Good evening,

I have a question on the approximation of holomorphic functions on a space of cartesian product type.

**Question:** Let $U,V$ be domains in $\mathbb{C}^n$ and $f\in \mathcal{O}(U\times V)$ a holomorphic function on $U\times V.$ Do we always have the following : $f$ can be approximated by holomorphic functions on $U\times V$ of the form $g(z,w) = \sum_{i=1}^N h_i(z)k_i(w)$ where $h_i\in \mathcal{O}(U)$ and $k_i\in\mathcal{O}(V)$ ? (N is arbitrary)

If it is not possible, can this be true if we put some conditions on $U$ and $V$? So what are the conditions?

Any help is appreciated. Thanks in advance.

Duc Anh