Suppose that for a given $b\in \mathbb{R}$
\begin{align}
0= \int_{-\infty}^\infty e^{-\frac{(bt+\omega)^2}{2}} f(t+\omega) \frac{1}{i \pi t} dt,  \forall \omega \in \mathbb{R}
\end{align} 
where $i =\sqrt{-1}$. 

**Question:** How to find a set of general solutions to this equation?   I tried to do the Fourier inversion but things didn't work out.