orthogonality of Chebyshev polynomials

By expanding the of de Moivre identity

$$\cos n\phi ={(\cos \phi +i\sin \phi )}^n$$

to sum, one obtains as real part certain terms containing power products of $\cos \phi$ and $\sin \phi$, the latter ones only with even exponents.  When these are expressed with cosines (${\sin }^2\phi =1-{\cos }^2\phi$), the real part becomes a polynomial $T_n$ of degree $n$ in the argument (http://planetmath.org/Argument2) $\cos \phi$:

$\cos n\phi =T_n(\cos \phi )$ (1)

This can be written equivalently (http://planetmath.org/Equivalent3)

$T_n(x)=\cos (n\arccos x).$ (2)

It’s a question of Chebyshev polynomial of first kind and of $n$ (cf. special cases of hypergeometric function).

For showing the orthogonality of $T_m$ and $T_n$ we start from the integral ${\displaystyle {\int }_0^{\pi }}\cos m\phi \cos n\phi d\phi$, which via the substitution

$$\cos \phi :=x,dx=-\sin \phi d\phi =-\sqrt{1-x^2}d\phi$$

changes to

${\displaystyle {\int }_0^{\pi }}\cos m\phi \cos n\phi d\phi =-{\displaystyle {\int }_1^{-1}}{T}_m(x)T_n(x){\displaystyle \frac{dx}{\sqrt{1-x^2}}}.$ (3)

The left of this equation is evaluated by using the product formula in the entry trigonometric identities:

${\displaystyle {\int }_0^{\pi }}\cos m\phi \cos n\phi d\phi ={\displaystyle \frac{1}{2}}{\displaystyle {\int }_0^{\pi }}(\cos (m-n)\phi +\cos (m+n)\phi )𝑑\phi =\{\begin{array}{cc}0\text{ for }m\ne n,\hfill & \\ \frac{\pi }{2}\text{ for }m=n\ne 0.\hfill & \end{array}$

By (3), we thus have

${\displaystyle {\int }_{-1}^1}{T}_m(x)T_n(x){\displaystyle \frac{dx}{\sqrt{1-x^2}}}=\{\begin{array}{cc}0\text{ for }m\ne n,\hfill & \\ \frac{\pi }{2}\text{ for }m=n\ne 0,\hfill & \end{array}$

which means the orthogonality of the polynomials $T_m(x)$ and $T_n(x)$ weighted by $\frac{1}{\sqrt{1-x^2}}$.

Any Riemann integrable real function $f$, defined on  $[-1,\mathrm{\hspace{0.17em}1}]$,  may be expanded to the series

$$f(x)=\frac{a_0}{2}{T}_0(x)+\sum _{j=1}^{\mathrm{\infty }}{a}_j{T}_j(x),$$

where

$$a_j=\frac{2}{\pi }{\int }_{-1}^{1}f(x)T_j(x)\frac{dx}{\sqrt{1-x^2}}\mathit{\hspace{1em}\hspace{1em}}(j=0,\mathrm{\hspace{0.17em}1},\mathrm{\hspace{0.17em}2},\mathrm{\dots })$$

This concerns especially the polynomials  $f(x):=x^n$,  for which we obtain

$x^n$ $\mathrm{\hspace{0.33em}}={\cos }^n\phi ={\cosh }^{n}i\phi ={\mathrm{\hspace{0.33em}2}}^{-n}(e^{i\phi }+e^{-i\phi })$

$\mathrm{\hspace{0.33em}}={\mathrm{\hspace{0.33em}2}}^{-n}\left[\left({\displaystyle \genfrac{}{}{0pt}{}{n}{0}}\right)(e^{ni\phi }+e^{-ni\phi })+\left({\displaystyle \genfrac{}{}{0pt}{}{n}{1}}\right)(e^{(n-2)i\phi }+e^{-(n-2)i\phi })+\mathrm{\dots }\right]$

$\mathrm{\hspace{0.33em}}=2^{1-n}\left[\left({\displaystyle \genfrac{}{}{0pt}{}{n}{0}}\right)T_n(x)+\left({\displaystyle \genfrac{}{}{0pt}{}{n}{1}}\right)T_{n-2}(x)+\left({\displaystyle \genfrac{}{}{0pt}{}{n}{2}}\right)T_{n-4}(x)+\mathrm{\dots }\right].$

(If $n$ is even, the last term contains $T_0(x)$ but its coefficient is only a half of the middle number of the Pascal’s triangle row in question.)  Explicitly:

$1=T_0$
$x=T_1$
$x^2={\mathrm{\hspace{0.33em}2}}^{-1}(T_2+T_0)$
$x^3={\mathrm{\hspace{0.33em}2}}^{-2}(T_3+3T_1)$
$x^4={\mathrm{\hspace{0.33em}2}}^{-3}(T_4+4T_2+3T_0)$
$x^5={\mathrm{\hspace{0.33em}2}}^{-4}(T_5+5T_3+10T_1)$
$x^6={\mathrm{\hspace{0.33em}2}}^{-5}(T_6+6T_4+15T_2+10T_0)$
$x^7={\mathrm{\hspace{0.33em}2}}^{-6}(T_7+7T_5+21T_3+35T_1)$
$x^8={\mathrm{\hspace{0.33em}2}}^{-7}(T_8+8T_6+28T_4+56T_2+36T_0)$
$x^9={\mathrm{\hspace{0.33em}2}}^{-8}(T_9+9T_7+36T_5+84T_3+126T_1)$
$\mathrm{\cdots }\mathit{\hspace{1em}\hspace{1em}}\mathrm{\cdots }$

Title	orthogonality of Chebyshev polynomials
Canonical name	OrthogonalityOfChebyshevPolynomials
Date of creation	2013-03-22 18:54:42
Last modified on	2013-03-22 18:54:42
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	16
Author	pahio (2872)
Entry type	Derivation
Classification	msc 33C45
Classification	msc 33D45
Classification	msc 42C05
Related topic	OrthogonalPolynomials
Related topic	LaguerrePolynomial
Related topic	ChangeOfVariableInDefiniteIntegral
Related topic	DeterminationOfFourierCoefficients
Related topic	OrthogonalityOfLegendrePolynomials
Related topic	PropertiesOfOrthogonalPolynomials