Distributions on non-test functions

[pellman]

The definitions of distributions that I have seen (for instance https://en.wikipedia.org/wiki/Distribution_(mathematics)#Distributions ) define a distribution as a map whose domain is a set of test functions. A defining quality of test functions is that they have compact support, which for most practical purposes means they are non-zero on only a finite interval (or intervals).

So why then are we comfortable writing something like $\int^{\infty}_{-\infty} \delta(x-a) x^2 dx = a^2$? In the language of distributions this would be $\delta_a(\phi) = a^2$ where $\phi(x) = x^2$. If we mean that $\phi(x) = x^2$ for all real x, is this a problem?

Is there an applied assumption here that we don't mean $\phi(x) = x^2$ on the entire real line but only on some compact subspace? If so, is this potentially a problem in physics which often deals with functions whose domain is explicitly infinite? I have never seen anyone applying the Dirac delta function have any concern that the function applied to is of compact support.

 

[mathman]

Not a problem as long as you know what you are doing. In example the integral is $a^2$ for all finite intervals containing the delta function, so passing to the limit is O.K.

 

[pellman]

Lol. But I don't know what I'm doing.

I'm guessing that for the specific case of the delta distribution that it is fine to act on functions which are not of compact support even though distributions in general are only defined to act on test functions. Is this the case?

 

[pwsnafu]

A defining quality of test functions is that they have compact support, which for most practical purposes means they are non-zero on only a finite interval (or intervals).

The set of compact supported smooth functions is one class of test functions, denoted $\mathcal{D}$. There are other classes including $\mathcal{E}$ which is the set of all smooth functions, and Dirac is defined as a functional of this space. Importantly, the distributions defined on $\mathcal{E}$ are characterized as those distributions with compact support (note I'm talking about the support of the distribution being compact, not the test function).

 

[pellman]

Thanks, pwsnafu.