Functions on real projective plane and convolutions.

Question

Say there are $N$ functions (indexed by $k$) defined as, $$F_k:\mathbb{R}P^2 \rightarrow\mathbb{R} $$ What is desired is an analog of convolution like we would have for functions $g_k : \mathbb{R} \rightarrow \mathbb{R}$ defined as $C_{mn}(\tau)=\int g_m (x) g_n(x-\tau) dx$. One way was to just take $F_k$ as parity symmetric functions on $S^2$, i.e., $$F_k(\theta,\phi) = F_k(\pi-\theta,\phi+\pi)$$ and then define convolution as, $$C_{mn}(R)=\int_{SO(3)}F_m(R^{-1}Q) F_n(Q) dQ $$ where $R,Q\in SO(3)$ and $dQ$ is an appropriate measure. Is there a better, simpler way to define a convolution which takes advantage of the fact that antipodal points on the sphere are identified?