Mean value theorems via spectral synthesis
type: Item
instance of: scholarly article
title: Mean value theorems via spectral synthesis
zbMATH Open document ID: 0846.39010
author: John A. Baker
published in: Journal of Mathematical Analysis and Applications
publication date: 1995-01-01T00:00:00Z
review text: Let \(v_1, v_2, \dots, v_n\) \((n\geq 2)\) be the vertices of some fixed regular simplex in \(\mathbb{R}^n\) with centre 0 and radius 1. Then the vertices of any regular simplex in \(\mathbb{R}^n\) with centre \(x\) and radius \(r>0\) are \(x+ rU v_0, x+rU v_1, \dots, x+rU v_n\) where \(U\) denotes an appropriately chosen orthogonal \(n\times n\) matrix. The author characterizes those continuous functions \(f: \mathbb{R}^n\to \mathbb{C}\) with the property that, for a fixed \(r>0\), \(\sum^n_{k=0} f(x+ rUv_k)= (n+1) f(x)\) for all \(x\in \mathbb{R}^n\) and all real orthogonal \(n\times n\) matrices \(U\). Indeed \(f\) is precisely a harmonic polynomial of degree at most 2. Another interesting result proved is the following: Suppose \(0\neq \alpha_2\in \mathbb{C}\), \(0\leq \nu\leq N\) (\(N\) is an integer \(\geq 1\)), \(\sum^N_{\nu=0} \alpha_\nu =0\) and \(r_1, r_2, \dots, r_N\) are distinct real numbers. (i) There exists a non-negative integer \(m<N\) such that \(\sum^N_{\nu=0} \alpha_\nu r^j_\nu= 0\) for \(0\leq j\leq m\) and \(\sum^N_{\nu=0} \alpha_\nu r_\nu^{m+1} \neq 0\). (ii) If \(f: \mathbb{R}^n\to \mathbb{C}\), \(n\geq 2\) and \(f\) is continuous then \(\sum^N_{\nu=0} \alpha_\nu f(x+ r_\nu u)=0\) for all \(x\in \mathbb{R}^n\) and all \(u\in S^{n-1}= \{u\in \mathbb{R}^n: |u|=1\}\) iff \(f\) is a polynomial function of degree at most \(m\).
Mathematics Subject Classification ID: 39B32
zbMATH DE Number: 811589
zbMATH Keywords: mean value theorems
zbMATH Keywords: spectral synthesis
zbMATH Keywords: equations for complex functions
zbMATH Keywords: regular simplex
zbMATH Keywords: harmonic polynomial
reviewed by: Harkrishan L. Vasudeva
MaRDI profile type: Publication
full work available at URL: jmaa.1995.1237
OpenAlex ID: W2012083269
DOI: 10.1006/JMAA.1995.1237
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description: scientific article; zbMATH DE number 811589