A note on uniqueness of extension for characteristic functions

TL;DR

This paper investigates whether characteristic functions of probability measures can be uniquely extended beyond their initial domain, focusing on the possibility of non-uniqueness when matching on certain intervals.

Contribution

It introduces a question about the uniqueness of extensions of characteristic functions on intervals not containing zero, exploring conditions for non-uniqueness.

Findings

Existence of characteristic functions matching on specific intervals but differing elsewhere.

Conditions under which non-unique extensions are possible.

Insights into the structure of characteristic functions and their extensions.

Abstract

Let $f:R\to C$ be the characteristic function of a probability measure. We study the following question: Is it true that for any closed interval $I$ on $R$, which does not contain the origin, there exists a characteristic function $g$ such that $g$ coincides with $f$ on $I$ but $g \not\equiv f$ on $R$?