D'Alembert's functional equation on groups

• Autorzy: Henrik Stetkær
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 22A25: Representations of general topological groups and semigroups
• 39B52: Equations for functions with more general domains and/or ranges

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Banach Center Publications
• Rocznik: 2013
• Tom: 99
• Numer: 1
• Strony: 173-191

Daty

wydano

2013

Twórcy

autor

Henrik Stetkær

• Department of Mathematical Sciences, University of Aarhus, Ny Munkegade 118, Building 1530, DK-8000 Aarhus C, Denmark

Identyfikatory

DOI

10.4064/bc99-0-11