4-1

$\displaystyle \Int f(t') K(t-t') dt' = 2\pi \Int \hat{f}(\omega)\hat{K} (\omega) e^{-i\omega t} d\omega$

(18)

$\displaystyle \left(\text{LHS}\right)$	$\displaystyle = \Int dt ' \left[ \left( \Int d\omega \hat{f}(\omega) e^{-i\omeg... ...+\infty} d\omega d\omega' \hat{f}(\omega)\hat{K}(\omega') e^{-\omega' t} \notag$
	$\displaystyle =2\pi \Int d\omega d\omega' \, \delta(\omega'-\omega) \hat{f}(\om... ... t} =2\pi \Int d\omega \, \hat{f}(\omega) \hat{K}(\omega) e^{-i\omega t} \notag$
	$\displaystyle = \left(\text{RHS}\right)$