On the imaginary part of the characteristic function

Abstract: Suppose that $f$ is the characteristic function of a probability measure on the real line $\R$. In this paper, we deal with the following problem posed by N.G. Ushakov: Is it true that $f$ is never determined by its imaginary part $\Im f$? In other words, is it true that for any characteristic function $f$ there exists a characteristic function $g$ such that $\Im f\equiv \Im g$ but $ f\not\equiv g$? We study this question in the more general case of the characteristic function defined on an arbitrary locally compact abelian group. A characterization of what characteristic functions are uniquely determined by their imaginary parts are given. As a consequence of this characterization, we obtain that several frequently used characteristic functions on the classical locally compact abelian groups are uniquely determined by their imaginary parts.