By $SO(3)$ I mean rotation matrices. Let $\cal{L}=\{f:[0,l]\to \rm{SO}(3)\} \cap L^1([0,l];\mathbb{R}^{3\times3})$. How to approximate funkctions from $\cal{L}$ with functions from $\cal{W}=\{f:[0,l]\to \rm{SO}(3)\}\cap W^{1,2}([0,l];\mathbb{R}^{3\times3})$?
So having $f\in\cal{L}$ I need a sequence $(g_n)_n \subset \cal{W}$ so that
$$ \| f - g_n \|_{L^1} \to 0.$$
My try #2: From $f$ go to $h$ such that $h(t)=\log(f(t))\in\mathbb{R}^{3\times 3}_\rm{skew}$ where $\log$ is matrix logarithm. If I would know that $h\in L^1([0,l];\mathbb{R}^{3\times 3}_\rm{skew})$ then I could use approximation in that normed space by molifiers to construct $h_n\in W^{1,2}([0,l];\mathbb{R}^{3\times 3}_\rm{skew})$ such that $h_n \to h$ in $L^1$, and then maybe for $g_n=\exp(h_n)$ we would have $g_n \to f$ in $L^1$.
My try #1 was directly molifying $f$, but the resulting molifications are not functions on $SO(3)$ anymore.