A tempered function on $\mathbb{R}^n$ is a locally integrable function that is tempered as a distribution, i.e. $L^1_{loc}\cap\mathcal{S}'$ is the space of tempered functions.
This MSE question asked for a characterization of tempered functions.
Not every tempered function has a tempered function as its Fourier transform, e.g. the constant function $1$ has the delta distribution $\delta_0$ as its Fourier transform.
My question is, is there a nice characterization of tempered functions whose Fourier transforms are also tempered functions, i.e.
$$ \{ f \in L^1_{loc}\cap\mathcal{S}' : \hat{f} \in L^1_{loc}\cap\mathcal{S}' \} \quad \text{?}$$
