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 , 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-bc99-0-11,
author = {Henrik Stetk\ae r},
title = {D'Alembert's functional equation on groups},
journal = {Banach Center Publications},
volume = {99},
year = {2013},
pages = {173-191},
zbl = {1281.39028},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc99-0-11}
}
Henrik Stetkær. D'Alembert's functional equation on groups. Banach Center Publications, Tome 99 (2013) pp. 173-191. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc99-0-11/