Fourier transform frequency shift proof from duality and translation property

Question

I read that you can prove the frequency shift property of the Fourier transform from the translation property using the duality property. I have tried substituting things into one another, transforming one into the other etc. but nothing seems to work. I feel I'm missing something simple. Can I have a hint please?

Answer 1

You know (at least up to different placements of the factor $2\pi$) that

$$ \mathcal{F}(T_x f) = e^{-2\pi i \langle x, \cdot \rangle} \cdot \widehat{f} =: M_{-x} \widehat{f}. \qquad (\dagger) $$

You also know that the inverse Fourier transform (at least for Schwartz functions or $L^2$ functions or $f \in L^1$ with $\widehat{f} \in L^1$) is given by

$$ (\mathcal{F}^{-1} f)(x) = \widehat{f} (- x). $$

Now apply the inverse Fourier transform to $(\dagger)$, where $g := \mathcal{F}^{-1} f$.

Finally, if you want to get the general case from that of Schwartz functions/..., use an approximation argument.