Fourier transforms of rapidly decreasing functions

If $f\in L^p(\mathbb R)$, $p\geqslant 2$, then the Fourier transform $F(z)$ of the function $\exp(-a|t|^\alpha)f(t)$, $a>0$, $\alpha>1$, belongs to the space of entire functions that are $p$-power integrable over the whole plane with some completely determined weight. Conversely, if $F(z)$ is an entire function in such a space, where $1\leqslant p\leqslant 2$, then $F(z)$ is a Fourier transform of the above form for some function $f\in L^p(\mathbb R)$.