Some generalization of the quadratic and Wilson's functional equation

Abstract

We find the solutions f, g, h: G → X, ϕ: G → K of each of the functional equation

λ∈K f(x + λy) = |K|ϕ(y)g(x) + |K|h(y), x,y∈ G, where (G, +) is an abelian group, K is a finite, abelian subgroup of the automorphism group of G, X is a linear space over the field K ∈ {R,C}.