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.

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