# Dirac sequence

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

1. 1.

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

2. 2.

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

3. 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.

Title Dirac sequence DiracSequence 2013-03-22 14:11:35 2013-03-22 14:11:35 mathwizard (128) mathwizard (128) 5 mathwizard (128) Definition msc 26A30 delta sequence DiracDeltaFunction FejerKernel