Rational approximation near zero sets of functions (Q582414)

The author shows that a Lipschitz function f approximable by rational functions on the set \(X\setminus \{x:f(x)=0\}\) where \(X\subset {\mathbb{C}}\) is a compact set, is approximable by rational functions on the entire set X provided that X does not have ``unstable'' points. (A point \(x\in X\) is called stable if e.g. \[ \underline{\lim}_{\delta \to 0} \log_{\delta}(\alpha (T(x,\delta)\setminus X))\geq 2, \] where T(x,\(\delta)\) is the square centered at x with side length \(\delta\) and \(\alpha\) denotes the continuous analytic capacity).