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To complement/supplement other answers, it may be worthwhile to note that the question itself blurs two substantially different mechanisms. Namely, there is, first, for any group representation of a topological group $G$ on a topological vector space $V$, there is an action of compactly-supported continuous functions on $G$ on $V$, by (e.g., Gelfand-Pettis/weak) integrals $f\cdot v=\int_G f(g)\cdot gv\,dg$. It is of some moment to note that this does not depend on $v$ being in any sort of natural function-space. The second point is that $f\cdot (g\cdot v)=(f*g)\cdot v$, where $*$ denotes the convolution. That is, the notion of convolution is externally determined by being what it has to be for (for example) compactly-supported continuous functions to act (associatively) on any representation space.

Depending on one's outlook, this may reduce some element of seeming whimsy in "defining" convolution, since, in a larger context, _there_is_no_choice_.

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To complement/supplement other answers, it may be worthwhile to note that the question itself blurs two substantially different mechanisms. Namely, there is, first, for any group representation of a topological group $G$ on a topological vector space $V$, there is an action of compactly-supported continuous functions on $G$ on $V$, by (e.g., Gelfand-Pettis/weak) integrals $f\cdot v=\int_G f(g)\cdot gv\,dg$. It is of some moment to note that this does not depend on $v$ being in any sort of natural function-space. The second point is that $f\cdot (g\cdot v)=(f*g)\cdot v$, where $*$ denotes the convolution. That is, the notion of convolution is externally determined by being what it has to be for (for example) compactly-supported continuous functions to act (associatively) on any representation space.

Depending on one's outlook, this may reduce some element of seeming whimsy in "defining" convolution, since, in a larger context, _there_is_no_choice_.