E. Elqorachi, M. Akkouchi, A. Bakali, B. Bouikhalene
Abstract:
This paper is mainly concerned with the following functional equation
$$\int_{G}\bigg\{\int_{K}f(xtk\cdot y)dk\bigg\}d\mu(t)=f(x)f(y),\quad x,y\in
G,$$
where $G$ is a locally compact group, $K$ a compact subgroup of its morphisms,
and $\mu$ is a generalized Gelfand measure. It is shown that continuous and
bounded solutions of this equation can be expressed in terms of $\mu$-spherical
functions. This extends the previous results obtained by Badora (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.
Keywords:
D'Alembert functional equation, Gelfand measure, $\mu$-spherical function, Lie
group.
MSC 2000: 39B05, 43A90