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D'Alembert's functional equation on groups

100%

Stetkær H.

Banach Center Publications

2013

tom 99

nr 1

173-191

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.

Ograniczanie wyników

1 Banach Center Publications

1 Stetkær H.

1 2013

Wyniki wyszukiwania

D'Alembert's functional equation on groups