how the yellow shades comes and what is the use of this theorem
and how can we say that $supp(U^i) \subseteq supp(W_i)$
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Is the simple case clear to you : let $U $ the unit ball in $\mathbb{R}^2$ and $u \in W^{k,2}(U)$ and $v_m(x) = u(\frac{m-1}{m} x)$ and $\phi \in C^\infty_c(U), \int \phi(x)dx = 1, \phi_m(x) = m^2 \phi(mx), u_m = v_m \ast \phi_m$ then $u_m \to u$ in $W^{k,2}(U)$. Then the yellow part is for an arbitrary complicated open set $U$, decomposing it as $U = \bigcup V_i$ using a $C^\infty_c$ partition of unity $ u = \sum_i u\xi_i$, then approximating each $u \xi_i$ is achieved using the same argument as in the simple case – reuns Feb 25 '19 at 14:09
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@reuns can u prove how this comes $supp(U^i) \subseteq supp(W_i)$ – Inverse Problem Feb 25 '19 at 14:34