Universal entire functions with gap power series
type: Item
instance of: scholarly article
title: Universal entire functions with gap power series
zbMATH Open document ID: 0914.30001
DOI: 10.1016/S0019-3577(98)80032-3
author: W. Luh
author: Valeri A. Martirosian
author: Jürgen Müller
published in: Indagationes Mathematicae. New Series
publication date: 1999-02-17T00:00:00Z
review text: The following result on ``lacunary universal functions'' is proved: Let \({\mathcal M}\) be the family of all compact sets \(K\) in \(\mathbb{C}\) which have connected complement and let \(A(K)\) denote the Banach space of all functions which are continuous on \(K\) and holomorphic in its interior \(K^\circ\). Theorem 1. Let \(Q\) be any subsequence of \(\mathbb{N}_0\) with density \(\Delta(Q)= 1\) and let \(\{z_n\}\) be any unbounded sequence of complex numbers. Then there exists an entire function \(\phi\) with lacunary power series \(\phi(z)= \sum^\infty_{v= 0}\phi_v z^v\), \(\phi_v= 0\) for \(v\not\in Q\) which is ``universal'' in the double sense that (A) the sequence \(\{\phi(z+ z_n)\}\) is dense in \(A(K)\), (B) the sequence \(\{\phi(zz_n)\}\) is dense in \(A(K)\) if \(0\not\in K\). The proof uses essentially the Riesz representation theorem and the Hahn-Banach theorem. Further results concern an investigation of the sharpness of the density property of \(Q\).
Mathematics Subject Classification ID: 30B60
Mathematics Subject Classification ID: 30B10
Mathematics Subject Classification ID: 30E10
Mathematics Subject Classification ID: 41A30
zbMATH DE Number: 1251211
zbMATH Keywords: lacunary universal functions
reviewed by: Rolf Wallisser
MaRDI profile type: Publication
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full work available at URL: s0019-3577(98)80032-3
OpenAlex ID: W1967380199
description: scientific article; zbMATH DE number 1251211