Almost periodicity and functional equations
type: Item
instance of: scholarly article
title: Almost periodicity and functional equations
zbMATH Open document ID: 0566.39010
DOI: 10.1007/BF02189679
published in: Aequationes Mathematicae
publication date: 1983-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/136997
review text: From author's introduction: ''In this note we present some applications of Fourier analysis in the theory of functional equations. The basic idea is that we try to prove the almost periodicity of the solution instead of differentiability, and the basic observation is that from this point of view ''boundedness'' is a good condition for the solution, playing the role of ''measurability''. In other words, it turns out that some types of functional equations on topological groups have the property that all bounded continuous solutions are almost periodic, and by Fourier transformation the solutions can be determined effectively from simple linear systems of equations.'' The author proves the following theorem. Let G be a topological group, \(F: G\times G\to {\mathbb{C}}\) be an almost periodic function, \(f,g_ i,k_ i: G\to K\) \((i=1,2,...,n)\) be mappings such that \((1)\quad f(xy)=\sum_{i}g_ i(x)h_ i(y)+F(x,y)\) holds. If each \(g_ i\) is bounded and continuous and each \(h_ i\) is bounded, then f is almost periodic. The author applies this theorem to obtain continuous bounded solutions of \(f(xy)=f(x)g(y)+f(y)g(x),\quad f(xy)=f(x)f(y)-g(x)g(y),\) and \(f(xy)+f(xy^{-1})=2f(x)f(y)\) on a topological Abelian group G with a separating family \(G^*\).
Mathematics Subject Classification ID: 39B62
Mathematics Subject Classification ID: 43A60
zbMATH DE Number: 3903218
zbMATH Keywords: almost periodicity
zbMATH Keywords: boundedness
zbMATH Keywords: measurability
zbMATH Keywords: topological groups
zbMATH Keywords: Fourier transformation
zbMATH Keywords: continuous bounded solutions
author: László Székelyhidi
reviewed by: Pl. Kannappan
MaRDI profile type: Publication
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OpenAlex ID: W2020663384
description: scientific article; zbMATH DE number 3903218