Trigonometry/For Enthusiasts/Chebyshev Polynomials

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The Chebyshev polynomials, named after Pafnuty Chebyshev,[1] are a sequence of polynomials related to the trigonometric multi-angle formulae.

We usually distinguish between

  • Chebyshev polynomials of the first kind, denoted Tn and are closely related to \cos and
  • Chebyshev polynomials of the second kind, denoted Un 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 Tn or Un are polynomials of degree n and the sequence of Chebyshev polynomials of either kind composes a 'polynomial sequence'.

Examples[edit]

The first few Chebyshev polynomials of the first kind in the domain −1 < x < 1: The flat T0, T1, T2, T3, T4 and T5.

The first few Chebyshev polynomials of the first kind are

 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, Un(1) = n + 1 and Un(−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 - 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. \,

Definition of Tn(x)[edit]

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

T_n(x) = \cos(n\arccos(x)),  x \in [-1,1]

In other words, Tn(x) is the polynomial that expresses \displaystyle \cos n \theta in therms of \displaystyle \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{align}
T_0(x) & = 1 \\
T_1(x) & = x \\
T_{n+1}(x) & = 2xT_n(x) - T_{n-1}(x).
\end{align}


The recurrence comes from the relation:

\displaystyle\cos((n+1)\theta) = 2\cos(\theta)\cos(n\theta) - \cos((n-1)\theta)

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

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

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

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

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

Uses[edit]

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_1T_1(x) + a_2T_2(x) + a_3T_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 Tn, 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 formulas[edit]

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 \ge 1 \\
(-1)^n \cosh(n \, \mathrm{arccosh}(-x)), & \ x \le -1 \\
\end{cases} \,\!



\begin{align}
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} \\
& = x^n \sum_{k=0}^{\lfloor n/2\rfloor} \binom{n}{2k} (1 - x^{-2})^k \\
& = \frac{n}{2}\sum_{k=0}^{\lfloor n/2\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)\\
& = \, _2F_1\left(-n,n;\frac 1 2; \frac{1-x} 2 \right) \\
\end{align}



\begin{align}
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} \\
& = x^n \sum_{k=0}^{\lfloor n/2\rfloor} \binom{n+1}{2k+1} (1 - x^{-2})^k \\
& =\sum_{k=0}^{\lfloor n/2\rfloor} \binom{2k-(n+1)}{k}~(2x)^{n-2k} \quad (n>0)\\
& =\sum_{k=0}^{\lfloor n/2\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) \, _2F_1\left(-n,n+2; \frac 3 2; \frac{1-x} 2 \right)
\end{align}

where _2F_1 is a hypergeometric function.

Uses[edit]

Chebyshev polynomials are important in approximation theory because the roots of the Chebyshev polynomials Tn, 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[edit]

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


\begin{align}
T_0(x) & = 1 \\
T_1(x) & = x \\
T_{n+1}(x) & = 2xT_n(x) - T_{n-1}(x).
\end{align}

The conventional generating function for Tn 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{align}
U_0(x) & = 1 \\
U_1(x) & = 2x \\
U_{n+1}(x) & = 2xU_n(x) - U_{n-1}(x).
\end{align}

One example of a generating function for Un is

\sum_{n=0}^{\infty}U_n(x) t^n = \frac{1}{1-2 t x+t^2}. \,\!

Trigonometric definition[edit]

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{align}
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{align}

Pell equation definition[edit]

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].[2] 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[edit]

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

\frac{d}{dx} \, T_n(x) = n U_{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

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

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{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}

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 Un (the polynomial of degree n) with Un+1 instead.


Properties[edit]

Roots and extrema[edit]

A Chebyshev polynomial of either kind with degree n has n different simple roots, called Chebyshev roots, in the interval [−1,1]. The roots 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 Tn are

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

Similarly, the roots of Un 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[edit]

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{d T_n}{d x} = n U_{n - 1}\,
\frac{d U_n}{d x} = \frac{(n + 1)T_{n + 1} - x U_n}{x^2 - 1}\,
\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}.\,

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}{d x^2} \Bigg|_{x = 1} \!\! = \frac{n^4 - n^2}{3},
\frac{d^2 T_n}{d x^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{n T_n - x U_{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{n T_n - x U_{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{n T_n - x U_{n - 1}}{(x + 1)(x - 1)} =
\lim_{x \to 1} n \frac{\frac{n T_n - x U_{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{n T_n - x U_{n - 1}}{x - 1}}{\lim_{x \to 1} (x + 1)} =
\frac{n}{2} \lim_{x \to 1} \frac{n T_n - x U_{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) = n T_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{align}
 T''_n(1) & = \frac{n}{2} \lim_{x \to 1} \frac{\frac{d}{dx}(n T_n - x U_{n - 1})}{\frac{d}{dx}(x - 1)} \\
 & = \frac{n}{2} \lim_{x \to 1} \frac{d}{dx}(n T_n - x U_{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}(n U_{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{align}

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}{d x^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 Tn 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{n T_{n + 1}}{n^2 - 1} - \frac{x T_n}{n - 1}.\,

Orthogonality[edit]

Both the Tn and the Un 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}}=
\begin{cases}
0 &: n\ne m \\
\pi &: n=m=0\\
\pi/2 &: n=m\ne 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_{-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}

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

The Tn also satisfy a discrete orthogonality condition:

 \sum_{k=0}^{N-1}{T_i(x_k)T_j(x_k)} = 
\begin{cases}
0 &: i\ne j \\
N &: i=j=0 \\
N/2 &: i=j\ne 0
\end{cases} \,\!

where the  x_k are the N GaussLobatto zeros of  T_N(x)

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

Minimal ∞-norm[edit]

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

f(x) = \frac1{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

\frac1{2^{n-1}}

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

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

Proof[edit]

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 \frac1{2^{n-1}}.

We define

f_n(x) = \frac1{2^{n-1}}T_n(x) - w_n(x)

Because at extreme points of T_n we have |w_n(x)| < |\frac1{2^{n-1}}T_n(x)|

f_n(x) > 0 \text{ for } x = \cos \frac{2k\pi}{n} \text{ where } 0 \le 2k \le n
f_n(x) < 0 \text{ for } x = \cos \frac{(2k + 1)\pi}{n} \text{ where } 0 \le 2k + 1 \le 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[edit]

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, Tn(x) and Un(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 Tn is 2n − 1 if 1 ≤ n, but 1 if 0 = n.

Tn are a special case of Lissajous curves with frequency ratio equal to n.

Several polynomial sequences like Lucas polynomials (Ln), Dickson polynomials(Dn), Fibonacci polynomials(Fn) are related to Chebyshev polynomials Tn and Un.

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\ge 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[edit]

File:ChebyshevExpansion.png
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:[3]

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 an can be determined easily through the application of an inner product. This sum is called a Chebyshev series or a Chebyshev expansion.

Since a Chebyshev series is related to a Fourier cosine series through a change of variables, all of the theorems, identities, etc. that apply to Fourier series have a Chebyshev counterpart.[3] 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,[3] often in favor of trigonometric series due to generally faster convergence for continuous functions (Gibbs' phenomenon is still a problem).

Example 1[edit]

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

 \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_{-1}^{+1}\frac{T_m(x)\log(1+x)}{\sqrt{1-x^2}}dx = \sum_{n=0}^{\infty}a_n\int_{-1}^{+1}\frac{T_m(x)T_n(x)}{\sqrt{1-x^2}}dx,

which gives


a_n=
\begin{cases}
-\log(2) &:n = 0 \\
\frac{-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[edit]

To provide another example:

\begin{align}(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{align}


Partial sums[edit]

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 an are through the use of the inner product as in Galerkin's method and through the use of collocation which is related to interpolation.

As an interpolant, the N coefficients of the (N − 1)th partial sum are usually obtained on the Chebyshev–Gauss–Lobatto[4] 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[edit]

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[edit]

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.


Notes[edit]

  1. Chebyshev polynomials were first presented in: P. L. Chebyshev (1854) "Théorie des mécanismes connus sous le nom de parallélogrammes," Mémoires des Savants étrangers présentés à l’Académie de Saint-Pétersbourg, vol. 7, pages 539-586.
  2. Jeroen Demeyer Diophantine Sets over Polynomial Rings and Hilbert's Tenth Problem for Function Fields, Ph.D. theses (2007), p.70.
  3. a b c Boyd, John P. (2001). Chebyshev and Fourier Spectral Methods (second ed.). Dover. ISBN 0486411834. http://www-personal.umich.edu/~jpboyd/aaabook_9500may00.pdf. 
  4. Chebyshev Interpolation: An Interactive Tour

Credits[edit]