论文标题
在特征函数的虚构部分
On the imaginary part of the characteristic function
论文作者
论文摘要
假设$ f $是实际行$ \ r $上概率度量的特征函数。在本文中，我们处理了N.G.提出的以下问题。 USHAKOV：$ F $从未由其虚构部分$ \ im f $确定的是真的吗？换句话说，对于任何特征函数$ f $是否确实存在特征函数$ g $，以便$ \ im f \ equiv \ im g $但是$ f \ f \ not \ equiv g $？我们在任意局部紧凑的Abelian组定义的特征函数的更一般情况下研究了这个问题。给出了哪些特征函数的特征是由其假想部分唯一决定的。由于这种表征，我们得到了经典局部紧凑的阿贝尔群体上的几种经常使用的特征功能由其虚构部分决定。
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.
