example of Dirac sequence
We can construct a Dirac sequence 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 any test function (a topological vector space of smooth functions with compact support). Let us take an arbitrary test function and assume that the closed and compact set is contained in some open interval ( and ). Using the triangle inequality and the fact that for all we can write
It is easy to see that , and therefore and . Finally we want to estimate when .
We now conclude that . This means that which shows that converges to the Dirac delta distribution .
|Title||example of Dirac sequence|
|Date of creation||2013-03-22 14:13:10|
|Last modified on||2013-03-22 14:13:10|
|Last modified by||Johan (1032)|