  We discuss Meyers-Serrin's type results for smooth approximations of
functions $b=b(t,x):\mathbb{R}\times\mathbb{R}^n\to\mathbb{R}^m$, with
convergence of an energy of the form \[ \int_{\mathbb{R}}\int_{\mathbb{R}^n}
w(t,x) \varphi\left(|Db(t,x)|\right)\mathrm{d} x \mathrm{d} t\,, \] where $w>0$
is a suitable weight function, and $\varphi:[0,\infty)\to [0,\infty)$ is a
convex function with $\varphi(0)=0$ having exponential or sub-exponential
growth.
