aequationes mathematicae

, Volume 67, Issue 1–2, pp 12–25 | Cite as

d’Alembert’s and Wilson’s equations on Lie groups

  • Peter de Place Friis
Research paper


Let G be a connected nilpotent Lie Group. We show that the solutions of the short and the long version of d’Alembert’s equation on G have the same form as on abelian groups. Furthermore we show that any solution of Wilson’s equation \( f(xy) + f(xy^{-1}) = 2f(x)g(y) \) in the case g ≠ 1 has the form \( f = A(m + \check{m})/2 + B(m-\check{m})/2 \) and \( g = (m + \check{m})/2 \) where \( m : G \rightarrow {\mathbb C}^* \) is a homomorphism and A and B are complex constants. Finally we solve Jensen’s equation \( f(xy) + f(xy^{-1}) = 2f(x) \) on a semidirect product of two abelian groups.

Mathematics Subject Classification (2000).



Functional equation d’Alembert Wilson Jensen Nilpotent Lie group. 


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Copyright information

© Birkhäuser-Verlag 2004

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsMississippi State UniversityUSA

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