= Solution
Using <Fourier inversion> with the stated convention,
$$
f(-\omega)=\frac1{2\pi}\int_{-\infty}^{\infty}\widetilde f(t)e^{-i\omega t}\,dt.
$$
Hence applying the transform twice gives the <duality property of the Fourier transform>:
$$
\boxed{\mathcal F[\widetilde f](\omega)=2\pi f(-\omega)}.
$$
The formula holds directly for rapidly decreasing smooth functions, and extends through the usual <L2 space> or <tempered distribution> formulations when absolute convergence is unavailable.
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