2025/11/30 by Fournodavlos, Grigorios, Nestoridis, Vassili, Pasias, Spyros
#30E10 #Complex Variables (math.CV) #FOS: Mathematics
paper · doi:10.48550/arxiv.2512.00802
Arakelian's classical approximation theorem \citeAr gives necessary and sufficient conditions such that functions can be uniformly approximated in (unbounded) closed sets F⊂ ℂ by entire functions. The conditions are purely topological and concern the connectedness of the complement of F. We give a new characterization of Arakelian sets in terms of logarithmic branches of functions f∈ A(F), which are continuous in F and holomorphic in its interior F^∘. Our proof is based on a contradiction argument and the counterexample function that we use is furnished by the Weierstrass factorization theorem.