Dirac comb

Example

Let $T > 0$. The Dirac comb of period $T$ is the sum of Dirac deltas at the points $kT$, $k \in \Z$: $$\operatorname{III}_T(\varphi) = \sum_{k \in \Z} \varphi(kT) \quad \text{for } \varphi \in \mathcal{S}(\R),$$ where $\mathcal{S}(\R)$ is the Schwartz space. It is a tempered distribution. Indeed, $\abs{\varphi(kT)} \le (1 + k^2 T^2)^{-1} \sup_x (1 + x^2) \abs{\varphi(x)}$, so $$\abs{\operatorname{III}_T(\varphi)} \le \Big( \sum_{k \in \Z} \frac{1}{1 + k^2 T^2} \Big) \big( p_{0,0}(\varphi) + p_{2,0}(\varphi) \big),$$ where $p_{a,b}(\varphi) = \sup_x \abs{x^a \varphi^{(b)}(x)}$. This is a bound by finitely many Schwartz seminorms.