Lipschitz approximation on closed sets (Q1803636)

Some of the classical results for uniform approximation of continuous functions on a closed subset \(F\) of a complex plane remain true if the uniform norm is replaced by a Lipschitz norm with exponent \(\alpha\) in the interval \(0<\alpha<1\). This is true for Runge's theorem for approximation of holomorphic functions by rational functions, for the fusion Lemma of Roth and the localization principle. Vitushkin's theorem characterizes those closed sets such that each continuous function \(f\) continuous on \(F\) and holomorphic in the interior of \(F\) can be approximated by a rational function. It contains a condition on the relative size of the singularities of \(f\) measured in terms of analytic capacity. For Lipschitz approximation the \((1+\alpha)\)-dimensional Hausdorff content replaces capacity as first shown by O'Farrell.