On Gajda’s type quadratic equation on a locally compact abelian group

Abstract

Let (G,+) be a locally compact abelian Hausdorff group, and let μ be a regular compactly supported complex-valued Borel measure on G such that μ(G)=12. We find the continuous solutions f,g:G→C of the functional equation ∫G{f(x+y−t)+f(x−y+t)}dμ(t)=f(x)+g(y),x,y∈G, in terms of quadratic and additive functions. This equation provides a common generalization of many functional equations (quadratic, Jensen’s, Drygas’ equations...).