Are Distributions just functions with infinitesimal coefficients?

Question

It is relatively well known that the Dirac delta, the classical example of a distribution, can be written as $$ \frac{a}{π(a^2+x^2)} $$ where $a$ is an infinitesimal such as a hyperreal. This can be confirmed by simply integrating and evaluating the integral symmetrically about the y axis which is simply $\frac{\arctan(x/a)}{π}$ from $x=-b$ to $x=b$ with $b$ real which is of course equal to $1$.

My question is if this holds generally. Are there any distributions of note that cannot be written as straightforward functions in a setting equipped with infinitesimals?

Answer 1

Every distribution can be written as a limit of smooth functions:

Let $u\in\mathcal{D}'(\mathbb R)$ and $\eta\in C^\infty_c(\mathbb R).$ Then let $\eta_\epsilon(x) = \frac1\epsilon\eta(\frac x \epsilon)$ and $u_\epsilon=u*\eta_\epsilon.$ Then $u_\epsilon\in C^\infty(\mathbb R)$ and $u_\epsilon\to u$ in $\mathcal{D}'(\mathbb R)$ as $\epsilon\to 0.$

Answer 2

I have several related remarks.

1. The (internal) function $\delta_a(x)=\frac{a}{π(a^2+x^2)}$ you mentioned (for $a$ infinitesimal) will have the property that, when integrating it against any continuous function $f$, one will get $f(0)$ but only up to an infinitesimal, so that $\int \delta_a(x) f(x)dx \approx f(0)$. This was already mentioned in Robinson's 1966 book.

2. There is later work whereby you can get a delta function satisfying equality on the nose, but this requires more effort.

3. An interesting question is to gauge how much choice one would need to get such a delta function via infinitesimals. This may depend on the boundary conditions one chooses and what class of functions exactly one is dealing with. For example, the theory SCOT is conservative over ZF+ADC (where ADC is axiom of dependent choice), but it may be too weak to construct such a delta function.