I have to find inverse Fourier transform of \begin{align*} F(\omega)=e^{-\frac{\omega^2}{2}}\frac{1}{\mathsf{sinc}( \omega )} \end{align*} I was wondering whether it exists maybe in a sense of distributions.
Using duality between Fourier transform and inverse Fourier transform. We can ask what is Fourier transform of \begin{align*} f(x)=e^{-\frac{x^2}{2}}\frac{1}{\mathsf{sinc}( x )} \end{align*} and whether it exits?
Note, that the function $f(t)$ has vertical asymptots at for $x=n \pi$ for $n \in \mathbb{Z}$ \begin{align} \mathsf{sinc}( x )=\frac{\sin(x)}{x} \end{align} equals $0$ at integer multiplies of $\pi$. Is this a problem?
One way to look at the problem is find convolution of $\mathcal{F}(e^{-\frac{x^2}{2}})$ and $\mathcal{F}(\frac{1}{\mathsf{sinc}( x )})$. However, this direction can be problematic since $\mathcal{F}(\frac{1}{\mathsf{sinc}( x )})$ might not exist.