Definition:Sampling Function

Definition
The sampling function is the distribution $\operatorname {III}_T: \map \DD \R \to \R$ defined as:


 * $\forall x \in \R: \map {\operatorname {III}_T } x := \ds \sum_{n \mathop \in \Z} \map \delta {x - T n}$

where:
 * $T \in \R_{\ne 0}$ is a non-zero real number
 * $\delta$ denotes the Dirac delta distribution.

When $T = 1$, it is usually omitted:
 * $\forall x \in \R: \map {\operatorname {III} } x := \ds \sum_{n \mathop \in \Z} \map \delta {x - n}$

Also see

 * Sampling Function is Tempered Distribution