The stability of the quadratic functional equation on amenable groups
type: Item
instance of: scholarly article
title: The stability of the quadratic functional equation on amenable groups
zbMATH Open document ID: 1040.43002
author: Dilian Yang
published in: Journal of Mathematical Analysis and Applications
publication date: 2004-06-15T00:00:00Z
review text: Let \(G\) be a group, and let \(E\) be a Banach space. The author defines \((G,E)\) to be QS (short for \((G,E)\) has the property of the stability of the quadratic functional equation) if, for every function \(f : G \to E\) such that \[ \sup_{x,y \in G} \| f(xy) + f(xy^{-1}) - 2f(x) -2f(y) \| := \delta < \infty, \] there are a function \(q : G \to E\) satisfying \[ q(xy) + q(xy^{-1}) - 2q(x)-2q(y) = 0 \qquad (x,y \in G) \] and \(\delta' > 0\) (depending only on \(\delta\)) such that \(\| f(x) - q(x) \| < \delta'\) for all \(x \in G\). There are two theorems in the paper. The first one asserts that if \((G,\mathbb{C})\) is QS, then \((G,E)\) is QS for every complex Banach space \(E\) (and analogously with \(\mathbb{C}\) replaced by \(\mathbb{R}\)). Secondly, the author proves that \((G,\mathbb{C})\) is QS for every amenable (discrete) group \(G\), so that \((G,E)\) is QS for every Banach space in this particular situation.
reviewed by: Volker Runde
Mathematics Subject Classification ID: 43A07
Mathematics Subject Classification ID: 22D99
zbMATH DE Number: 2075650
zbMATH Keywords: amenability
zbMATH Keywords: quadratic functional equation
zbMATH Keywords: stability
MaRDI profile type: Publication
full work available at URL: j.jmaa.2003.11.021
OpenAlex ID: W2058682521
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DOI: 10.1016/J.JMAA.2003.11.021
description: scientific article; zbMATH DE number 2075650