Triangular mean value theorems and Fréchet's equation
type: Item
instance of: scholarly article
title: Triangular mean value theorems and Fréchet's equation
zbMATH Open document ID: 0880.39020
DOI: 10.1007/BF01874613
author: John A. Baker
published in: Acta Mathematica Hungarica
publication date: 1998-01-08T00:00:00Z
review text: The author offers the following two main results (the first is somewhat more general in the paper than stated here). 1). Theorem 1. If the \(m\)-th iterate of the difference of a complex valued continuous function \(f\) of \(n (\geq 2)\) real variables is zero for all spans of length 1 then it is zero for all spans of any length in the \(n\)-dimensional real space and thus \(f\) is a polynomial of degree at most \(m-1\). 2). A real valued function of two real variables has the triangular mean value property for radius \(r\) if its value at the center of every equilateral (``equivalent'' in Theorem 4 is a misprint) triangle of radius \(r\) equals the arithmetic mean of its values at the vertices. Theorem 4. If a real valued function of two real variables is continuous and has the triangular mean value property for a fixed radius then it is a harmonic polynomial of at most second degree.
Mathematics Subject Classification ID: 39B32
Mathematics Subject Classification ID: 39B22
Mathematics Subject Classification ID: 42B10
Mathematics Subject Classification ID: 46F05
zbMATH DE Number: 878092
zbMATH Keywords: functional equations
zbMATH Keywords: Fourier transformation
zbMATH Keywords: distributions
zbMATH Keywords: differences
zbMATH Keywords: triangular mean value properties
reviewed by: János Aczél
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01874613
OpenAlex ID: W2066002263
description: scientific article; zbMATH DE number 878092