[Math] Fourier transform of the Fourier transform

Question

Can someone refer me on the Fourier transform of the fourier transform or clarify it for me?

It is known that the F.T. of the F.T. is some small modification of the original function itself, but I can't find anything about this online.

Answer 1

The Fourier transform of $f$ is: $$F(\omega)=\mathscr{F}(f(t))\{\omega\}=\int_{-\infty}^{\infty} \mathrm{d}t f(t) e^{-i \omega t}$$ Transforming it again: $$g(\tau)=\mathscr{F}(F(\omega))\{\tau\}=\int_{-\infty}^{\infty} \mathrm{d} \omega e^{-i \tau \omega}\int_{-\infty}^{\infty} \mathrm{d}t f(t) e^{-i \omega t}$$ Changing the order of integrations: $$=\int_{-\infty}^{\infty} \mathrm{d}t f(t) \int_{-\infty}^{\infty} \mathrm{d} \omega e^{-i \omega t} e^{-i \omega \tau}$$ And $\mathscr{F}(e^{i a t})\{\omega\}=2 \pi \delta(\omega -a)$: $$=\int_{-\infty}^{\infty} \mathrm{d}t f(t) 2 \pi \delta(t + \tau)$$ $$=2 \pi\int_{-\infty}^{\infty} \mathrm{d}t f(t) \delta(t + \tau)$$ $$=2 \pi f(-\tau)$$ But you might get a different result with a different definition of the FT.