example of Dirac sequence

We can construct a Dirac sequence {δn}n+ by choosing


To show that conditions 1 and 3 in the definition of a Dirac sequence are satisfied is trivial and condition 2 is also fulfilled since


for all n+, hence {δn}n+ is a Dirac sequence.

To prove that it actually converges in 𝒟() (the space of all distributionsDlmfPlanetmath on 𝒟()) to the Dirac delta distribution δ, we must show that


for any test function φ𝒟() (a topological vector spaceMathworldPlanetmath of smooth functions with compact support). Let us take an arbitrary test function φ𝒟() and assume that the closed and compact set supp(φ) is contained in some open intervalDlmfPlanetmath (a,b) (a<0 and b>0). Using the triangle inequalityMathworldMathworldPlanetmath and the fact that δn(x)𝑑x=1 for all n+ we can write


It is easy to see that limnδn(x)=0, x(-,a][b,) and therefore limnI1=0 and limnI3=0. Finally we want to estimate I2 when n.


We now conclude that limnI2=0. This means that limnI1+I2+I3=0 which shows that {δn}n+ converges to the Dirac delta distribution δ.

Title example of Dirac sequence
Canonical name ExampleOfDiracSequence
Date of creation 2013-03-22 14:13:10
Last modified on 2013-03-22 14:13:10
Owner Johan (1032)
Last modified by Johan (1032)
Numerical id 8
Author Johan (1032)
Entry type Example
Classification msc 46F05
Related topic Distribution4
Related topic DeltaDistribution
Related topic DiracDeltaFunction
Related topic ConstructionOfDiracDeltaFunction