Suppose that we have a sequence on non-positive functions $\{f_i(x)\}\in C({\mathbb R})$ such that $f(x)=\sum_{i}f_i(x)$ has a moderate growth rate, e.g. polynomial. Let us denote the corresponding tempered distribution by $T_{f_i}$, i.e. $\langle T_{f_i}, \phi\rangle = \int f_i \phi$. Is it true that
$$ \sum_{i=1}^N \mathcal{F}[T_{f_i}]\rightarrow \mathcal{F}[f] $$ in weak topology of ${\mathbb S}'({\mathbb R})$? Here $\mathcal{F}$ is the Fourier transform.