The imaginary part of the characteristic function

We consider the conditions under which a continuous function \({\varphi \colon {\mathbb{R}}^n \to \mathbb {R}}\) is the imaginary part \({\Im f}\) of the characteristic function f of a probability measure on \({{\mathbb{R}}^n}\). A similar problem about such an \({\varphi}\) that it is the argument of the characteristic function was solved by Ilinskii Theory Probab. Appl. 20 (1975), 410–415]. In this paper, a characterization of what \({\varphi}\) might serve as the imaginary part of the characteristic function f is given. As a consequence, we provide an answer to the following question posed by N. G. Ushakov 7]: 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 prove that the answer to this question is negative. In addition, several examples of characteristic functions which are uniquely determined by their imaginary parts are given.