An uniqueness theorem for characteristic functions.

Suppose that f is the characteristic function of a probability measure on the real line $$\mathbb R$$ . 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 $$\mathfrak {I}f$$ ? In other words, is it true that for any characteristic function f there exists a characteristic function g such that $$\mathfrak {I}f\equiv \mathfrak {I}g$$ 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. [ABSTRACT FROM AUTHOR]