An entire holomorphic function associated to an entire harmonic function
type: Item
instance of: scholarly article
title: An entire holomorphic function associated to an entire harmonic function
zbMATH Open document ID: 0971.30018
published in: Journal of Approximation Theory
publication date: 2001-10-30T00:00:00Z
review text: Given a harmonic function \(h\) on \(\mathbb{R}^N\), \(N\geq 2\), \((h\in {\mathcal H}_N)\) there is a unique holomorphic function \(f\) on \(\mathbb{C}\), \((f\in{\mathcal E})\) such that \(f(t)=h(t,0,\dots,0)\) for \(t\in\mathbb{R}\). This paper has two purposes. Firstly to obtain theorems on the connection between the growth rates of \(h\) and \(f\) which are more comprehensive and precise than those obtained formerly, and secondly to apply these results to deduce uniqueness theorems for harmonic functions. One of them is given below as theorem 1. Let \(M_2(g,r)= (\int_S |g(rx) |^2 d\sigma(x))^{1/2}\), where \(S\) is the unit sphere in \(R^N\) if \(g=h\in{\mathcal H}\), or the unit circle in \(\mathbb{C}\) if \(g=f\in{\mathcal E}\). The maximum value of \(|g|\) on \(S\) is denoted by \(M_\infty(g,r)\). Let \(\rho(g)\) and \(\tau(g)\) be the order and type of \(g\) respectively. It is said that \(g\) is of growth \((\rho,\tau)\) if \(\rho(g) <\rho\) or \(\rho(g)=\rho\), \(\tau (g)\leq\tau\). The values of \(\rho(g)\) and \(\tau(g)\) are unaffected if \(M_\infty (g,r)\) is replaced by \(M_2(g,r)\) in their definitions. Proposition 1: If \(h\) is of growth \((\rho,\tau)\), then so also is \(f\). For certain class of functions of order l the following result is proved. Proposition 2: If \(M_2(h,r)=O(r^p e^{\lambda r})\), \((r\to+ \infty)\), \(p\in\mathbb{R}\), \(\lambda>0\), then \(M_\infty (f, r)=O(r^{p+(2N-3)/4} e^{\lambda r})\), \((r\to+\infty\). Theorem 1. Suppose that \(h\in{\mathcal H}_N\) and \(h(m,0, \dots,0)=0\) for all \(m\in\mathbb{Z}\). (i) If \(M_2 (h,r)=O(r^p e^{\pi r})\), \((r\to+ \infty)\), for some \(p\geq(3-2N)/4\), then \(h(t,0,\dots, 0)=P(t)\sin (\pi t)\) for all real \(t\), where \(P\) is a polynomial of degree at most \(p+(2N-3)/4\). (ii) If \(M_2(h,r)=o(r^{3-2N)/4}e^{\pi r})\), \((r\to +\infty)\), then \(h(t,0, \dots,0)=0\) for all real \(t\).
Mathematics Subject Classification ID: 30D15
Mathematics Subject Classification ID: 31A05
zbMATH DE Number: 1346406
zbMATH Keywords: harmonic functions
zbMATH Keywords: entire functions
zbMATH Keywords: uniqueness theorems
zbMATH Keywords: growth rates
author: D. H. Armitage
reviewed by: J. Illán
MaRDI profile type: Publication
full work available at URL: jath.1999.3338
OpenAlex ID: W2048639644
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DOI: 10.1006/JATH.1999.3338
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description: scientific article; zbMATH DE number 1346406