Mazur I. The Skitovich-Darmois functional equation on the locally compact abelian groups

The thesis is devoted to the study of solutions of the Skitovich-Darmois functional equation in a class of continuous positive definite functions on various locally compact Abelian groups. We prove that solutions of the Skitovich-Darmois functional equation for n functions of n variables on finite Abelian groups in the class of positive definite functions are the characteristic functions of the idempotent distributions. We describe solutions of the Skitovich-Darmois functional equation for n functions of n variables on the product of k-dimensional vector space, arbitrary compact Abelian group and a discrete periodic group of special form. We describe discrete periodic Abelian groups for which all solutions of the Skitovich-Darmois functional equation for n functions of n variables in the class of normalized positive definite functions are characteristic functions of the idempotent distributions. We describe a-adic solenoids for which the Skitovich - Darmois theorem holds in the case n = 3. We obtain a generalization of the Stapleton theorem that characterizes the Haar distribution on a compact Abelian group.