A note on simultaneous approximation on Vitushkin sets

Abstract

Given a planar Jordan domain $G$ with rectifiable boundary, it is well known that smooth functions on the closure of $G$ do not always admit smooth extensions to $\mathbb C$. Further conditions on the boundary are necessary to guarantee such extensions. On the other hand, Weierstrass’ approximation theorem yields polynomials converging uniformly to $f \in C(\overline{G}, \mathbb C)$. In this note we show that for Vitushkin sets $K$ with $K = \overline{K^\circ}$ it is always possible to uniformly approximate on $K$ the smooth function $f \in C^1(K, \mathbb C)$ by smooth functions $f_n$ in $\mathbb C$ so that also $\overline{\partial}f_n$ converges uniformly to $\overline{\partial}f$ on $K$. As a byproduct we deduce from its ‘‘smooth in a neighborhood version’’ the general Gauss integral theorem for functions whose partial derivatives in $G$ merely admit continuous extensions to its boundary.