# Definition:Tempered Dirac Delta Distribution

## Definition

Let $a \in \R^d$ be a real vector.

Let $\phi \in \map \DD {\R^d}$ be a Schwartz test function.

Let $\delta_a \in \map {\DD'} {\R^d}$ be a tempered distribution.

Suppose $\delta_a$ is such that:

$\map {\delta_a} \phi = \map \phi a$

Then $\delta_a$ is known as the tempered Dirac delta distribution.