Non-extendability of holomorphic functions with bounded or continuously extendable derivatives

2017/09/01 by D. Moschonas, Moschonas, D., V. Nestoridis +1
Mathematics · #Advanced Topology and Set Theory #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Meromorphic and Entire Functions

paper · pdf · doi:10.48550/arxiv.1709.00276

openalex publication_date 2017/09/01 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28

Abstract

We consider the spaces H_F^∞(Ω) and A_F(Ω) containing all holomorphic functions f on an open set Ω⊆ ℂ, such that all derivatives f^(l), l∈ F ⊆ ℕ₀=\ 0,1,...\, are bounded on Ω, or continuously extendable on Ω, respectively. We endow these spaces with their natural topologies and they become Fréchet spaces. We prove that the set S of non-extendable functions in each of these spaces is either void, or dense and G_δ. We give examples where S=\varnothing or not. Furthermore, we examine cases where F can be replaced by \widetildeF=\ l∈ ℕ₀:min F \leqslant l \leqslant sup F\, or \widetildeF₀= \ l∈ ℕ₀:0\leqslant l \leqslant sup F\ and the corresponding spaces stay unchanged.

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