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Abstract
This paper is mainly concerned with the following functional equation
where 𝐺 is a locally compact group, 𝐾 a compact subgroup of its morphisms, and μ is a generalized Gelfand measure. It is shown that continuous and bounded solutions of this equation can be expressed in terms of μ-spherical functions. This extends the previous results obtained by Badora (Aequationes Math. 43: 72–89, 1992) on locally compact abelian groups. In the case where 𝐺 is a connected Lie group, we characterize solutions of the equation in question as joint eigenfunctions of certain operators associated to the left invariant differential operators.
Key words and phrases:: D'Alembert functional equation; Gelfand measure; μ-spherical function; Lie group
Received: 2003-10-25
Revised: 2004-04-22
Published Online: 2010-02-26
Published in Print: 2004-September
© Heldermann Verlag
