# The proof of the dual convolution/multiplication properties? [closed]

I've been trying to find a rigorous proof of the dual convolution / multiplication, but I found nothing, can you give me a hand with this?

\begin{align} f(t) * g(t) &\overset{\mathcal F}{\iff} F(j\omega)G(j\omega)\\ f(t)g(t) &\overset{\mathcal F}{\iff}\frac1{2\pi} F(j\omega) * G(j\omega)\\ \end{align}

• Have you checked Wikipedia? – Tendero Dec 5 '18 at 15:09

\begin{align*}\mathscr{F}\left\{f(t) * g(t)\right\} &= \mathscr{F}\left\{\int_{-\infty}^\infty f(\tau)g(t-\tau)d\tau\right\} \\ \\ &= \int_{-\infty}^\infty\left[\int_{-\infty}^\infty f(\tau)g(t-\tau)d\tau\right]e^{-j\omega t}dt\\ \\ &= \int_{-\infty}^\infty f(\tau)\left[\int_{-\infty}^\infty g(t-\tau)e^{-j\omega t}dt\right]d\tau\\ \\ &= \int_{-\infty}^\infty f(\tau)e^{-j\omega \tau}G(\omega)d\tau\\ \\ &= F(\omega)G(\omega)\\ \end{align*}