Let $f \in C^{\infty}(\mathbb{R}^2)$ be smooth and compactly supported. Can we approximate $f(x,y)$ by sums of the form $\sum\_{i=1}^m g\_i(x) h\_i (y)$ where $g\_i, h\_i \in C^{\infty}(\mathbb{R})$ are smooth with compact support.


Exact formulation:
Suppose $f \in C^{\infty}(\mathbb{R}^2)$ with $supp(f)\subseteq [a,b] \times [c,d]$, and $\varepsilon > 0$. Can we find a sum $\sum\_{i=1}^m g\_i(x) h\_i (x)$ such that $supp(g\_i) \subseteq [a,b]$ and $supp(h\_i) \subseteq [c,d]$, with
$\| f(x,y)- \sum\_{i=1}^m g\_i(x) h\_i (y)\|\_\infty < \varepsilon$ ?



Approximation of smooth compactly supported functions on $\mathbb{R}^2$ using sums of products of one variable functions
The situation is even much better, because $C^\infty([a,b]\times [c,d])=C^\infty([a,b])\tilde{\otimes}\_\pi C^\infty([c,d])$ (the completed projective tensor product) and due to a celebrated result of Grothendieck every Element of $X\tilde{\otimes}\_\pi Y$ for two Frechet spaces is even a series $\sum\limits\_{n=0}^\infty x\_n\otimes y\_n$ which converges in the topology of the projective tensor product. In the concerte situation this implies uniform convergence not only of the functions but of all partial derivatives.


