A Mergelyan-Vitushkin approximation theorem for rational modules (Q2639988)

Let \({\bar \partial}\) denote the Cauchy-Riemann operator in the complex plane \({\mathbb{C}}\). The inner boundary of a compact subset X of \({\mathbb{C}}\) is the set of boundary points of X not belonging to the boundary of a component of the complement of X. For a compact set X denote by \(R(X,\bar z)\) the rational module which consists of all functions of type \(r_ 0(z)+r_ 1(z)\bar z\) where the \(r_ i\) are rational functions with poles off X. The author uses techniques of Mergelyan and Vitushkin for the proof of the main Theorem: Let X be a compact subset of the complex plane such that the inner boundary of X is empty. Then a continuous function f on X can be uniformly approximated on X by functions in \(R(X,\bar z)\) if and only if \({\bar \partial}^ 2f=0\) at all interior points of X. By a localization argument this is proved also for sets X with countable inner boundary. A similar approximation theorem has been obtained by \textit{J. J. Carmona} [J. Approximation Theory 44, 113-126 (1985; Zbl 0574.30041)] for compact sets X whose complement has only finitely many components.