Giörgy Gát

Badora'S Equation on Non-Abelian Locally Compact Groups

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