The high level question is: Just as the Fourier transform of a Gaussian is a Gaussian, is the Fourier Transform of a sub-Gaussian also a sub-Gaussian?

Let $x \in \mathbf{R}^n$ denote some sub-Gaussian random vector, i.e. there exists $b>0$ such that for any $t\in \mathbf{R}^n$ its Laplace transform is upper-bounded as follows: $$ \mathbf{E}[ e^{t \cdot x} ] \leq e^{b^2 \|t\|^2}. $$

What is the Fourier Transform of the PDF of $x$, is it also sub-Gaussian?

It seems that just substituting $t = 2\pi i$ would immediately imply that, but is this substitution justified?