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.

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.

Let S be a non empty set. We prove the stability (in the sense of Ulam) of the functional equation: f(t)=F(t,f (?(t))), where ? is a given function of S into itself and F is a function satisfying a contraction of Ciric type ([5]). Our analysis is based on the use of a fixed point theorem of Ciric (see [5] and [4]). In particular our result provides a generalization and a natural continuation of a paper of Baker (see [3]).

We generalize the well-known Baker's superstability result for the d'Alembert functional equation with values in the field of complex numbers to the case of the integral equation where 𝐺 is a locally compact group, μ is a generalized Gelfand measure and σ is a continuous involution of 𝐺.

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