Let $f:\mathbb{R}^n \to \mathbb{R}$ be of the form $f(x_1, ..., x_n) = f_1(x_1)\cdots f_n(x_n)$ where each $f_i$ is a Schwartz function on $\mathbb{R}$. Then the poisson summation formula says
$$\sum_{u \in \mathbb{Z}^n}f(u) = \sum_{u \in \mathbb{Z}^n}\widehat{f}(u).$$
Also,
$$\sum_{k \in \mathbb{Z}}f_i(k) = \sum_{k \in \mathbb{Z}}\widehat{f}_i(k).$$
For each $1 \leq i \leq n$. My question is, do we know anything about the sum along the diagonal? In particular, do we have $$\sum_{k \in \mathbb{Z}}f(k, ..., k) = \sum_{k \in \mathbb{Z}}\widehat{f}(k, ..., k).$$