For $l \in\mathbb{N} $ the space of spherical harmonics $ Y_{l}^{m} $forms a irreduceble representation of SO(3). One can hence write: $$\forall l\in\mathbb{N}\,\forall R\in SO\left(3\right)\,\exists A\in Mat\left(2l+1,2l+1\right):\forall m':Y_{l}^{m'}\circ R=\sum_{m=-l}^{l}A_{m',m}Y_{l}^{m}$$
Can one recover an approximation of $R \in SO(3)$ from an approximation for $A_{m',m}$?