We define $W^{1,p}_{0}(\Omega)$ as the closure of $C_c^{\infty}(\Omega)$ in the $W^{1,p}$-norm (or equivalently as the closure of the $W^{1,p}$-functions with compact support). Given $u\in W^{1,p}_{0}(\Omega)$, we know by definition that there exists a sequence of functions of compact support, that are in $W^{1,p}(\Omega)$ and converge to $u$. Is it possible to explicitely construct such a sequence?
My first thought was to use standard convolution arguments, but it doesn't seem to work that easy.