The local structure of the solutions of the multidimensional translation equation
type: Item
instance of: scholarly article
title: The local structure of the solutions of the multidimensional translation equation
zbMATH Open document ID: 0793.39004
DOI: 10.1007/BF01834005
author: Lothar Berg
published in: Aequationes Mathematicae
publication date: 1994-08-21T00:00:00Z
full work available at URL: https://eudml.org/doc/137547
review text: This paper continues, in a manner of speaking, work by the author, the reviewer and \textit{Z. Moszner} [Result Math. 19, No. 3/4, 195-201 (1991; Zbl 0724.39008)] on the translation equation \(F(F(z,s),t)=F(z,s+t)\), with the initial condition \(F(z,0)=0\), where \(s,t\) are \(m\)-component, \(z\) and the function values \(n\)-component real vectors. Here, however, local solutions are offered on neighbourhoods, with the possible exception of sets nowhere dense in them, and continuous partial differentiability is assumed. Note: In Remark (v) of section 1, the author gives the example \(F(z,t)=z/(1-zt)\) \((m=n=1)\) as not contained in the general solution of the translation equation given by \textit{Z. Moszner} [Aequationes Math. 9, 46-59 (1973; Zbl 0263.94016); ibid. 37, No. 2/3, 267-278 (1989; Zbl 0672.39005)]. This is not defined for \(t=1/z\); however, with \(f(z)=1- (1/z)\), it is elsewhere of the subform \(f^{-1} (f(z)+t)\) analogous to Moszner's solution.
reviewed by: János Aczél
Mathematics Subject Classification ID: 39B22
Mathematics Subject Classification ID: 39B42
Mathematics Subject Classification ID: 26B05
zbMATH DE Number: 493241
zbMATH Keywords: local structure
zbMATH Keywords: multidimensional translation equation
zbMATH Keywords: translation functional equations
zbMATH Keywords: local solutions
zbMATH Keywords: general solution
MaRDI profile type: Publication
cites work: Q5514010
cites work: Sur l'équation de translation multidimensionnelle. (On the multidimensional translation equation)
cites work: On transformations with several parameters and operations in multidimensional spaces
cites work: The Pexider Functional Equations in Distributions
cites work: Q3785320
cites work: On the structure of the solutions of the Jabotinsky equations in Banach spaces
cites work: Structure de l'automate plein, reduit et inversible
cites work: Une généralisation d'un résultat de J. Aczél et M. Hosszú sur l'équation de translation. (A generalization of a result by J. Aczél and M. Hosszú on the translation equation)
cites work: Inverse systems and the translation equation on topological spaces
cites work: Pexider's equation and aggregation of allocations
cites work: Q4051278
OpenAlex ID: W2048034190
description: scientific article; zbMATH DE number 493241