Proposition 13

— If $f \chi^{-1} \in L^1(G, \beta)$ and $g \in \overline{\mathscr{K}(G)}$, or if $f \in \mathscr{K}(G)$ and $g {\chi'}^{-1} \in L^1(G, \beta)$, then $f$ and $g$ are convolvable, and (15) defines for every $x \in G$ a product $f * g$ that belongs to $\mathscr{K}(G)$.

This follows from Prop. 5 of No. 2, and the remarks in No. 4.