Functional equations on abelian groups with involution. (English) Zbl 0899.39007

The author investigates generalizations of several functional equations such as d’Alembert’s, Wilson’s, Jensen’s and the functional equation of quadratic functions. The main point is that he replaces terms of the form \(x-y\) by \(x+ \sigma (y)\), where \(\sigma\) is an involution. Moreover a general framework is used which is interesting for itself. The main tool is the notion of a \(K\)-spherical function, where \(K\) is a compact transformation group of a topological group. The paper is very interesting, unifies and generalizes earlier results and shows new perspectives in the considered area.
Reviewer: J.Schwaiger (Graz)


39B52 Functional equations for functions with more general domains and/or ranges
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