{2n}(x).
\end{align}</math>
Partial sumsEdit
The partial sums of: <math display="block">f(x) = \sum_{n = 0}^\infty a_n T_n(x)</math> 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 Template:Mvar 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 Template:Mvar coefficients of the Template:Mathst partial sum are usually obtained on the Chebyshev–Gauss–Lobatto<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</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 display="block">x_k = -\cos\left(\frac{k \pi}{N - 1}\right); \qquad k = 0, 1, \dots, N - 1.</math>
Polynomial in Chebyshev formEdit
An arbitrary polynomial of degree Template:Mvar can be written in terms of the Chebyshev polynomials of the first kind.Template:Sfn Such a polynomial Template:Math is of the form: <math display="block">p(x) = \sum_{n=0}^N a_n T_n(x).</math>
Polynomials in Chebyshev form can be evaluated using the Clenshaw algorithm.
Edit
Polynomials denoted <math>C_n(x)</math> and <math>S_n(x)</math> closely related to Chebyshev polynomials are sometimes used. They are defined by:Template:Sfn <math display="block">C_n(x) = 2T_n\left(\frac{x}{2}\right),\qquad S_n(x) = U_n\left(\frac{x}{2}\right)</math> and satisfy: <math display="block">C_n(x) = S_n(x) - S_{n-2}(x).</math> A. F. Horadam called the polynomials <math>C_n(x)</math> Vieta–Lucas polynomials and denoted them <math>v_n(x)</math>. He called the polynomials <math>S_n(x)</math> Vieta–Fibonacci polynomials and denoted them Template:Nowrap<ref>Template:Citation</ref> Lists of both sets of polynomials are given in Viète's Opera Mathematica, Chapter IX, Theorems VI and VII.<ref>Template:Cite book</ref> The Vieta–Lucas and Vieta–Fibonacci polynomials of real argument are, up to a power of <math>i</math> and a shift of index in the case of the latter, equal to Lucas and Fibonacci polynomials Template:Math and Template:Math of imaginary argument.
Shifted Chebyshev polynomials of the first and second kinds are related to the Chebyshev polynomials by:Template:Sfn <math display="block">T_n^*(x) = T_n(2x-1),\qquad U_n^*(x) = U_n(2x-1).</math>
When the argument of the Chebyshev polynomial satisfies Template:Math the argument of the shifted Chebyshev polynomial satisfies Template:Math. Similarly, one can define shifted polynomials for generic intervals Template:Closed-closed.
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."<ref name=MasonElliott1993>Template:Citation</ref> The Chebyshev polynomials of the third kind are defined as: <math display="block">V_n(x)=\frac{\cos\left(\left(n+\frac{1}{2}\right)\theta\right)}{\cos\left(\frac{\theta}{2}\right)}=\sqrt\frac{2}{1+x}T_{2n+1}\left(\sqrt\frac{x+1}{2}\right)</math> and the Chebyshev polynomials of the fourth kind are defined as: <math display="block">W_n(x)=\frac{\sin\left(\left(n+\frac{1}{2}\right)\theta\right)}{\sin\left(\frac{\theta}{2}\right)}=U_{2n}\left(\sqrt\frac{x+1}{2}\right),</math> where <math>\theta=\arccos x</math>.<ref name=MasonElliott1993/><ref name=DesmaraisBland1995>Template:Citation</ref> They coincide with the Dirichlet kernel.
In the airfoil literature <math>V_n(x)</math> and <math>W_n(x)</math> are denoted <math>t_n(x)</math> and <math>u_n(x)</math>. The polynomial families <math>T_n(x)</math>, <math>U_n(x)</math>, <math>V_n(x)</math>, and <math>W_n(x)</math> are orthogonal with respect to the weights: <math display="block">\left(1-x^2\right)^{-1/2},\quad\left(1-x^2\right)^{1/2},\quad(1-x)^{-1/2}(1+x)^{1/2},\quad(1+x)^{-1/2}(1-x)^{1/2}</math> and are proportional to Jacobi polynomials <math>P_n^{(\alpha,\beta)}(x)</math> with:<ref name="DesmaraisBland1995" /> <math display="block">(\alpha,\beta)=\left(-\frac{1}{2},-\frac{1}{2}\right),\quad(\alpha,\beta)=\left(\frac{1}{2},\frac{1}{2}\right),\quad(\alpha,\beta)=\left(-\frac{1}{2},\frac{1}{2}\right),\quad(\alpha,\beta)=\left(\frac{1}{2},-\frac{1}{2}\right).</math>
All four families satisfy the recurrence <math>p_n(x)=2xp_{n-1}(x)-p_{n-2}(x)</math> with <math>p_0(x) = 1</math>, where <math>p_n = T_n</math>, <math>U_n</math>, <math>V_n</math>, or <math>W_n</math>, but they differ according to whether <math>p_1(x)</math> equals <math>x</math>, <math>2x</math>, <math>2x-1</math>, or Template:Nowrap<ref name=MasonElliott1993/>
Even order modified Chebyshev polynomialsEdit
Some applications rely on Chebyshev polynomials but may be unable to accommodate the lack of a root at zero, which rules out the use of standard Chebyshev polynomials for these kinds of applications. Even order Chebyshev filter designs using equally terminated passive networks are an example of this.<ref name=":022">Template:Cite book</ref> However, even order Chebyshev polynomials may be modified to move the lowest roots down to zero while still maintaining the desirable Chebyshev equi-ripple effect. Such modified polynomials contain two roots at zero, and may be referred to as even order modified Chebyshev polynomials. Even order modified Chebyshev polynomials may be created from the Chebyshev nodes in the same manner as standard Chebyshev polynomials. <math display="block">P_N = \prod_{i=1}^N(x-C_i) </math>
where
- <math>P_N</math> is an N-th order Chebyshev polynomial
- <math>C_i</math> is the i-th Chebyshev node
In the case of even order modified Chebyshev polynomials, the even order modified Chebyshev nodes are used to construct the even order modified Chebyshev polynomials. <math display="block">Pe_N = \prod_{i=1}^N(x-Ce_i) </math>
where
- <math>P e_N</math> is an N-th order even order modified Chebyshev polynomial
- <math>Ce_i</math> is the i-th even order modified Chebyshev node
For example, the 4th order Chebyshev polynomial from the example above is <math>X^4-X^2+.125 </math>, which by inspection contains no roots of zero. Creating the polynomial from the even order modified Chebyshev nodes creates a 4th order even order modified Chebyshev polynomial of <math>X^4-.828427X^2 </math>, which by inspection contains two roots at zero, and may be used in applications requiring roots at zero.
See alsoEdit
- Chebyshev rational functions
- Function approximation
- Discrete Chebyshev transform
- Markov brothers' inequality
ReferencesEdit
SourcesEdit
- Template:Cite book Reprint: 1983. New York: Dover. Template:Isbn.
- Template:Cite book Reprint: 1981. Melbourne, FL: Krieger. Template:ISBN.
- Template:Cite book
Further readingEdit
- Template:Cite journal
- Template:Cite journal
- Template:Cite journal
- Template:Cite journal
- Template:Cite book
- Template:Dlmf
- {{#invoke:citation/CS1|citation
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External linksEdit
- Template:Commons category-inline
- {{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web
|_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:ChebyshevPolynomialoftheFirstKind%7CChebyshevPolynomialoftheFirstKind.html}} |title = Chebyshev polynomial[s] of the first kind |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}
- {{#invoke:citation/CS1|citation
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