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 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 A CðG;CÞ. In this note we show that for Vitushkin sets K with K 1⁄4 K it is always possible to uniformly approximate on K the smooth function f A C ðK ;CÞ by smooth functions fn in C so that also qfn converges uniformly to qf 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.