A note on n-divisible positive definite functions

Abstract: Let $PD(\mathbb{R})$ be the family of continuous positive definite functions on $\mathbb{R}$. For an integer $n>1$, a $f\in PD(\mathbb{R})$ is called $n$-divisible if there is $g\in PD(\mathbb{R})$ 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 ${g\in PD(\mathbb{R}): g^n=f}$ for $n$-divisible $f\in PD(\mathbb{R})$ and obtain precise estimate for the cardinality of this class.