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 analysisMathworldPlanetmath, the Dirac delta function admits a completely legitimate construction as an actual function. We give this construction here.

Choose any positivePlanetmathPlanetmath infinitesimalMathworldPlanetmathPlanetmath ε and define the hyperreal valued function δ:** by


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


so the integral property is satisfied. Finally, for any continuousMathworldPlanetmathPlanetmath real function f:, choose an infinitesimal z>0 such that |f(x)-f(0)|<z for all |x|<ε/2; then


which implies that -δ(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 DerivationMathworldPlanetmath
Classification msc 34L40
Classification msc 26E35