I am trying to apply the duality property of the Fourier Transform to the time and frequency shifts, using the definitions which yield the following properties:
$G(t) = f(t) \rightarrow F(\omega) = g(\omega) \Rightarrow g(t) \rightarrow 2 \pi G(-\omega)$
Then:
If $f(t) e^{i \omega _ 0 t} \rightarrow F(\omega - \omega _ 0)$ then using duality $f(t- t _ 0) \rightarrow 2 \pi F(-\omega) e^{i (-\omega) t _0} = 2 \pi F(-\omega) e^{-i \omega t _0}$ which is wrong
Also in the other case:
If $f(t- t _ 0) \rightarrow F(\omega) e^{i \omega t _0}$ then using duality $f(t) e^{i \omega _ 0 t} \rightarrow 2 \pi F(-\omega + \omega _0)$ which is wrong
It is off by a factor of $2 \pi$ and the transform is inverted. What am I doing wrong?
P.S: I know how to derive them from the definition. I am trying to understand in depth the use of the duality property.