So I am new to the idea of convolution of sheaves, and I am trying to understand the convolution of two sheaves on $S^1$. Let $\pi:\mathbb{R}\to S^1$ be the universal covering. Consider $\mathcal{F(n)}=\pi_{*}\underline{\mathbb{C}}_{(0,n)}, n\ge 0$, similarly with $(n,0)$ for $n<0$. Now let $p_i$ be the projection $S^1\times S^1\to S^1$ from the $i$-th coordinate. Let $m:S^1\times S^1\to S^1$ be the multiplication map. Then the convolution is defined as
$$\mathcal{F}(n)\star\mathcal{F}(n^\prime):=m_!(p_1^\ast\mathcal{F}(n)\otimes p_2^\ast\mathcal{F}(n^\prime))$$
I am trying to show $\mathcal{F}(n)\star\mathcal{F}(n^\prime)\cong \mathcal{F}(n+n^\prime)$. Even when $m=n=0$ I can't do it. Then both sheaves are skyscraper sheaves, but I don't quite understand what's happening with $m_!$. I worked with $m_\ast$ because here the domain is compact hence $m$ is proper. Now if I take any open set $U$ in $S^1$, consider $p_i^\ast \mathcal{F}(n)(m^{-1}(U))=\mathcal{F}(n)(p_i(m^{-1}(U))$ but $p_i(m^{-1}(U))=S^1$. This doesn't make sense to me, and this isn't giving me the right answer. Could someone show me how to solve this problem, and hopefully point me towards some good references? I haven't found any references for computations and I think some computed examples of convolution product would help a lot. Thanks in advance.