A Runge type theorem for product of planar domains

Abstract

We consider \(A(\Omega )\), the Banach algebra of all functions f from \(\overline{\Omega }=\prod _{i\in I}\overline{U_{i}}\) 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. We show that finite sums of finite products of rational functions of one variable with prescribed poles off \(\overline{U_{i}}\) are uniformly dense in \(A(\Omega )\). This generalizes previous results where \(U_{i}=\mathbb D\) is the open unit disc in \(\mathbb C\) or \(\overline{U_{i}}^{c}\) is connected.