construction of Dirac delta function

The Dirac delta function is notorious in mathematical circles for having no actual as a function. However, a little known secret is that in the domain of nonstandard analysis, the Dirac delta function admits a completely legitimate construction as an actual function. We give this construction here.

Choose any positive infinitesimal $\varepsilon$ and define the hyperreal valued function $\delta:\,^{*}\mathbb{R}\longrightarrow\,^{*}\mathbb{R}$ by

\delta(x):=\begin{cases}1/\varepsilon&-\varepsilon/2<x<\varepsilon/2,\\ 0&\text{otherwise.}\end{cases}

We verify that the above function satisfies the required properties of the Dirac delta function. By definition, $\delta(x)=0$ for all nonzero real numbers $x$ . Moreover,

\int_{-\infty}^{\infty}\delta(x)\ dx=\int_{-\varepsilon/2}^{\varepsilon/2}% \frac{1}{\varepsilon}\ dx=1,

so the integral property is satisfied. Finally, for any continuous real function $f:\mathbb{R}\longrightarrow\mathbb{R}$ , choose an infinitesimal $z>0$ such that $|f(x)-f(0)|<z$ for all $|x|<\varepsilon/2$ ; then

\varepsilon\cdot\frac{f(0)-z}{\varepsilon}<\int_{-\infty}^{\infty}\delta(x)f(x% )\ dx<\varepsilon\cdot\frac{f(0)+z}{\varepsilon}

which implies that $\int_{-\infty}^{\infty}\delta(x)f(x)\ dx$ is within an infinitesimal of $f(0)$ , and thus has real part equal to $f(0)$ .

Title	construction of Dirac delta function
Canonical name	ConstructionOfDiracDeltaFunction
Date of creation	2013-03-22 12:35:48
Last modified on	2013-03-22 12:35:48
Owner	djao (24)
Last modified by	djao (24)
Numerical id	5
Author	djao (24)
Entry type	Derivation
Classification	msc 34L40
Classification	msc 26E35