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  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.

Few details:
- the integral above is performed in a sense of Cauchy principal value.   
- Note that the division by the imaginary number is not necessary. However, I keep it so that the final solution is real-valued. (At least I think it guarantees that).  One can sertaily remove it.