# another example of Dirac sequence

Let $\displaystyle A_{n}=\left[\frac{-1}{2^{n}},\frac{1}{2^{n}}\right]$ and $\delta_{n}=2^{n-1}\chi_{A_{n}}$ for every positive integer $n$, where $\chi_{S}$ denotes the characteristic function of the set $S$. Then $\{\delta_{n}\}$ is a Dirac sequence:

1. 1.

$\delta_{n}(t)\geq 0$ for every positive integer $n$ and every $t\in\mathbb{R}$.

2. 2.

Let $n$ be a positive integer. Then $\displaystyle\int\limits_{-\infty}^{\infty}\delta_{n}(t)\,dt=\int\limits_{% \frac{-1}{2^{n}}}^{\frac{1}{2^{n}}}2^{n-1}\,dt=1$.

3. 3.

Let $r>0$. Then there exists a positive integer $N$ such that, for every integer $k>N$, we have $\displaystyle\frac{1}{2^{k}}. Thus, for every integer $k>N$, we have $\displaystyle\int\limits_{\mathbb{R}\setminus\left[-r,r\right]}d_{k}(t)\,dt=0$.

Title another example of Dirac sequence AnotherExampleOfDiracSequence 2013-03-22 17:19:50 2013-03-22 17:19:50 Wkbj79 (1863) Wkbj79 (1863) 8 Wkbj79 (1863) Example msc 26A30