On extensions of characteristic functions

Abstract

Here we deal with the following question: Is it true that, for any closed interval on the real line R that does not contain the origin, there exists a characteristic function f such that f(x) coincides with the normal characteristic function e−x2/2 on this interval but f(x) 6≡ e−x2/2 on R? The answer to this question is positive. We study a more general case of an arbitrary characteristic function g of a continuous probability density, instead of e−x2/2.