Funciones ortogonales. Fourier.

Un conjunto de funciones $\phi_k (t)$ es ortogonal en un intervalo $a<t<b$ si para dos funciones cualesquiera $\phi_m (t)$ y $\phi_n (t)$ pertenecientes al conjunto $\phi_k (t)$ , cumple:

$\displaystyle \int_{a}^{b}{\phi_m (t) \phi_n (t) dt} = \left\{ \begin{matrix} 0, \quad para \quad m \ne n \\ r_n, \quad para \quad m=n \end{matrix} \right.$

Considerando como ejemplo, un conjunto de funciones senoidales, mediante el cálculo elemental se puede demostrar que

$\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\cos{m\omega_o t}dt}=0$ para $m \ne 0$

$\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\sin{m\omega_o t}dt}=0$ para todo valor de $m$

$\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\cos{m\omega_o t} \cos{n\omega_o t} dt} = \left\{ \begin{matrix} 0, \quad \quad \quad para \quad m \ne n \\ \frac{T}{2}, \quad para \quad m=n\ne 0 \end{matrix} \right.$

$\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\sin{m\omega_o t} \sin{n\omega_o t} dt} = \left\{ \begin{matrix} 0, \quad \quad \quad para \quad m \ne n \\ \frac{T}{2}, \quad para \quad m=n\ne 0 \end{matrix} \right.$

$\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\sin{m\omega_o t}\cos{n\omega_o t}dt}=0$ para todo valor de $m$ y $n$

donde $\displaystyle \omega_o=\frac{2\pi}{T}$ .

Estas relaciones demuestran que las funciones 1, $\cos{\omega_o t}$ , $\cos{2\omega_o t}$ , $\cos{3\omega_o t}$ , …, $\cos{n\omega_o t}$ , …, $\sin{\omega_o t}$ , $\sin{2\omega_o t}$ , $\sin{3\omega_o t}$ , …, $\sin{n\omega_o t}$ , …, forman un conjunto de funciones ortogonales en el intervalo $-\frac{T}{2}<t<\frac{T}{2}$ .

Demostraciones de las integrales mencionadas

Demostración de la primera integral

$\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\cos{m\omega_o t}dt} = \left[\frac{1}{m\omega_o} \sin{m\omega_o t}\right]_{-\frac{T}{2}}^{\frac{T}{2}}$

$\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\cos{m\omega_o t}dt} = \left[\frac{1}{m\omega_o} \sin{m\omega_o \left(-\frac{T}{2}\right)}\right] - \left[\frac{1}{m\omega_o} \sin{m\omega_o \left(\frac{T}{2}\right)}\right]$

$\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\cos{m\omega_o t}dt} = \left[\frac{1}{m\omega_o} \sin{m \left(\frac{2\pi}{T} \right) \left(-\frac{T}{2}\right)}\right] - \left[\frac{1}{m\omega_o} \sin{m \left(\frac{2\pi}{T} \right) \left(\frac{T}{2}\right)}\right]$

$\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\cos{m\omega_o t}dt} = \left[\frac{1}{m\omega_o} \sin{(-m\pi)}\right] - \left[\frac{1}{m\omega_o} \sin{(m\pi)} \right] = 0$

Finalmente

$\displaystyle \therefore \int_{-\frac{T}{2}}^{\frac{T}{2}}{\cos{m\omega_o t}dt}=0$

Demostración de la segunda integral

$\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\sin{m\omega_o t}dt} = \left[-\frac{1}{m\omega_o} \cos{m\omega_o t}\right]_{-\frac{T}{2}}^{\frac{T}{2}}$

$\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\sin{m\omega_o t}dt} = \left[-\frac{1}{m\omega_o} \cos{m\omega_o \left(-\frac{T}{2}\right)}\right] - \left[-\frac{1}{m\omega_o} \cos{m\omega_o \left(\frac{T}{2}\right)}\right]$

$\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\sin{m\omega_o t}dt} = \left[-\frac{1}{m\omega_o} \cos{m \left(\frac{2\pi}{T} \right) \left(-\frac{T}{2}\right)}\right] - \left[-\frac{1}{m\omega_o} \cos{m \left(\frac{2\pi}{T} \right) \left(\frac{T}{2}\right)}\right]$

$\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\sin{m\omega_o t}dt} = \left[-\frac{1}{m\omega_o} \cos{(-m\pi)}\right] - \left[-\frac{1}{m\omega_o} \cos{(m\pi)} \right]$

$\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\sin{m\omega_o t}dt} = - \frac{1}{m\omega_o} \cos{(m\pi)} + \frac{1}{m\omega_o} \cos{(m\pi)} = 0$

Finalmente

$\displaystyle \therefore \int_{-\frac{T}{2}}^{\frac{T}{2}}{\sin{m\omega_o t}dt}=0$

Demostración de la tercera integral

Si $m=n \ne 0$

$\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\cos{m\omega_o t} \cos{n\omega_o t} dt} = \int_{-\frac{T}{2}}^{\frac{T}{2}}{\frac{1}{2} \left[\cos{(m+n)\omega_o t} + \cos{(m-n)\omega_o t}\right] dt}$

$\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\cos{m\omega_o t} \cos{n\omega_o t} dt} = \frac{1}{2} \left[\frac{1}{(m+n)\omega_o} \sin{(m+n)\omega_o t} + \frac{1}{(m-n)\omega_o} \sin{(m-n)\omega_o t} \right]_{-\frac{T}{2}}^{\frac{T}{2}}$

$\displaystyle = \frac{1}{2} \left[\frac{1}{(m+n)\omega_o} \sin{(m+n)\left(\frac{2\pi}{T}\right) \left(\frac{T}{2}\right)} + \frac{1}{(m-n)\omega_o} \sin{(m-n)\left(\frac{2\pi}{T}\right)\left(\frac{T}{2}\right)} \right] - \frac{1}{2} \left[\frac{1}{(m+n)\omega_o} \sin{(m+n)\left(\frac{2\pi}{T}\right) \left(-\frac{T}{2}\right)} + \frac{1}{(m-n)\omega_o} \sin{(m-n)\left(\frac{2\pi}{T}\right)\left(-\frac{T}{2}\right)} \right]$

$\displaystyle = \frac{1}{2} \left[\frac{1}{(m+n)\omega_o} \sin{(m+n)\left(\pi \right)} + \frac{1}{(m-n)\omega_o} \sin{(m-n)\left(\pi \right)} \right] - \frac{1}{2} \left[\frac{1}{(m+n)\omega_o} \sin{(m+n)\left(-\pi \right)} + \frac{1}{(m-n)\omega_o} \sin{(m-n)\left(-\pi \right)} \right]$

$\displaystyle = \frac{1}{2(m+n)\omega_o} \sin{(m+n)\left(\pi \right)} + \frac{1}{2(m-n)\omega_o} \sin{(m-n)\left(\pi \right)} + \frac{1}{2(m+n)\omega_o} \sin{(m+n)\left(\pi \right)} + \frac{1}{2(m-n)\omega_o} \sin{(m-n)\left(\pi \right)}$

$\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\cos{m\omega_o t} \cos{n\omega_o t} dt} = \frac{1}{(m+n)\omega_o} \sin{(m+n)\left(\pi \right)} + \frac{1}{(m-n)\omega_o} \sin{(m-n)\left(\pi \right)}$

$\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\cos{m\omega_o t} \cos{n\omega_o t} dt}= \frac{1}{(m+n)\omega_o} \sin{2m\left(\pi \right)} =0$

Si $m = n \ne 0$

$\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\cos{m\omega_o t} \cos{n\omega_o t} dt} = \int_{-\frac{T}{2}}^{\frac{T}{2}}{\cos^2{m\omega_o t} \, dt}$

$\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\cos{m\omega_o t} \cos{n\omega_o t} dt} = \int_{-\frac{T}{2}}^{\frac{T}{2}}{\left(\frac{1}{2} + \frac{1}{2}\cos{2mt}\right) \, dt} = \left[\frac{1}{2}t + \frac{1}{4m}\sin{2mt}+C \right]_{-\frac{T}{2}}^{\frac{T}{2}}$

$\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\cos{m\omega_o t} \cos{n\omega_o t} dt} = \left[\frac{1}{2} \left( \frac{T}{2} \right) + \frac{1}{4m}\sin{2m\left( \frac{T}{2} \right)} \right] - \left[\frac{1}{2}\left( -\frac{T}{2} \right) + \frac{1}{4m}\sin{2m\left( -\frac{T}{2} \right)} \right]$

$\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\cos{m\omega_o t} \cos{n\omega_o t} dt} = \left[\frac{T}{4} + \frac{1}{4m}\sin{(mT)} \right] - \left[-\frac{T}{4} - \frac{1}{4m}\sin{(mT)} \right]$

$\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\cos{m\omega_o t} \cos{n\omega_o t} dt} = \frac{T}{4} + \frac{1}{4m}\sin{(mT)} + \frac{T}{4} + \frac{1}{4m}\sin{(mT)}$

$\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\cos{m\omega_o t} \cos{n\omega_o t} dt} = \frac{T}{2} + \frac{1}{2m}\sin{(mT)} = \frac{T}{2}$

donde $m$ es un número entero.

Finalmente

$\displaystyle \therefore \int_{-\frac{T}{2}}^{\frac{T}{2}}{\cos{m\omega_o t} \cos{n\omega_o t} dt} = \left\{ \begin{matrix} 0, \quad \quad \quad \text{para} \quad m \ne n \\ \frac{T}{2}, \quad \text{para} \quad m=n\ne 0 \end{matrix} \right.$

Demostración de la cuarta integral

Si $m=n \ne 0$

$\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\sin{m\omega_o t} \sin{n\omega_o t} dt} = \int_{-\frac{T}{2}}^{\frac{T}{2}}{\frac{1}{2} \left[-\cos{(m+n)\omega_o t} + \cos{(m-n)\omega_o t}\right] dt}$

$\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\sin{m\omega_o t} \sin{n\omega_o t} dt} = \frac{1}{2} \left[-\frac{1}{(m+n)\omega_o} \sin{(m+n)\omega_o t} + \frac{1}{(m-n)\omega_o} \sin{(m-n)\omega_o t} \right]_{-\frac{T}{2}}^{\frac{T}{2}}$

$\displaystyle = \frac{1}{2} \left[-\frac{1}{(m+n)\omega_o} \sin{(m+n)\left(\frac{2\pi}{T}\right) \left(\frac{T}{2}\right)} + \frac{1}{(m-n)\omega_o} \sin{(m-n)\left(\frac{2\pi}{T}\right)\left(\frac{T}{2}\right)} \right] - \frac{1}{2} \left[-\frac{1}{(m+n)\omega_o} \sin{(m+n)\left(\frac{2\pi}{T}\right) \left(-\frac{T}{2}\right)} + \frac{1}{(m-n)\omega_o} \sin{(m-n)\left(\frac{2\pi}{T}\right)\left(-\frac{T}{2}\right)} \right]$

$\displaystyle = \frac{1}{2} \left[-\frac{1}{(m+n)\omega_o} \sin{(m+n)\left(\pi \right)} + \frac{1}{(m-n)\omega_o} \sin{(m-n)\left(\pi \right)} \right] - \frac{1}{2} \left[-\frac{1}{(m+n)\omega_o} \sin{(m+n)\left(-\pi \right)} + \frac{1}{(m-n)\omega_o} \sin{(m-n)\left(-\pi \right)} \right]$

$\displaystyle = - \frac{1}{2(m+n)\omega_o} \sin{(m+n)\left(\pi \right)} + \frac{1}{2(m-n)\omega_o} \sin{(m-n)\left(\pi \right)} - \frac{1}{2(m+n)\omega_o} \sin{(m+n)\left(\pi \right)} + \frac{1}{2(m-n)\omega_o} \sin{(m-n)\left(\pi \right)}$

$\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\sin{m\omega_o t} \sin{n\omega_o t} dt} = - \frac{1}{(m+n)\omega_o} \sin{(m+n)\left(\pi \right)} + \frac{1}{(m-n)\omega_o} \sin{(m-n)\left(\pi \right)}$

$\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\sin{m\omega_o t} \sin{n\omega_o t} dt} = - \frac{1}{(m+n)\omega_o} (0) + \frac{1}{(m-n)\omega_o} (0) =0$

Si $m = n \ne 0$

$\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\sin{m\omega_o t} \sin{n\omega_o t} dt} = \int_{-\frac{T}{2}}^{\frac{T}{2}}{\sin^2{m\omega_o t} \, dt}$

$\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\sin{m\omega_o t} \sin{n\omega_o t} dt} = \int_{-\frac{T}{2}}^{\frac{T}{2}}{\left(\frac{1}{2} - \frac{1}{2}\cos{2mt}\right) \, dt} = \left[\frac{1}{2}t - \frac{1}{4m}\sin{2mt}+C \right]_{-\frac{T}{2}}^{\frac{T}{2}}$

$\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\sin{m\omega_o t} \sin{n\omega_o t} dt} = \left[\frac{1}{2} \left( \frac{T}{2} \right) - \frac{1}{4m}\sin{2m\left( \frac{T}{2} \right)} \right] - \left[\frac{1}{2}\left( -\frac{T}{2} \right) - \frac{1}{4m}\sin{2m\left( -\frac{T}{2} \right)} \right]$

$\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\sin{m\omega_o t} \sin{n\omega_o t} dt} = \left[\frac{T}{4} - \frac{1}{4m}\sin{(mT)} \right] - \left[-\frac{T}{4} + \frac{1}{4m}\sin{(mT)} \right]$

$\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\sin{m\omega_o t} \sin{n\omega_o t} dt} = \frac{T}{4} - \frac{1}{4m}\sin{(mT)} + \frac{T}{4} - \frac{1}{4m}\sin{(mT)}$

$\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\sin{m\omega_o t} \sin{n\omega_o t} dt} = \frac{T}{2} - \frac{1}{2m}\sin{(mT)} = \frac{T}{2}$

donde $m$ es un número entero.

Finalmente

$\displaystyle \therefore \int_{-\frac{T}{2}}^{\frac{T}{2}}{\sin{m\omega_o t} \sin{n\omega_o t} dt} = \left\{ \begin{matrix} 0, \quad \quad \quad \text{para} \quad m \ne n \\ \frac{T}{2}, \quad \text{para} \quad m=n\ne 0 \end{matrix} \right.$

Demostración de la quinta integral

$\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\sin{m\omega_o t} \cos{n\omega_o t} dt} = \int_{-\frac{T}{2}}^{\frac{T}{2}}{\frac{1}{2} \left[\sin{(m+n)\omega_o t} + \sin{(m-n)\omega_o t}\right] dt}$

$\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\sin{m\omega_o t} \cos{n\omega_o t} dt} = \frac{1}{2} \left[-\frac{1}{(m+n)\omega_o} \cos{(m+n)\omega_o t} - \frac{1}{(m-n)\omega_o} \cos{(m-n)\omega_o t} \right]_{-\frac{T}{2}}^{\frac{T}{2}}$

$\displaystyle = \frac{1}{2} \left[-\frac{1}{(m+n)\omega_o} \cos{(m+n)\left(\frac{2\pi}{T}\right) \left(\frac{T}{2}\right)} - \frac{1}{(m-n)\omega_o} \cos{(m-n)\left(\frac{2\pi}{T}\right)\left(\frac{T}{2}\right)} \right] - \frac{1}{2} \left[-\frac{1}{(m+n)\omega_o} \cos{(m+n)\left(\frac{2\pi}{T}\right) \left(-\frac{T}{2}\right)} - \frac{1}{(m-n)\omega_o} \cos{(m-n)\left(\frac{2\pi}{T}\right)\left(-\frac{T}{2}\right)} \right]$

$\displaystyle = \frac{1}{2} \left[-\frac{1}{(m+n)\omega_o} \cos{(m+n)\left(\pi \right)} - \frac{1}{(m-n)\omega_o} \cos{(m-n)\left(\pi \right)} \right] - \frac{1}{2} \left[-\frac{1}{(m+n)\omega_o} \cos{(m+n)\left(-\pi \right)} - \frac{1}{(m-n)\omega_o} \cos{(m-n)\left(-\pi \right)} \right]$

$\displaystyle = - \frac{1}{2(m+n)\omega_o} \cos{(m+n)\left(\pi \right)} - \frac{1}{2(m-n)\omega_o} \cos{(m-n)\left(\pi \right)} + \frac{1}{2(m+n)\omega_o} \cos{(m+n)\left(\pi \right)} + \frac{1}{2(m-n)\omega_o} \cos{(m-n)\left(\pi \right)}$

$\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\sin{m\omega_o t} \cos{n\omega_o t} dt} = 0$

Finalmente

$\displaystyle \therefore \int_{-\frac{T}{2}}^{\frac{T}{2}}{\sin{m\omega_o t}\cos{n\omega_o t}dt}=0$

Funciones ortogonales. Fourier.