Take the 2-minute tour ×   (a) Show $f$ is a tempered
distribution.  (b) Compute $\hat{f}$ using the convention
$\int_{\mathbb{R}}f(x)e^{-ix\xi}\;dx$ convention for
$\mathcal{F}$.  (c) Let $\{e_{k}\}_{k\leq d}$ is a basis for
$\mathbb{R}^{d}$ and compute
$\hat{f}\in\mathscr{S}'(\mathbb{R}^{d})$ where $$f(x):=\sum_{n\i
n\mathbb{Z}^{d}}\delta(x-[n_{1}e_{1}+\ldots+n_{d}e_{d}]),$$ and
formulate the result in terms of the dual basis.  Part (a) is
straightforward:  Let $f_{n}$ denote the nth symmetric partial
sum of the series defining $f$. Then
$\{f_{n}\}_{n\in\mathbb{N}}$ defines a sequence of tempered
distributions (as they are compactly supported) whereby for each
$\phi\in\mathscr{D}$ we have
$(f_{n},\phi)\to\sum_{n\in\mathbb{Z}}\phi(n)$ as $n\to\infty$.
The righthand sum is finite since $\phi$ is rapidly decreasing
(in particular, for large $n$, $\phi(n)$ is majorized by
$|n|^{-k}$ for every $k>0$) and thus $f_{n}\to f$ in the
distributional sense. Since $\mathscr{S}'$ is closed in
$\mathscr{D}'$, the claim fol