I was wondering if the Fourier transform relation can be viewed as evaluation in zero of the convolution between $x(t)$ and $e^{j\omega t}$, so if $*$ is the convolution operator
$$X(f) = \left.x(t) * e^{j\omega t}\right|_{t=0}$$
If I'm right will you be able to prove the Convolution Theorem for Fourier Transform with this approach ?