A Note on n -Divisible Positive Definite Functions

Abstract

Let $PD\left(ℝ\right)$ be the family of continuous positive definite functions on $ℝ$ . For an integer $n>1$ , a $f\in PD\left(ℝ\right)$ is called $n$ -divisible if there is $g\in PD\left(ℝ\right)$ such that $g^n=f$ . Some properties of infinite-divisible and $n$ -divisible functions may differ in essence. Indeed, if $f$ is infinite-divisible, then for each integer $n>1$ , there is an unique $g$ such that $g^n=f$ , but there is a $n$ -divisible $f$ such that the factor $g$ in $g^n=f$ is generally not unique. In this paper, we discuss about how rich can be the class $\left\{g\in PD\left(ℝ\right)\text{:}{g}^n=f\right\}$ for $n$ -divisible $f\in PD\left(ℝ\right)$ and obtain precise estimate for the cardinality of this class.