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.