Endows $C^\infty_c(\mathbb{R}^2)$ with the test functions convergence, i.e. $f_n\to f,n\to+\infty$ iff there exists a compact set $C\subset \mathbb{R}^n$ such that for all $n\in\mathbb{N}$ we have that the support of $f_n$ lies in $C$ and for all $k\in\mathbb{N_0}$ the convergence of $D^kf_n$ to $D^kf$ is uniform, where $D^k$ denotes the $k$-derivative.
If $f\in C^\infty_c(\mathbb{R}^2)$, can we find $(f_n)_{n\in\mathbb{N}}, (g_n)_{n\in\mathbb{N}}\subset C^\infty_c(\mathbb{R})$ such that $$(x,y)\mapsto\sum_{n=1}^N f_n(x)g_n(y)$$ converges to $f$ in $C^\infty_c(\mathbb{R}^2)$ as $N\to+\infty$?
Pick some $T$ such that $f \in C^\infty_c([-T/4,T/4]^2)$, let $\varphi \in C^\infty_c([-1,1]), \int \varphi = 1$, $1/k < T/4$, $\phi = 1_{[-T/2,T/2]} \ast k \varphi(k.)$ then
$$f(x,y) = h(x,y) \phi(x)\phi(y), \qquad h(x,y) = \sum_{n,m} f(x+2nT,y+2mT)$$ $$h(x,y) = \sum_{n,m} \hat{h}(n,m) e^{i \pi x/T} e^{i \pi y/T}$$
$$f(x,y) =\sum_{n,m} \hat{h}(n,m) e^{i \pi x/T} e^{i \pi y/T} \phi(x)\phi(y)$$