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.