4-2

$\displaystyle \Int f(\tau) f(t+\tau) d\tau = 2\pi \Int \left\vert \hat{f}(\omega)\right\vert^2 e^{-i\omega t} d\omega$

(19)

$\displaystyle \left(\text{LHS}\right)$	$\displaystyle = \Int d\tau \left[ \left( \Int d\omega' \, \hat{f}(\omega') e^{-... ...ty}^{+\infty} d\omega d\omega'\, \hat{f}(\omega')\hat{f}(\omega) e^{-i\omega t}$
	$\displaystyle =2\pi \iint_{-\infty}^{+\infty} d\omega d\omega'\, \delta(\omega'... ...a t}d \omega =2\pi \Int \hat{f}^*(\omega)\hat{f}(\omega) e^{-i\omega t}d \omega$
	$\displaystyle =2\pi \Int \left\vert \hat{f}(\omega)\right\vert^2 e^{-i\omega t}d \omega =\left(\text{RHS}\right)$