Harmonic analogues of G. R. MacLane's universal functions. II
type: Item
instance of: scholarly article
title: Harmonic analogues of G. R. MacLane's universal functions. II
zbMATH Open document ID: 0945.31001
author: M. P. Aldred
author: D. H. Armitage
published in: Journal of Mathematical Analysis and Applications
publication date: 1999-10-06T00:00:00Z
review text: [For part I see J. Lond. Math. Soc. (2) 57, No. 1, 148-156 (1998; Zbl 0922.31004).] Let \(H_N\) be the space of all harmonic functions on \(\mathbb{R}^n\) with the topology of local uniform convergence. The authors investigate the question of the existence of a harmonic function \(h\in H_N\), such that \[ \{\partial^{\alpha_1+\cdots+ \alpha_N}/\partial_{x_1}^{\alpha_1}\dots \partial x_N^{\alpha_N} h: (\alpha_1,\dots, \alpha_N)\in J\subset \mathbb{N}^N\} \] is dense in \(H_N\) (a function with this property is called \(J\)-universal function). One of the results obtained is: Theorem 1. If \(J\) is an infinite subset of \(\mathbb{N}^N\), then there exists a \(J\)-universal function of exponential type \(c_N\) where \(c_2=1\) and \[ c_N= N\Biggl( \prod_{j=1}^{N-1} \frac{(2j)^{2j}} {(2j+1)^{2j+1}} \Biggr)^{1/(2N)}, \quad N\geq 3. \] In the case where \(J= \{(m,\dots, m)\in \mathbb{N}^N\}\) there is no \(J\)-universal function of exponential type less than \(\sqrt{N/2}\).
Mathematics Subject Classification ID: 31B05
Mathematics Subject Classification ID: 30D15
zbMATH DE Number: 1215730
zbMATH Keywords: universal function
zbMATH Keywords: universal function of exponential type
zbMATH Keywords: harmonic functions
reviewed by: P. Z. Agranovich
MaRDI profile type: Publication
full work available at URL: jmaa.1997.5892
OpenAlex ID: W2072676973
cites work: Harmonic Analogues of G. R. Maclane's Universal Functions
cites work: Q4023465
cites work: A Universal Entire Function
cites work: Q3342857
cites work: Q3733130
cites work: Growth of entire harmonic functions in \(R^n,\;n\geq 2\)
cites work: On the universal functions of G.R. Maclane
cites work: On Brelot-Choquet Axial Polynomials
cites work: Sequences of derivatives and normal families
DOI: 10.1006/JMAA.1997.5892
description: scientific article; zbMATH DE number 1215730