A Class of Functional Equations on a Locally Compact Group

Abstract

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.

Functional equations and matrix-valued spherical functions

“…In the same setup as ours Benson, Jenkins and Ratcliff [5,6,7] have found the bounded K-spherical functions on certain Gelfand pairs. For results on bounded spherical functions, also matrixvalued, on abelian groups see Chojnacki [8,9], and on generalized Gelfand pairs see Akkouchi, Bakali and Khalil [2]. For operator cosine functions with exponential type of growth see Niechwiej [21].…”

On the Operator-valued $μ$-cosine functions

“…Proof. Let ξ ∈ H. By the same way as in the proof of Theorem 2.2 in [1], we can suppose that the vector ξ is cyclic Let now ϕ(x) = Φ(x)ξ, ξ for all x ∈ G. For all x, y ∈ G we have K ϕ (x, y) = Φ(x)ξ, Φ(y)ξ . So that K ϕ is a positive definite kernel.…”

Superstability problem for a large class of functional equations

“…Therefore, our approach provide a unified treatment of the superstability of Cauchy's, d'Alembert's, Wilson's, spherical type functional equations and others. For informations about these equations we refer to [1,2,5,11,[23][24][25].…”