On a uniqueness theorem for characteristic functions
Articles
Saulius Norvidas
Vilnius University; Vilnius Gediminas Technical University, Lithuania
Published 2017-05-10
https://doi.org/10.15388/NA.2017.3.9
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Keywords

Bochner's theorem
characteristic function
Fourier algebra
positive definite function
imaginary part of the characteristic function

How to Cite

Norvidas S. (2017) “On a uniqueness theorem for characteristic functions”, Nonlinear Analysis: Modelling and Control, 22(3), pp. 412-420. doi: 10.15388/NA.2017.3.9.

Abstract

Suppose that f is the characteristic function of a probability measure on the real line R. We deal with the following open 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 = Im g, but f ≠ g? The answer to this question is no. We give a characterization of those characteristic functions, which are uniquely determined by their imaginary parts. Also, several examples of characteristic functions, which are uniquely determined by their imaginary parts, are given.

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