# D'Alembert's functional equation on groups

• Volume: 99, Issue: 1, page 173-191
• ISSN: 0137-6934

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## Abstract

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Given a (not necessarily unitary) character μ:G → (ℂ∖0,·) of a group G we find the solutions g: G → ℂ of the following version of d’Alembert’s functional equation $g\left(xy\right)+\mu \left(y\right)g\left(x{y}^{-1}\right)=2g\left(x\right)g\left(y\right)$, x,y ∈ G. (*) The classical equation is the case of μ = 1 and G = ℝ. The non-zero solutions of (*) are the normalized traces of certain representations of G on ℂ². Davison proved this via his work [20] on the pre-d’Alembert functional equation on monoids. The present paper presents a detailed exposition of these results working directly with d’Alembert’s functional equation. In the process we find for any non-abelian solution g of (*) the corresponding solutions w: G → ℂ of w(xy) + w(yx) = 2w(x)g(y) + w(y)g(x), x,y ∈ G. (**) A novel feature is our use of the theory of group representations and their matrix-coefficients which simplifies some arguments and relates the results to harmonic analysis on groups.

## How to cite

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Henrik Stetkær. "D'Alembert's functional equation on groups." Banach Center Publications 99.1 (2013): 173-191. <http://eudml.org/doc/281683>.

@article{HenrikStetkær2013,
abstract = {Given a (not necessarily unitary) character μ:G → (ℂ∖0,·) of a group G we find the solutions g: G → ℂ of the following version of d’Alembert’s functional equation $g(xy) + μ(y)g(xy^\{-1\}) = 2g(x)g(y)$, x,y ∈ G. (*) The classical equation is the case of μ = 1 and G = ℝ. The non-zero solutions of (*) are the normalized traces of certain representations of G on ℂ². Davison proved this via his work [20] on the pre-d’Alembert functional equation on monoids. The present paper presents a detailed exposition of these results working directly with d’Alembert’s functional equation. In the process we find for any non-abelian solution g of (*) the corresponding solutions w: G → ℂ of w(xy) + w(yx) = 2w(x)g(y) + w(y)g(x), x,y ∈ G. (**) A novel feature is our use of the theory of group representations and their matrix-coefficients which simplifies some arguments and relates the results to harmonic analysis on groups.},
author = {Henrik Stetkær},
journal = {Banach Center Publications},
keywords = {d'Alembert's functional equation; group; representation; matrix coefficient},
language = {eng},
number = {1},
pages = {173-191},
title = {D'Alembert's functional equation on groups},
url = {http://eudml.org/doc/281683},
volume = {99},
year = {2013},
}

TY - JOUR
AU - Henrik Stetkær
TI - D'Alembert's functional equation on groups
JO - Banach Center Publications
PY - 2013
VL - 99
IS - 1
SP - 173
EP - 191
AB - Given a (not necessarily unitary) character μ:G → (ℂ∖0,·) of a group G we find the solutions g: G → ℂ of the following version of d’Alembert’s functional equation $g(xy) + μ(y)g(xy^{-1}) = 2g(x)g(y)$, x,y ∈ G. (*) The classical equation is the case of μ = 1 and G = ℝ. The non-zero solutions of (*) are the normalized traces of certain representations of G on ℂ². Davison proved this via his work [20] on the pre-d’Alembert functional equation on monoids. The present paper presents a detailed exposition of these results working directly with d’Alembert’s functional equation. In the process we find for any non-abelian solution g of (*) the corresponding solutions w: G → ℂ of w(xy) + w(yx) = 2w(x)g(y) + w(y)g(x), x,y ∈ G. (**) A novel feature is our use of the theory of group representations and their matrix-coefficients which simplifies some arguments and relates the results to harmonic analysis on groups.
LA - eng
KW - d'Alembert's functional equation; group; representation; matrix coefficient
UR - http://eudml.org/doc/281683
ER -

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