# D'Alembert's functional equation on groups

Banach Center Publications (2013)

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

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topHenrik 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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