Carleman approximation by non-critical functions on Riemann surfaces

TL;DR

This paper characterizes sets on Riemann surfaces where non-critical holomorphic functions can approximate continuous functions, expanding understanding of Carleman approximation in complex analysis.

Contribution

It introduces semi-admissible sets and provides new conditions for non-critical Carleman approximation on Riemann surfaces.

Findings

Characterization of semi-admissible sets for approximation

Conditions for approximation on sets with empty interior

Alternative approach for uniform approximation on broader sets

Abstract

We present the class of semi-admissible subsets of an open Riemann surface on which Carleman approximation by non-critical holomorphic functions is possible. In particular we characterize closed sets with empty interior on which continuous functions can be approximated by non-critical holomorphic ones. We also consider a different approach, which in some cases gives uniform approximation by non-critical holomorphic functions on more general sets than semi-admissible ones.