= Duality property of the Fourier transform
{title2=$\mathcal F^2f(\omega)=2\pi f(-\omega)$}
For the <Fourier transform> convention $\mathcal Ff(\omega)=\int_{\mathbb R}f(t)e^{-i\omega t}\,dt$, <Fourier inversion> at $-\omega$ gives $f(-\omega)=(2\pi)^{-1}\int_{\mathbb R}(\mathcal Ff)(t)e^{-i\omega t}\,dt$. 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.
Back to article page
