Pointwise approximation on the Alice Roth’s Swiss cheese

Abstract

We show that the complex conjugate function \(z\mapsto \overline{z}\) cannot be pointwise approximated by holomorphic polynomials on the Alice Roth’s Swiss cheese \(Q_R\subset \mathbb {C}\). Moreover, under some extra hypotheses, we also show that the complex conjugate cannot be pointwise approximated either by functions holomorphic on \(Q_R\).