On the Bernstein functional equation

We give two proofs of the Bernstein theorem about characterization of the Gaussian distribution by the independence of the sum and the difference of independent random variables. These proofs use neither the Cramer theorem about decomposition of a Gaussian distribution nor the finite difference method. Due to this fact our proofs without changes are carried over to the case of a locally compact Abelian group with single-valued division by two, provided that the characteristic functions of the considering distributions do not vanish. We use the last result for the description of all locally compact Abelian groups for which the Bernstein theorem is valid.