Abstract
Let G be a locally compact group not necessarily unimodular. Let
μ be a regular and bounded measure
on G. We study, in this paper, the following integral equation, formula here This equation generalizes the functional equation for spherical functions
on a Gel'fand pair. We seek
solutions ϕ in the space of continuous and bounded functions on G.
If π is a continuous unitary
representation of G such that π(μ) is of rank one, then
tr(π(μ)π(x)) is a solution of [Escr ](μ). (Here, tr means trace). We give some conditions under which all solutions are of that form.
We show that [Escr ](μ) has
(bounded and) integrable solutions if and only if G admits integrable,
irreducible and continuous unitary
representations. We solve completely the problem when G is compact.
This paper contains also a list of
results dealing with general aspects of [Escr ](μ) and properties of
its solutions. We treat examples and give some
applications.