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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 ( µ). (Here, tr means trace). We give some conditions under which all solutions are of that form. We show that ( 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 ( µ) and properties of its solutions. We treat examples and give some applications.

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