Holomorphic functions with universal boundary behaviour
type: Item
instance of: scholarly article
title: Holomorphic functions with universal boundary behaviour
zbMATH Open document ID: 1460.30020
published in: Journal of Approximation Theory
publication date: 2020-04-07T00:00:00Z
review text: The paper contains a number of results concerning the generic boundary behaviour of functions in \(H(\mathbb{D})\), the Fréchet space of functions holomorphic in the unit open disc \(\mathbb{D}\), and of functions in the analytic Wiener algebra \(W^+\). Theorem 1 says that for a dense \(G_\delta\)-set of functions \(f\) in \(H(\mathbb{D})\) the following property holds: Given any compact set \(K\) strictly contained in \(\mathbb{T}:=\partial \mathbb{D}\), any function \(\varphi\) which is continuous on \(K\), any compact set \(L \subset \mathbb{D}\) and any \(\ell \in \mathbb{Z}\), there exists an increasing sequence \((r_n)\) with \(0\le r_n \to 1\) and such that \(f^{(\ell)} (r_n\zeta+(1-r_n)z)\) converges to \(\varphi(\zeta)\) as \(n \to \infty\), uniformly with respect to \((\zeta, z) \in K\times L\). Also, the radial growth of such functions \(f\) and connections to universal Taylor series are investigated. In particular, the functions necessarily have to grow fast near \(\mathbb{T}\), in the sense that \(\sup_\mathbb{D}|f(z)|/v_\phi(z)=\infty\) for all radial weights \(v_\phi(z)=\exp(\phi(1/(1-|z|))\) with \(\int_1^\infty \log\phi(t)t^{-2} dt< \infty\). On the other hand, in Theorem 2 it is shown that the derivatives of functions from the analytic Wiener algebra generically satisfy a weakened kind of universality: There is a residual set of functions in \(W^+\) having the property that for any family \((\varphi_\ell)_{\ell=1}^\infty\) of measurable functions on \(\mathbb{T}\) there exist a sequence \((r_n)\) as above and a set \(E \subset \mathbb{T}\) of full arc length measure such that, for any \(\ell\in \{1,2, \ldots\}\), any \(\zeta \in E\) and any \(z \in \mathbb{D}\), the sequence \(f^{(\ell)}(r_n \zeta +(1-r_n)z)\) tends to \(\varphi_\ell(\zeta)\) as \(n \to \infty\).
Mathematics Subject Classification ID: 30K05
Mathematics Subject Classification ID: 30E10
zbMATH DE Number: 7187596
zbMATH Keywords: universal functions
zbMATH Keywords: boundary behavior
author: Stéphane Charpentier
reviewed by: Jürgen Müller
MaRDI profile type: Publication
full work available at URL: j.jat.2020.105391
OpenAlex ID: W3007734022
cites work: Some boundary properties of analytic functions
cites work: Linearity of sets of strange functions
cites work: UNIVERSAL RADIAL LIMITS OF HOLOMORPHIC FUNCTIONS
cites work: Abstract theory of universal series and applications
cites work: Disjoint hypercyclic operators
cites work: Maximal cluster sets of \(L\)-analytic functions along arbitrary curves
cites work: Universal Taylor series with maximal cluster sets
cites work: Maximal cluster sets along arbitrary curves
cites work: Simultaneously maximal radial cluster sets
cites work: Large subspaces of compositionally universal functions with maximal cluster sets
cites work: Disjointness in hypercyclicity
cites work: Approximation on Closed Sets by Analytic or Meromorphic Solutions of Elliptic Equations and Applications
cites work: On countably universal series in the complex plane
cites work: Universal Taylor series and summability
cites work: On the boundary behaviour of derivatives of functions in the disc algebra
cites work: On the radial behavior of universal Taylor series
cites work: Doubly universal Taylor series
cites work: Extension à la dimension n d'un théorème de Ortel et Schneider. (Extension of a theorem of Ortel and Schneider to n dimensions)
cites work: On the existence of O-universal functions
cites work: On the radial cluster sets for holomorphic functions in the unit ball of \({\mathbb{C}}^ n\)
cites work: Q5610507
cites work: Holomorphic monsters
cites work: On the Peano Curves Associated with Some Conformal Maps
cites work: On the growth of universal functions
cites work: Growth of coefficients of universal Taylor series and comparison of two classes of functions
cites work: Hypercyclic subspaces and weighted shifts
cites work: On doubly universal functions
cites work: Universal Taylor series
cites work: Conformal mappings and Peano curves
cites work: Maximal cluster sets on spaces of holomorphic functions
cites work: Lacunary power series and Peano curves
cites work: Power series and Peano curves
cites work: Disjoint universality for families of Taylor-type operators
DOI: 10.1016/J.JAT.2020.105391
Recommended article: On boundary behaviours of holomorphic functions
Recommended article: The boundary behavior of holomorphic functions
Recommended article: Q3807503
Recommended article: Holomorphic functions with highly noncontinuable boundary behavior
Recommended article: Q4039110
Recommended article: Universality of holomorphic functions bounded on closed sets
Recommended article: Boundary behaviour of holomorphic functions of bounded type
Recommended article: Q3995812
Recommended article: Q3333561
Recommended article: Boundary behavior of holomorphic functions of several complex variables
description: scientific article; zbMATH DE number 7187596