Levi-Civita functional equations and the status of spectral synthesis on semigroups
type: Item
instance of: scholarly article
title: Levi-Civita functional equations and the status of spectral synthesis on semigroups
zbMATH Open document ID: 1482.43005
author: Che Tat Ng
author: Bruce Ebanks
published in: Semigroup Forum
publication date: 2021-09-14T00:00:00Z
review text: The Levi-Civita functional equation on a semigroup \(S\) is the equation \(f(xy)=\sum_{i=1}^n g_i(x)\, h_i(y)\), \(x, y\in S\), for any positive integer \(n\) and unknown functions \(f, g_1,\dots, g_n, h_1, \dots, h_n : S\to {\mathbb C}\). One of the major results concerning the Levi-Civita equation is that on abelian groups all of its solutions are exponential polynomials. In the paper under review the authors show that on many semigroups (even commutative ones with identity) there exist solutions of the Levi-Civita equation which are not exponential polynomials. Also they identify conditions on commutative monoids insuring that all solutions of the Levi-Civita equation are exponential polynomials.
reviewed by: Sergei S. Platonov
Mathematics Subject Classification ID: 43A45
zbMATH DE Number: 7395081
zbMATH Keywords: semigroup
zbMATH Keywords: prime ideal
zbMATH Keywords: semigroup representation
zbMATH Keywords: topological semigroup
zbMATH Keywords: monoid
zbMATH Keywords: Levi-Civita equation
zbMATH Keywords: spectral synthesis
MaRDI profile type: Publication
full work available at URL: s00233-021-10211-z
OpenAlex ID: W3183630809
cites work: Q5514010
cites work: D'Alembert's other functional equation on monoids with an involution
cites work: Q5614003
cites work: The Levi-Civita equation in function classes
cites work: The matrix equation \(a(x\cdot y)=a(x)+a(x)a(y)+a(y)\)
cites work: Equations of the form \(H(x\cdot y)=\sum_if_i(x)g_i(y)\)
cites work: Functional Equations on Groups
cites work: Q4002300
DOI: 10.1007/S00233-021-10211-Z
Recommended article: Levi-Civita functional equations and the status of spectral synthesis on semigroups. II
Recommended article: Some Levi-Civita functional equations on semigroups
Recommended article: A Levi-Civita functional equation on semigroups
Recommended article: The Levi-Civita equation, vector modules and spectral synthesis
Recommended article: Q3827009
Recommended article: Stability of a simple Levi-Civitá functional equation on non-unital commutative semigroups
Recommended article: A note on the Levi-Civita functional equation
Recommended article: On certain generalizations of the Levi-Civita and Wilson functional equations
Recommended article: A Levi-Civitá equation on compact groups and nonabelian Fourier analysis
Recommended article: Q3989428
description: scientific article; zbMATH DE number 7395081