What question did this study set out to answer?

The research aims to identify subsets of open Riemann surfaces where Carleman approximation by non-critical holomorphic functions is achievable.

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February 5, 2026Open Access

Carleman approximation by non-critical functions on Riemann surfaces

Key Points

The research aims to identify subsets of open Riemann surfaces where Carleman approximation by non-critical holomorphic functions is achievable.
Identification of semi-admissible subsets on Riemann surfaces
Characterization of closed sets with empty interior
Analysis of approximation methods for continuous functions
Exploration of alternative approaches for broader sets
Characterization of closed sets where continuous functions can be approximated by non-critical holomorphic functions
Uniform approximation achieved on certain broader sets than just semi-admissible ones

Abstract

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.

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