On a functional equation appearing in characterization of distributions by the optimality of an estimate

Abstract

Let $X$ be a second countable locally compact Abelian group containing no subgroup topologically isomorphic to the circle group $\mathbb{T}$. Let $\mu$ be a probability distribution on $X$ such that its characteristic function $\hat{\mu }(y)$ does not vanish and $\hat{\mu }(y)$ for some $n\ge 3$ satisfies the equation $$\prod _{j=1}^n\hat{\mu }(y_j+y)=\prod _{j=1}^n\hat{\mu }(y_j-y),\sum _{j=1}^n{y}_j=0,y_1,\dots ,y_n,y\in Y.$$ Then $\mu$ is a convolution of a Gaussian distribution and a distribution supported in the subgroup of $X$ generated by elements of order 2.