I Square of Dirac delta function

[Happiness]

Is the square of a Dirac delta function, $(\delta(x))^2$, still a Dirac delta function, $\delta(x)$?

A Dirac delta function peaks at one value of $x$, say 0. If it is squared, it still peaks at the same value, so it seems like the squared Dirac delta function is still a Dirac delta function, $\delta(x)$, or some multiple of it, $k\delta(x)$, where $k>1$, since the area under graph seems larger.

How about the square root of a Dirac delta function?

 

[scottdave]

It is something totally different than just multiplying by a constant.One thing to think about. The Del function is sometimes described as a rectangle of width d, and height (1/d) then take the limit as d->0. (so height approaches infinity)
At all values of d, you get an area of (d/d) = 1. But if you have (Del)^2, the width is essentially the same as Del, but the height is infinity^2 ?

WolframAlpha produced a surprising result for this. http://www.wolframalpha.com/input/?i=(DiracDelta[t])*(DiracDelta[t])

 

[FactChecker]

The Dirac delta "function" is defined by its behavior inside an integral: ∫f(x)δ(x)dx = f(0). It is a generalized function, not a function.
I think that your comment about an increased area of δ² is confusing the delta function with an approximation of the δ function.
One approach to multiplying generalized functions is to separate them into their "smooth" and "singular" parts. In that, δ(x)² = 0. (see https://en.wikipedia.org/wiki/Generalized_function#Algebras_of_generalized_functions)

 

[JJacquelin]

One of the best answer in :
https://mathoverflow.net/questions/48067/is-square-of-delta-function-defined-somewhere