On the imaginary part of the characteristic function

TL;DR

This paper investigates whether the imaginary part of a characteristic function uniquely determines the function itself, providing a characterization and showing that on many groups, it does.

Contribution

It offers a characterization of characteristic functions determined by their imaginary parts, extending the analysis to general locally compact abelian groups.

Findings

Certain characteristic functions are uniquely determined by their imaginary parts.

The study generalizes the problem to arbitrary locally compact abelian groups.

Several classical characteristic functions are shown to be determined by their imaginary parts.

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.