Dirac sequence

A Dirac sequence is a sequence $(\delta_{k})$ of functions $\delta_{k}$ , which satisfies the following conditions:

1.

$\delta_{k}\geq 0$ for all $k$ .
2.

$\int_{-\infty}^{\infty}\delta_{k}(t)dt=1$ for all $k$ .
3.

For every $r>0$ and $\varepsilon>0$ there is an $N\in\mathbb{N}$ , such that for all $k>N$ we have

$\int_{\mathbb{R}\backslash[-r,r]}\delta_{k}(t)dt<\varepsilon.$

These functions “converge” to the Dirac delta function.