Duality property of the Fourier transform (${\mathcal{F}}^{2}f(\omega )=2\pi f(-\omega )$)

For the Fourier transform convention $\mathcal{F}f(\omega )={\int }_{\mathbb{R}}f(t)e^{-i\omega t}dt$, Fourier inversion at $-\omega$ gives $f(-\omega )=(2\pi {)}^{-1}{\int }_{\mathbb{R}}(\mathcal{F}f)(t)e^{-i\omega t}dt$. Thus transforming twice reflects the argument and multiplies by $2\pi$. The identity holds directly on the Schwartz space, and extends to the L2 space and to tempered distributions. It lets an already known transform pair be used in reverse without another integration.