On a quadratic type functional equation on locally compact abelian groups

Let (G,+) be a locally compact abelian Hausdorff group, k is a finite automorphism group of G, κ = cardK and let μ be a regular compactly supported complex-valued Borel measure on G such that . We find the continuous solutions f, g : G → C of the functional equation in terms of k-additive mappings. This equations provides a common generalization of many functional equations (quadratic, Jensen’s, Cauchy equations).