Convolution between two distributions

Question

I want to define the convolution $*$ between two distributions $S$ and $T$. For a test function $\varphi$, can I say:

$$\langle S * T, \varphi \rangle \doteqdot \langle S, T*\varphi \rangle $$

where the convolution between a distribution and a test function is a function that I define as:

$$ T*\varphi \doteqdot x \mapsto \langle T,\tau_x \varphi \rangle $$

With $\tau$ the translation operator, i.e., $\tau_x (t \mapsto \varphi(t))\doteqdot t \mapsto \varphi(t-x) $ .

Does this make any sense? I'm trying to follow what my textbook says but the author is not exactly clear.

Answer 1

In general, convolutions of distributions cannot be defined. (It's possible with some extra conditions, for example that at least one of the distributions has compact support.)

The problem with your approach is that $T*\phi$ is not necessarily a test function.

Answer 2

In addition to the good answer by mrf, it is surely important to realize that even when convolutions of distributions seem to be "well-defined", there are some problems in a naive conception. Consider the classic non-associativity $$ (H * \delta')*1 \;=\; \delta * 1 \;=\; 1 \;\not=\; 0 \;=\; H * 0 \;=\; H * (\delta' * 1) $$

I can't help but mention that the phrase "convolution action" is misleading, for the following reason: when a group $G$ acts on a nice (e.g., quasi-complete, locally convex) topological vector space $V$, the compactly-supported continuous functions $\varphi$ on $G$ act by the natural integrated action: $\varphi\cdot v=\int_G \varphi(g)\,g\cdot v\;dg$ with Haar measure. Then convolution of continuous compactly-supported functions $\varphi, \psi$ is completely characterized by the associativity $\varphi(\psi\cdot v)=(\varphi*\psi)\cdot v$ (for all actions on all topological vector spaces $V$, etc.)

One point is that the action of $G$ on $V$ need not be "convolution" or any particular formula.

So, to have a good convolution of classes (or "generalized functions", e.g., distributions) of functions on $G=\mathbb R$, we'd want an action of that class of functions on representation spaces of $G$. Not just a formula.