Definition:Even Impulse Pair Function

Definition

The even impulse pair function is the real function $\operatorname {II}: \R \to \R$ defined as:

$\forall x \in \R: \map {\operatorname {II} } x := \dfrac 1 2 \map \delta {x + \dfrac 1 2} + \dfrac 1 2 \map \delta {x - \dfrac 1 2}$

where $\delta$ denotes the Dirac delta function.

The graph of the even impulse pair function is illustrated below:

It is to be understood that the blue arrows represent rays from the $x$-axis for constant $n \in \set {-\dfrac 1 2, \dfrac 1 2}$.

Let $\operatorname {II}: \R \to \R$ denote the even impulse pair function.

The $2$-dimensional form of $\operatorname {II}$ is defined and denoted:

$\forall x, y \in \R: \map {\operatorname { {}^2 II} } {x, y} := \map {\operatorname {II} } x \map {\operatorname {II} } y$