Chebyshev polynomials

thumb|Plot of the first five Chebyshev polynomials (first kind)

thumb|Plot of the first five Chebyshev polynomials (second kind)

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

T_n(\cos \theta) = \cos(n\theta).

</math>

Similarly, the Chebyshev polynomials of the second kind <math>U_n</math> are defined by

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 are polynomials with the largest possible leading coefficient whose absolute value on the interval is bounded by 1. They are also the "extremal" polynomials for many other properties.

In 1952, Cornelius Lanczos showed that the Chebyshev polynomials are important in approximation theory for the solution of linear systems; the roots of , 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. The letter is used because of the alternative transliterations of the name Chebyshev as , (French) or (German).

Definitions

Recurrence definition

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 definition

The Chebyshev polynomials of the first and second kind can be defined as the unique polynomials satisfying

T_n(\cos\theta) = \cos(n\theta) \quad

</math>

and

U_n(\cos\theta) = \frac\right).

</math>

The ordinary generating function for is

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

</math>

and the exponential generating function is

\sum_{n=0}^\infty U_n(x) \frac{t^n}{n!}

= e^{tx} \biggl( {\cosh}\bigl({\textstyle t\sqrt{x^2 - 1} ~\!}\bigr) + {\frac{x}{\sqrt{x^2 - 1 \sinh}\bigl({\textstyle t\sqrt{x^2 - 1}~\!}\bigr)\biggr).

</math>

Relations between the two kinds of Chebyshev polynomials

The Chebyshev polynomials of the first and second kinds correspond to a complementary pair of Lucas sequences <math>\tilde V_n(P,Q)</math> and <math>\tilde U_n(P,Q)</math> with parameters <math>P=2x</math> and <math>Q=1</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:

<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:

T_n(x) = \tfrac12 \big(U_n(x) - U_{n-2}(x)\big).

</math>

Using this formula iteratively gives the sum formula:

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>:

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),

</math>

for <math>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:

<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 are

<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), \\[3mu]

\int_{-1}^1\frac{U_{n-1}(y)}{y-x}\, {\textstyle \sqrt{1 - y^2} }\, \mathrm{d}y &= -\pi\,T_n(x)

\end{align}</math>

where integrals are considered as principal value.

Explicit expressions

Using the complex number exponentiation definition of the Chebyshev polynomial, one can derive the following expressions, valid for any real :

<math display=block>\begin{align}

T_n(x)

&= \tfrac12 \Bigl(

\bigl( {\textstyle x - \sqrt{ x^2 - 1}\!~} \bigr)^n +

\bigl( {\textstyle x + \sqrt{x^2 - 1}\!~} \bigr)^n \Bigr) \\[5mu]

&= \tfrac12 \Bigl( \bigl( {\textstyle x - \sqrt{ x^2 - 1 } \!~} \bigr)^n + \bigl( {\textstyle x - \sqrt{x^2 - 1 }\!~} \bigr)^{-n} \Bigr).

\end{align}</math>The two are equivalent because <math>\textstyle \bigl(x + \sqrt{x^2 - 1 }\!~\bigr)^{\pm 1} = \bigl(x - \sqrt{x^2 - 1}\!~\bigr)^{\mp 1}.</math>

An explicit form of the Chebyshev polynomial in terms of monomials <math>\textstyle x^k</math> can be obtained as follows. Letting <math>\mathfrak{R}</math> denote the real part of a complex number, the following equalities, in order, follow by the definition of <math>T_n</math>, the definition of <math>\mathfrak{R}</math>, de Moivre's formula, and the binomial theorem:<math display="block">\begin{align}

T_n \bigl(\cos(\theta)\bigr) &= \cos(n\theta) \\

&= \mathfrak{R} \bigl( \cos(n\theta) + i \sin(n\theta) \bigr) \\

&= \mathfrak{R} \bigl( ( \cos(\theta) + i \sin(\theta))^n \bigr) \\

&= \mathfrak{R} \left( \sum_{j=0}^n \binom{n}{j}\, i^j \sin^j(\theta)\, \cos^{n-j}(\theta) \right).

\end{align}</math>Because of the factor of <math>i^j</math>, the even-indexed terms are purely real, while the odd-indexed terms are purely imaginary; furthermore, <math>\sin^{2j} \theta = \left(1 - \cos^2 \theta\right)^j,</math> so<math display="block">T_n \bigl(\cos(\theta)\bigr) = \sum_{j=0}^{\lfloor n/2 \rfloor} \binom{n}{2j}\, (-1)^j (1 - \cos^2(\theta))^j \cos^{n-2j}(\theta).</math>Finally, substituting <math>x = \cos(\theta)</math> yields<math display="block">

T_n(x) = \sum\limits_{j=0}^{\lfloor n/2 \rfloor} \binom{n}{2j} \left(x^2 - 1 \right)^j x^{n-2j}.

</math>This can be written as a hypergeometric function:

<math display="block">\begin{align}

T_n(x)

&= \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 \\

&= \tfrac12 n \sum_{k=0}^{ \lfloor n/2 \rfloor} (-1)^k \frac{ (n-k-1)! }{ k! (n-2k)! } ( 2 x)^{n-2k} \qquad \text{ for } n > 0 \\

& = n \sum_{k=0}^{n}(-2)^{k} \frac{ (n+k-1)! }{ (n-k)! (2k)! } ( 1 - x )^k \qquad \text{ for } n > 0 \\

& = {}_2F_1 \bigl( {-n}, n ; \tfrac12 ; \tfrac12 (1 - x) \bigr) \\

\end{align}</math>with inverse<math display="block">

x^n = \frac{1}{2^{n-1 \mathop{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 ( indeterminate form, specifically) at <math>x=1</math> and <math>x=-1</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\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 is to be halved, if it appears.

Concerning integration, the first derivative of the 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 <math>n\geq 2</math>:<math display="block">

\int T_n\, \mathrm{d}x

= \frac12 \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 <math>T_n</math> 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}

\dfrac{(-1)^n + 1}{1 - n^2} & \text{ if } n \ne 1 \\[3mu]

0 & \text{ if } n = 1.

\end{cases}</math>

Products of Chebyshev polynomials

The Chebyshev polynomials of the first kind satisfy the relation<math display=block>

T_m(x)\,T_n(x) = \tfrac12 {\left(T_{m+n}(x) + T_{|m-n|}(x)\right)},

</math>for all non-negative values of and , 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 <math>n=1</math> this results in the already-known recurrence formula, just arranged differently, and with <math>n=2</math> it forms the recurrence relation for all even or all odd indexed Chebyshev polynomials (depending on the parity of the lowest ) 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, \\[3mu]

T_{2n+1}(x) &= 2 T_{n+1}(x)\,T_n(x) - T_1(x) &&= 2 T_{n+1}(x)\,T_n(x) - x, \\[3mu]

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}

\frac12\bigl(U_{m+n}(x) + U_{n-m}(x)\bigr), & \text{ if } n \ge m-1,\\[5mu]

\frac12\bigl(U_{m+n}(x) - U_{m-n-2}(x)\bigr), & \text{ if } n \le m-2.

\end{cases}</math>(with the definition <math>U_{-1}\equiv 0</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 <math>m\geq n</math>. For <math>n=2</math> this recurrence reduces to:<math display="block">\begin{align}

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),

\end{align}</math>which establishes the evenness or oddness of the even or odd indexed Chebyshev polynomials of the second kind depending on whether <math>m</math> starts with 2 or 3.

Composition and divisibility properties

The trigonometric definitions of <math>T_n</math> and <math>U_n</math> imply the composition or nesting properties:<math display=block>\begin{align}

T_{mn}(x) &= T_m \bigl(T_n(x)\bigr),\\[3mu]

U_{mn-1}(x) &= U_{m-1}\bigl(T_n(x)\bigr)\,U_{n-1}(x).

\end{align}

</math>For <math>T_{mn}</math> the order of composition may be reversed, making the family of polynomial functions <math>T_n</math> a commutative semigroup under composition.

Since <math>T_m(x)</math> is divisible by <math>x</math> if <math>m</math> is odd, it follows that <math>T_{mn}(x)</math> is divisible by <math>T_n(x)</math> if <math>m</math> is odd. Furthermore, <math>U_{mn-1}(x)</math> is divisible by <math>U_{n-1}(x)</math>, and in the case that <math>m</math> is even, divisible by <math>T_n(x)\,U_{n-1}(x)</math>.

Orthogonality

Both <math>T_n</math> and <math>U_n</math> form a sequence of orthogonal polynomials. The polynomials of the first kind <math>T_n</math> are orthogonal with respect to the weight:<math display=block>

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

</math>on the interval , 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 <math>x=\cos(\theta)</math> and using the defining identity <math>T_n(\cos(\theta))=\cos(n\theta)</math>.

Similarly, the polynomials of the second kind are orthogonal with respect to the weight<math display=block>

\sqrt{1-x^2}

</math>on the interval , 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 <math>\sqrt{1-x^2}\, \mathrm{d}x</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 <math>T_n</math> 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 <math>N</math> is any integer greater than <math>\max(i,j)</math>, and the <math>x_k</math> are the <math>N</math> Chebyshev nodes (see above) of <math>T_N(x)</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 <math>N>i+j</math> with the same Chebyshev nodes <math>x_k</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 <math>N>i+j</math>, based on the <math>N</math>} zeros of <math>U_N(x)</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 -norm

For any given <math>n\geq 1</math>, among the polynomials of degree <math>n</math> with leading coefficient 1 (monic polynomials):

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

</math>

is the one of which the maximal absolute value on the interval is minimal.

This maximal absolute value is:

\frac1{2^{n-1

</math>

and <math>|f(x)|</math> reaches this maximum exactly <math>n+1</math> times at:

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

</math>

T_n(x) - w_n(x)

</math>

Because at extreme points of we have

<math display=block>\begin{align}

|w_n(x)| &< \left|\frac1{2^{n-1T_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, has at least roots. However, this is impossible, as is a polynomial of degree , so the fundamental theorem of algebra implies it has at most roots.

Remark

By the equioscillation theorem, among all the polynomials of degree , the polynomial minimizes on if and only if there are points such that .

Of course, the null polynomial on the interval can be approximated by itself and minimizes the -norm.

Above, however, reaches its maximum only times because we are searching for the best polynomial of degree (therefore the theorem evoked previously cannot be used).

Chebyshev polynomials as special cases of more general polynomial families

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(-\frac12, -\frac12 \right)}(x)

= \frac{2^{2n{\binom{2n}{n P_n^{\left(-\frac12, -\frac12\right)}(x), \\[2ex]

U_n(x)

&= C_n^{(1)}(x) \\

&= \frac{n+1}{\binom{n+\frac12}{n P_n^{\left(\frac12, \frac12 \right)}(x)

= \frac{2^{2n+1{\binom{2n+2}{n+1 P_n^{\left(\frac12, \frac12\right)}(x).

\end{align}</math>

Chebyshev polynomials are also a special case of Dickson polynomials:

<math display=block>\begin{align}

D_n(2x\alpha,\alpha^2) &= 2\alpha^{n}T_n(x), \\

E_n(2x\alpha,\alpha^2) &= \alpha^{n}U_n(x).

\end{align}</math>

In particular, when <math>\alpha = \tfrac12</math>, they are related by <math>D_n\bigl(x,\tfrac14\bigr) = 2^{1-n}T_n(x)</math> and <math>E_n\bigl(x,\tfrac14\bigr) = 2^{-n}U_n(x)</math>.

Other properties

The curves given by , or equivalently, by the parametric equations , , are a special case of Lissajous curves with frequency ratio equal to .

Similar to the formula:

T_n(\cos\theta) = \cos(n\theta),

</math>

we have the analogous formula:

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

</math>

For :

T_n\!\left(\frac{x + x^{-1{2}\right) = \frac{x^n+x^{-n{2}

</math>

and:

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 .

There are relations between Legendre polynomials and Chebyshev polynomials

<math display=block>\begin{align}

\sum_{k=0}^{n}P_{k}(x)\,T_{n-k}(x) &= \left(n+1\right)P_{n}(x), \\

\sum_{k=0}^{n}P_{k}(x)\,P_{n-k}(x) &= U_{n}(x).

\end{align}</math>

These identities can be proven using generating functions and discrete convolution.

Chebyshev polynomials as determinants

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>:

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>.

Examples

First kind

[[File:Chebyshev Polynomials of the 1st Kind (n=0-5, x=(-1,1)).svg|thumb|300px|The first few Chebyshev polynomials of the first kind in the domain :

The flat , , , , and .]]

The first few Chebyshev polynomials of the first kind are

<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 kind

[[File:Chebyshev Polynomials of the 2nd Kind (n=0-5, x=(-1,1)).svg|thumb|300px|The first few Chebyshev polynomials of the second kind in the domain : The flat , , , , and .

Although not visible in the image, and .]]

The first few Chebyshev polynomials of the second kind are

<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 set

thumb|right|240px|The non-smooth function (top) , where 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 an orthonormal basis, so that a function in the same space can, on , be expressed via the expansion:

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

</math>

Polynomial in Chebyshev form

An arbitrary polynomial of degree can be written in terms of the Chebyshev polynomials of the first kind. Such a polynomial is of the form:

p(x) = \sum_{n=0}^N a_n T_n(x).

</math>

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

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:

C_n(x) = 2T_n\left(\frac{x}{2}\right),\qquad S_n(x) = U_n\left(\frac{x}{2}\right)

</math>

and satisfy:

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 All of these polynomials have 1 as their leading coefficient. Lists of both sets of polynomials are given in Viète's Opera Mathematica, Chapter IX, Theorems VI and VII. 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 and of imaginary argument.

Shifted Chebyshev polynomials of the first and second kinds are related to the Chebyshev polynomials by:

{T}_n^*(x) = T_n(2x-1),\qquad {U}_n^*(x) = U_n(2x-1).

</math>

When the argument of the Chebyshev polynomial satisfies the argument of the shifted Chebyshev polynomial satisfies . Similarly, one can define shifted polynomials for generic intervals .

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." The Chebyshev polynomials of the third kind are defined as:

V_n(x) = \frac{\cos\left(\left(n+\frac12\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:

W_n(x) = \frac{\sin\left(\left(n+\frac12\right)\theta\right)}

Definitions

Recurrence definition

Trigonometric definition

Relations between the two kinds of Chebyshev polynomials

Explicit expressions

Products of Chebyshev polynomials

Composition and divisibility properties

Orthogonality

Minimal -norm

Remark

Chebyshev polynomials as special cases of more general polynomial families

Other properties

Chebyshev polynomials as determinants

Examples

First kind

Second kind

As a basis set

Polynomial in Chebyshev form

Families of polynomials related to Chebyshev polynomials