From here; $\hat f=\mathcal{F}(f)$, bar = complex conjugate:

- Time-shift property: $x(t-b) \Leftrightarrow e^{-j\omega b}{\bf X} (\omega)$, so why is it $+$ (red)?
- What at all is happening? Looks like convolution theorem, taking $f$ and $\psi$'s time-domain multiplication into frequency domain, except don't we need a second integral for conv.?