CHEBYSHEV POLYNOMIALS

In mathematics the 'Chebyshev polynomials', named after Pafnuty Chebyshev, are a sequence of orthogonal polynomials which are related to de Moivre's formula and which are easily defined recursively, like Fibonacci or Lucas numbers. One usually distinguishes between 'Chebyshev polynomials of the first kind' which are denoted ''T''_''n'' and 'Chebyshev polynomials of the second kind' which are denoted ''U''_''n''. The letter T is used because of the alternative transliterations of the name ''Chebyshev'' as ''Tchebyshef'' or ''Tschebyscheff''.
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.
Chebyshev polynomials are important in approximation theory because the roots of the Chebyshev polynomials of the first kind, which are also called Chebyshev nodes, are used as nodes in polynomial interpolation. The resulting interpolation polynomial minimizes the problem of Runge's phenomenon and provides an approximation that is close to the polynomial of best approximation to a continuous function under the maximum norm. This approximation leads directly to the method of Clenshaw-Curtis quadrature.
In the study of differential equations they arise as the solution to the
'Chebyshev differential equations'
:$(1-x^2),y\text{'}\text{'}\; -\; x,y\text{'}\; +\; n^2,y\; =\; 0\; ,!$
and
:$(1-x^2),y\text{'}\text{'}\; -\; 3x,y\text{'}\; +\; n(n+2),y\; =\; 0\; ,!$
for the polynomials of the first and second kind, respectively. These equations are special cases of the Sturm-Liouville differential equation.

Contents

Definition

Trigonometric definition

Pell equation definition

Mutual recurrence

Orthogonality

Minimal $infty$-norm

Relation between Chebyshev polynomials of the first and second kind

Other properties

Examples

Polynomial in Chebyshev form

Chebyshev roots

Spread polynomials

See also

References

External links

Definition

The 'Chebyshev polynomials of the first kind' are defined by the recurrence relation
:$T\_0(x)\; =\; 1\; ,!$
:$T\_1(x)\; =\; x\; ,!$
:$T\_\{n+1\}(x)\; =\; 2xT\_n(x)\; -\; T\_\{n-1\}(x).\; ,!$
One example of a generating function for T_n is
:
The 'Chebyshev polynomials of the second kind' are defined by the recurrence relation
:$U\_0(x)\; =\; 1\; ,!$
:$U\_1(x)\; =\; 2x\; ,!$
:$U\_\{n+1\}(x)\; =\; 2xU\_n(x)\; -\; U\_\{n-1\}(x).\; ,!$
One example of a generating function for U_n is
:

Trigonometric definition

The Chebyshev polynomials of the first kind can be defined by the trigonometric identity:
:
whence:
:$T\_n(cos(\; heta))=cos(n\; heta)\; ,!$
for ''n'' = 0, 1, 2, 3, ..., while the polynomials of the second kind satisfy:
:
which is structurally quite similar to the Dirichlet kernel.
That cos(''nx'') is an ''n''th-degree polynomial in cos(''x'') can be seen by observing that cos(''nx'') is the real part of one side of de Moivre's formula, and the real part of the other side is a polynomial in cos(''x'') and sin(''x''), in which all powers of sin(''x'') are even and thus replaceable via the identity 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 heta)=2cos heta cos heta - cos(0 heta) = 2cos^{2}, heta - 1 ,!

:$$
cos(3 heta)=2cos heta cos(2 heta) - cos heta = 4cos^3, heta - 3cos heta ,!

and so forth. To trivially check whether the results seem reasonable, sum the coefficients on both sides of the equals sign (that is, setting theta equal to zero, for which the cosine is unity), and one sees that 1 = 2 - 1 in the former expression and 1 = 4 - 3 in the latter.
An immediate corollary is the composition identity (or the "nesting property")
:$T\_n(T\_m(x))\; =\; T\_\{ncdot\; m\}(x).,!$
Written explicitly
:$T\_n(x)\; =$
egin{cases}
cos(nrccos(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} ,!

(not forgetting that the inverse hyperbolic cosines of ''x'' and −''x'' differ by the constant π). From reasoning similar to that above, one can develop a closed-form generating formula for Chebyshev polynomials of the first kind:
:$$
cos(n heta)=rac{e^{i n heta}+e^{-i n heta}}{2}=rac{(e^{i heta})^n+(e^{i heta})^{-n}}{2} ,!

which, combined with DeMoivre's formula:
:$$
! e^{i heta}=cos heta+i sin heta=cos heta+i sqrt{1-cos^2 heta}=cos heta+sqrt{cos^2 heta-1} ,!

yields:
:$$
cos(n heta)=rac{left(cos heta+ sqrt{cos^2 heta-1}
ight)^n+left(cos heta+ sqrt{cos^2 heta-1},
ight)^{-n}}{2} ,!

which, of course, is far more expedient for determining the cosine of N times a given angle than is cranking through almost N rounds of the recursive generator calculation. Finally, if we replace $cos(\; heta)$ with ''x'', we can alternatively write:
:$$
T_n(x)=rac{left(x+ sqrt{x^2-1}
ight)^n+left(x+ sqrt{x^2-1}
ight)^{-n}}{2} ,!

Pell equation definition

The Chebyshev polynomials can also be defined as the solutions to the Pell equation
:$T\_i^2\; -\; (x^2-1)\; U\_\{i-1\}^2\; =\; 1\; ,!$
in a ring R[''x''] (e.g., see Demeyer (2007), p.70). Thus, they can be generated by the standard technique for Pell equations of taking powers of a fundamental solution:
:$T\_i\; +\; U\_\{i-1\}\; sqrt\{x^2-1\}\; =\; (x\; +\; sqrt\{x^2-1\})^i.\; ,!$

Mutual recurrence

Equivalently, the two sequences can also be defined at once 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 , then
:
::
::
::$=\; xT\_n(x)\; -\; (1\; -\; x^2)U\_\{n-1\}(x).\; ,!$
(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.)

Orthogonality

Both the ''T''_''n'' and the ''U''_''n'' form a sequence of orthogonal polynomials. The polynomials of the first kind are orthogonal with respect to the weight
:
on the interval [−1,1], i.e. we have:
:
egin{matrix}
0 &: n
e m~~~~~\
pi &: n=m=0\
pi/2 &: n=m
e 0
end{matrix}
ight. ,!

This can be proven by letting ''x='' cos(θ) and using the identity
''T_n'' (cos(θ))=cos(nθ). 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:
:$int\_\{-1\}^1\; U\_n(x)U\_m(x)sqrt\{1-x^2\},dx\; =$
egin{cases}
0 &: n
e m\
pi/2 &: n=m
end{cases} ,!

(which, when normalized to form a probability measure, is the Wigner semicircle distribution).

Minimal $infty$-norm

For any given $1\; le\; n$, among the polynomials of degree $n$ with leading coefficient 1, is the one of which the maximal absolute value on the interval $[-1,\; 1]$ is minimal.
This maximal absolute value is and $|f(x)|$ reaches this maximum exactly $n+1$ times: in $-1$ and $1$ and the other $n\; -\; 1$ extremal points of $f$.

Relation between Chebyshev polynomials of the first and second kind

The Chebyshev polynomials of the first and second kind are closely related by the following equations
:
:
:$T\_\{n+1\}(x)\; =\; xT\_n(x)\; -\; (1\; -\; x^2)U\_\{n-1\}(x),$
:$T\_n(x)\; =\; U\_n(x)\; -\; x\; ,\; U\_\{n-1\}(x).$
The recurrence relationship of the derivative of Chebyshev polynomials can be derived from these relations
:
This relationship is used in the Chebyshev spectral method of solving differential equations.

Other properties

The Chebyshev polynomials are a special case of the ultraspherical or Gegenbauer polynomials, which themselves are a special case of the Jacobi polynomials.
For every nonnegative integer $n$, $T\_n(x)$ 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 resp.
The leading coefficient of $T\_n$ is $2^\{n-1\}$ if $1\; le\; n$, but
$1$ if $0\; =\; n$.

Examples

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 are
:$U\_0(x)\; =\; 1\; ,$
:$U\_1(x)\; =\; 2x\; ,$
:$U\_2(x)\; =\; 4x^2\; -\; 1\; ,$
:$U\_3(x)\; =\; 8x^3\; -\; 4x\; ,$
:$U\_4(x)\; =\; 16x^4\; -\; 12x^2\; +\; 1\; ,$
:$U\_5(x)\; =\; 32x^5\; -\; 32x^3\; +\; 6x\; ,$
:$U\_6(x)\; =\; 64x^6\; -\; 80x^4\; +\; 24x^2\; -\; 1.\; ,$

Polynomial in Chebyshev form

A polynomial of degree ''N'' in Chebyshev form is a polynomial ''p''(''x'') of the form
:$p(x)\; =\; sum\_\{n=0\}^\{N\}\; a\_n\; T\_n(x)$
where ''T''_''n'' is the ''n''th Chebyshev polynomial.
Polynomials in Chebyshev form can be evaluated using the Clenshaw algorithm.

Chebyshev roots

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
:
ight)=0
one can easily prove that the roots of ''T''_''n'' are
:
ight) mbox{ , } k=1,ldots,n.
Similarly, the roots of ''U''_''n'' are
:
ight) mbox{ , } k=1,ldots,n.

Spread polynomials

The spread polynomials are in a sense equivalent to the Chebyshev polynomials of the first kind, but enable one to avoid square roots and conventional trigonometric functions in certain contexts, notably in rational trigonometry.

See also

★ Chebyshev nodes

★ Chebyshev filter

★ Chebyshev cube root

★ Legendre polynomials

★ Hermite polynomials

★ Chebyshev rational functions

★ Clenshaw-Curtis quadrature

★ Approximation theory

References

★

External links

★

★ Module for Chebyshev Polynomials by John H. Mathews