If we have a Gaussian pulse $f(t)$ in time , its Fourier Transform $F(\omega)$ is a Gaussian pulse. Consider the following polynomial phase function in the frequency domain, that is $$G(\omega) = e^{j(a_0+a_1\omega+a_2\omega^2+...a_n \omega^n)}$$
EDIT: Note that $a_0,..,a_n \in \mathbb{R}$ with $a_n \neq 0$
Now if we take the product of the Fourier Transforms $F(\omega)$ and $G(\omega)$, and then take the inverse Fourier of this product, will we get a Gaussian time pulse for any order $n$?
Is $h(t) = \mathcal{F}^{-1}\{F(\omega) \cdot G(\omega)\}(t) = f(t) * g(t)$ Gaussian?
I know for sure this is true for polynomials of order 0(due to linearity), 1(due to the time shift property) and 2(due to some notes in Optical Communications).
Any help is appreciated. Thanks in advance!