A note on powers of the characteristic function

2020/09/06 by Саулюс Норвидас, Norvidas, Saulius
Computer Science · Mathematics · #42A38 - 42A82 - 60E10 #FOS: Mathematics #Iterative Methods for Nonlinear Equations #Mathematical functions and polynomials #Matrix Theory and Algorithms #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2009.02777

openalex publication_date 2020/09/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let CH(R) denote the family of characteristic functions of probability measures (distributions) on the real line R. We study the following question: given an integer n>1, do there exist two different f, g∈ CH(R) such that fⁿ≡ gⁿ? For positive even n, well-known examples answer this question in the affirmative. It turns out that the same is true also for any odd n>1. For f∈ CH(R) and integer n>1, set C_n(f)=\g∈ CH(R): gⁿ≡ fⁿ\. In this paper, we give an estimate for cardinality (or cardinal number) of C_n(f). In addition, we describe such f for which our estimate is sharp.

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