If $\forall\ g \in \{\phi, \phi', \phi''\}$, $g \in L^1(\mathbb{R})$ and $\lim_{|x|\to\infty} g(x) = 0$ then $\exists\ f \in L^1(\mathbb{R})$ such that $\hat{f} = \phi$.
I could only think of using the inverse tranform for functions in $L^2$ (integrating against $e^{2\pi i xy}$) and check if that works, but I couldn't get to anything.
I also tried applying Integration by Parts to $\int \phi'' e^{2\pi i xy}$ or something similar so that I could use the properties above, but with no such luck. Any ideas? Thanks!