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We establish certain properties for the class $𝒰 (Ω, ζ_{0})$ of universal functions in $Ω$ with respect to the center $ζ_{0} \in Ω$ , for certain types of connected non-simply connected domains $Ω$ . In the case where $ℂ / Ω$ is discrete we prove that this class is $G_{δ}$ -dense in $H (Ω)$ , depends on the center $ζ_{0}$ and that the analog of Kahane’s conjecture does not hold.

Universal Taylor series, conformal mappings and boundary behaviour

Stephen J. Gardiner (2014)

Annales de l’institut Fourier

A holomorphic function $f$ on a simply connected domain $Ω$ is said to possess a universal Taylor series about a point in $Ω$ if the partial sums of that series approximate arbitrary polynomials on arbitrary compacta $K$ outside $Ω$ (provided only that $K$ has connected complement). This paper shows that this property is not conformally invariant, and, in the case where $Ω$ is the unit disc, that such functions have extreme angular boundary behaviour.

Universal Taylor series with maximal cluster sets.

L. Bernal-González, A. Bonilla, M.C. Calderón-Moreno, J.A. Prado-Bassas (2009)

Revista Matemática Iberoamericana

Universally divergent Fourier series via Landau's extremal functions

Gerd Herzog, Peer Chr. Kunstmann (2015)

Commentationes Mathematicae Universitatis Carolinae

We prove the existence of functions $f \in A (𝔻)$ , the Fourier series of which being universally divergent on countable subsets of $𝕋 = \partial 𝔻$ . The proof is based on a uniform estimate of the Taylor polynomials of Landau’s extremal functions on $𝕋 ∖ {1}$ .

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