The definition on the top of page 195 in B-J's article (in Corvallis 1) is ok, because $f\ast\xi$ is well-defined for any smooth $f:G(\mathbf{A})\to\mathbf{C}$ and any $\xi\in H$.
It suffices to verify the claim for pure tensors $\xi=\xi_\infty\otimes\xi_f$, in which case $f\ast\xi$ is $f\ast\xi_\infty$ convolved with $\xi_f\in H_f$ in the usual sense. The function $f\ast\xi_\infty$ exists by the smoothness assumption, while the integral defining the convolution converges, because $\xi_f:G(\mathbf{A}_f)\to\mathbf{C}$ is compactly supported and locally constant by definition.