Trigonometry/For Enthusiasts/Chebyshev Polynomials

The Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of polynomials related to the trigonometric multi-angle formulae.

We usually distinguish between

• Chebyshev polynomials of the first kind, denoted T_n and are closely related to $\cos$ and
• Chebyshev polynomials of the second kind, denoted U_n which are closely related to $\sin$

The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebycheff (French) or Tschebyschow (German).

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'.

Examples

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

$T_{0}(x)=1$

$T_{1}(x)=x$

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

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

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

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

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

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

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

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

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, U_n(1) = n + 1 and U_n(−1) = (n + 1)(−1)ⁿ.

The first few Chebyshev polynomials of the second kind are

$U_{0}(x)=1$

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

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

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

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

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

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

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

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

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

Definition of $T_{n}(x)$

There are many alternative ways to define $T_{n}(x)$ which lead to the same polynomials. The definition we'll use for $T_{n}(x)$ is:

$T_{n}(x)=\cos {\big (}n\arccos(x){\big )},\quad x\in [-1,1]$

In other words, $T_{n}(x)$ is the polynomial that expresses $\cos(n\theta )$ in terms of $\cos(\theta )$ .

For example:

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

comes directly from:

$\cos(3\theta )=4\cos ^{3}(\theta )-3\cos(\theta )$

Using this definition the recurrence relationship for Chebyshev polynomials follows immediately:

${\begin{aligned}T_{0}(x)&=1\\T_{1}(x)&=x\\T_{n+1}(x)&=2xT_{n}(x)-T_{n-1}(x)\end{aligned}}$

The recurrence comes from the relation:

$\cos {\big (}(n+1)\theta {\big )}=2\cos(\theta )\cos(n\theta )-\cos {\big (}(n-1)\theta {\big )}$

Which is a rearrangement of a relation derived from the addition formula for cosines:

$\cos(n\theta +\theta )+\cos(n\theta -\theta )=2\cos(n\theta )\cos(\theta )$

a special case where $\phi =n\theta$ of

$\cos(\phi +\theta )+\cos(\phi -\theta )=2\cos(\phi )\cos(\theta )$

That we saw with beat frequencies when adding two waves together.

Uses

If approximating some function on the interval $\displaystyle (0,1)$ one can use a polynomial approximation like:

$\displaystyle 0.7123+1.543x+0.311x^{2}+0.0172x^{3}$

to approximate its values. Because $\displaystyle x$ is between 0 and 1 the $\displaystyle x^{n}$ terms get smaller from left to right. With a sufficiently large number of terms, usually more than we have here, we can typically get very good approximations to well behaved functions. In the example we've truncated the series at the fourth term and are ignoring terms of $\displaystyle x^{4}$ and higher.

In the example polynomial above the actual numbers are just made up numbers, as an example, and have no special significance - in case you are wondering why those particular numbers were used.

It turns out that truncating a series is in a certain sense not the best possible way to approximate the original function's actual values with a polynomial of degree three. A more complex but better way uses Chebyshev polynomials.

If instead we can express the function as:

$\displaystyle a_{0}+a_{1}T_{1}(x)+a_{2}T_{2}(x)+a_{3}T_{3}(x)+\cdots$

where $\displaystyle a_{0},a_{1},a_{2},a_{3}\cdots$ are constants then if we truncate this series at the fourth term, i.e T₃, we again have a polynomial with no terms $\displaystyle x^{4}$ or higher. After we've expanded it out and collected the coefficients together, the coefficients will not generally be the same as when just truncating the series. It may sound crazy, so a worked example can show this more clearly.

Worked Example: Expansion of (2-x)-1 $\displaystyle {\frac {1}{2-x}}={\frac {1}{2}}+{\frac {x}{4}}+{\frac {x^{2}}{8}}+{\frac {x^{3}}{16}}+...$ A quick check... does this formula look reasonable? Put $\displaystyle x=0$ and we get $\displaystyle 1/2$. Put $\displaystyle x=1$ and we get $\displaystyle {\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+\cdots =1$. The formula looks reasonable. If we go to just four terms the error at $\displaystyle x=1$ is $\displaystyle {\frac {1}{16}}$. Now expressing this in Chebyshev polynomials: $\displaystyle {\frac {1}{1-x}}={\frac {1}{2}}+{\frac {T_{1}(x)}{4}}+{\frac {T_{2}(x)}{16}}+{\frac {T_{3}(x)}{64}}+...$ Expanding this out we get: $\displaystyle 0.53125+0.203125x+0.125x^{2}+0.0625x^{3}$ to-do: add graph showing worst error is better. Unfortunately it isn't with the current calculation. Hmm... still looks like I need to recompute the Chebyshev coefficients.

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

Explicit formulae

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

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

${\begin{aligned}T_{n}(x)&={\frac {{\big (}x-{\sqrt {x^{2}-1}}{\big )}^{n}+{\big (}x+{\sqrt {x^{2}-1}}{\big )}^{n}}{2}}\\&=\sum _{k=0}^{\left\lfloor {\frac {n}{2}}\right\rfloor }{\binom {n}{2k}}(x^{2}-1)^{k}x^{n-2k}\\&=x^{n}\sum _{k=0}^{\left\lfloor {\frac {n}{2}}\right\rfloor }{\binom {n}{2k}}(1-x^{-2})^{k}\\&={\frac {n}{2}}\sum _{k=0}^{\left\lfloor {\frac {n}{2}}\right\rfloor }(-1)^{k}{\frac {(n-k-1)!}{k!(n-2k)!}}~(2x)^{n-2k}\quad (n>0)\\&=n\sum _{k=0}^{n}(-2)^{k}{\frac {(n+k-1)!}{(n-k)!(2k)!}}(1-x)^{k}\quad (n>0)\\&=\,_{2}F_{1}\left(-n,n,{\frac {1}{2}},{\frac {1-x}{2}}\right)\\\end{aligned}}$

${\begin{aligned}U_{n}(x)&={\frac {{\big (}x+{\sqrt {x^{2}-1}}{\big )}^{n+1}-{\big (}x-{\sqrt {x^{2}-1}}{\big )}^{n+1}}{2{\sqrt {x^{2}-1}}}}\\&=\sum _{k=0}^{\left\lfloor {\frac {n}{2}}\right\rfloor }{\binom {n+1}{2k+1}}(x^{2}-1)^{k}x^{n-2k}\\&=x^{n}\sum _{k=0}^{\left\lfloor {\frac {n}{2}}\right\rfloor }{\binom {n+1}{2k+1}}(1-x^{-2})^{k}\\&=\sum _{k=0}^{\left\lfloor {\frac {n}{2}}\right\rfloor }{\binom {2k-(n+1)}{k}}~(2x)^{n-2k}\quad (n>0)\\&=\sum _{k=0}^{\left\lfloor {\frac {n}{2}}\right\rfloor }(-1)^{k}{\binom {n-k}{k}}~(2x)^{n-2k}\quad (n>0)\\&=\sum _{k=0}^{n}(-2)^{k}{\frac {(n+k+1)!}{(n-k)!(2k+1)!}}(1-x)^{k}\quad (n>0)\\&=(n+1)\,_{2}F_{1}\left(-n,n+2,{\frac {3}{2}},{\frac {1-x}{2}}\right)\end{aligned}}$

where $_{2}F_{1}$ is a hypergeometric function.

Uses

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

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

${\begin{aligned}T_{0}(x)&=1\\T_{1}(x)&=x\\T_{n+1}(x)&=2xT_{n}(x)-T_{n-1}(x).\end{aligned}}$

The conventional generating function for T_n is

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

The exponential generating function is

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

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

${\begin{aligned}U_{0}(x)&=1\\U_{1}(x)&=2x\\U_{n+1}(x)&=2xU_{n}(x)-U_{n-1}(x).\end{aligned}}$

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 $D_{n}(x)\,\!$:

$D_{n}(x)={\frac {\sin((2n+1)(x/2))}{\sin(x/2)}}=U_{2n}(\cos(x/2))\,\!$

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.

Two immediate corollaries are the composition identity (or the "nesting property")

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

and the expression of complex exponentiation in terms of Chebyshev polynomials: given z = a + bi,

${\begin{aligned}z^{n}&=|z|^{n}\left(\cos \left(n\arccos {\frac {a}{|z|}}\right)+i\sin \left(n\arccos {\frac {a}{|z|}}\right)\right)\\&=|z|^{n}T_{n}\left({\frac {a}{|z|}}\right)+ib\ |z|^{n-1}\ U_{n-1}\left({\frac {a}{|z|}}\right).\end{aligned}}$

Pell equation definition

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

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

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

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

Relation between Chebyshev polynomials of the first and second kinds

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.

Properties

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:

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

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

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

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

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, i.e. $U_{n-1}(1)=nT_{n}(1)=n$ 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

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

Orthogonality

Both the $T_{N}(x)$ and the $U_{N}(x)$ 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:

$\int \limits _{-1}^{1}T_{n}(x)T_{m}(x)\,{\frac {dx}{\sqrt {1-x^{2}}}}={\begin{cases}0&:n\neq m\\\pi &:n=m=0\\{\frac {\pi }{2}}&:n=m\neq 0\end{cases}}$

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:

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

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

The $T_{n}$ also satisfy a discrete orthogonality condition:

$\sum _{k=0}^{N-1}{T_{i}(x_{k})T_{j}(x_{k})}={\begin{cases}0&:i\neq j\\N&:i=j=0\\{\frac {N}{2}}&:i=j\neq 0\end{cases}}$

where the $x_{k}$ are the $N$ Gauss–Lobatto zeros of $T_{N}(x)$

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

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

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

Proof

Let's assume that $w_{n}(x)$ is a polynomial of degree n with leading coefficient 1 with maximal absolute value on the interval [−1, 1] less than ${\frac {1}{2^{n-1}}}$.

We define

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

Because at extreme points of $T_{n}$ we have $|w_{n}(x)|<|{\frac {1}{2^{n-1}}}T_{n}(x)|$

$f_{n}(x)>0{\text{ for }}x=\cos {\frac {2k\pi }{n}}{\text{ where }}0\leq 2k\leq n$

$f_{n}(x)<0{\text{ for }}x=\cos {\frac {(2k+1)\pi }{n}}{\text{ where }}0\leq 2k+1\leq n$

$f_{n}(x)$ is a polynomial of degree n - 1, so from the intermediate value theorem it has at least n roots which is impossible for polynomial of degree n - 1.

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:

• $T_{n}(x)={\frac {1}{n-{\frac {1}{2}} \choose n}}P_{n}^{-{\frac {1}{2}},-{\frac {1}{2}}}(x)={\frac {n}{2}}C_{n}^{0}(x),$
• $U_{n}(x)={\frac {1}{2{n+{\frac {1}{2}} \choose n}}}P_{n}^{{\frac {1}{2}},{\frac {1}{2}}}(x)=C_{n}^{1}(x).$

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. In fact,

$T_{n}\left(1-2x^{2}\right)=(-1)^{n}T_{2n}(x)$

and

$U_{n}\left(1-2x^{2}\right)x=(-1)^{n}U_{2n+1}(x).$

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 equal to n.

Several polynomial sequences like Lucas polynomials (L_n), Dickson polynomials(D_n), Fibonacci polynomials(F_n) are related to Chebyshev polynomials T_n and U_n.

The Chebyshev polynomials of the first kind satisfy the relation

$T_{j}(x)T_{k}(x)={\frac {1}{2}}\left(T_{j+k}(x)+T_{|j-k|}(x)\right),\quad \forall j,k\geq 0,\,$

which is easily proved from the product-to-sum formula for the cosine. The polynomials of the second kind satisfy the similar relation

$T_{j}(x)U_{k}(x)={\frac {1}{2}}\left(U_{j+k}(x)+U_{k-j}(x)\right),\quad \forall j,k$.

Similar to the formula

$T_{n}(\cos \theta )=\cos(n\theta )$

we have the analogous formula

$T_{2n+1}(\sin \theta )=(-1)^{n}\sin((2n+1)\theta )$.

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).

Example 1

Consider the Chebyshev expansion of $\log(1+x)$ . One can express

$\log(1+x)=\sum _{n=0}^{\infty }a_{n}T_{n}(x)$

One can find the coefficients $a_{n}$ either through the application of an inner product or by the discrete orthogonality condition. For the inner product,

$\int \limits _{-1}^{1}{\frac {T_{m}(x)\log(1+x)}{\sqrt {1-x^{2}}}}dx=\sum _{n=0}^{\infty }a_{n}\int \limits _{-1}^{1}{\frac {T_{m}(x)T_{n}(x)}{\sqrt {1-x^{2}}}}dx$

which gives

$a_{n}={\begin{cases}-\log(2)&:n=0\\{\dfrac {-2(-1)^{n}}{n}}&:n>0\end{cases}}$

Alternatively, when you cannot evaluate the inner product of the function you are trying to approximate, the discrete orthogonality condition gives

$a_{n}={\frac {2-\delta _{0n}}{N}}\sum _{k=0}^{N-1}T_{n}(x_{k})\log(1+x_{k})$

where $\delta _{ij}$ is the Kronecker delta function and the $x_{k}$ are the $N$ Gauss–Lobatto zeros of $T_{N}(x)$

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

This allows us to compute the coefficients $a_{n}$ very efficiently through the discrete cosine transform

$a_{n}={\frac {2-\delta _{0n}}{N}}\sum _{k=0}^{N-1}\cos \left({\frac {n\pi \left(k+{\frac {1}{2}}\right)}{N}}\right)\log(1+x_{k})$

Example 2

To provide another example:

${\begin{aligned}(1-x^{2})^{\alpha }=&-{\frac {1}{\sqrt {\pi }}}{\frac {\Gamma ({\frac {1}{2}}+\alpha )}{\Gamma (\alpha +1)}}+2^{1-2\alpha }\sum _{n=0}(-1)^{n}{2\alpha \choose \alpha -n}T_{2n}(x)\\=&2^{-2\alpha }\sum _{n=0}(-1)^{n}{2\alpha +1 \choose \alpha -n}U_{2n}(x).\end{aligned}}$

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.

Credits

• This page was originally created from Chebyshev polynomials at wikipedia