A Runge type theorem for product of planar domains (Q1674111)

Let \(A(\Omega) \) be the Banach algebra of all functions \(f\) from \(\overline{\Omega}=\Pi_{i\in I}U_i\) to \(\mathbb C\) that are continuous on \(\overline{\Omega}\) with respect to the product topology and separately holomorphic in \(\Omega\), where \(I\) is an arbitrary set and \(U_i\) are planar domains of some type. The main result of this article: Theorem. Let \(U_i\), \(i\in I\), be an arbitrary family of bounded planar domains such that \(\mathrm{int}(\overline{U}_i)=U_i\) and \((\mathbb C\cup \{\infty\})\setminus \overline{U_i}\) has finitely many components \(V_j^i\), \(j=1, \ldots, n_i\), \(i\in I\). Let \(\alpha_i^j\in V^j_i \) be fixed. If \(\overline{\Omega}=\Pi_{i\in I}U_i\), then, for every function \(f\in A(\Omega)\) and every \(\epsilon>0\), there exists a function \(g\in A(\Omega)\) depending on a finite number of coordinates \(F\subset I\) such that \(||f-g||<\epsilon\) and the function \(g\) is a finite sum of finite products of rational functions of one variable \(z_i\), \(i\in F\), with poles in the set \(S_i=\{\alpha^j_i : j=1, \ldots, n_i,\}\), \(i\in F\).