Functional equations on abelian groups with involution. II (Q1390820)

The author continues his investigations started in the paper [Aequationes Math. 54, No. 1-2, 144-172 (1997; Zbl 0899.39007)]. Here, using again the fruitfull notion of \(K\)-spherical function, he solves the functional equation \[ f(x+y)+ f(x+\sigma y)= 2f(x)+ h_1(y)+ g(x)h_2(y), \quad x,y \in G, \] where \(\sigma:G\to G\) is an automorphism of the abelian topological group \(G\) such that \(\sigma^2=\text{id}\) and \(f,g,h_1, h_2:G\to \mathbb{C}\) are continuous. This functional equation contains as a special case for example Swiatak's equation \((h_1=\), \(h_2=g)\) or the quadratic equation \((h_1=2f\), \(g=h_2=0)\).