A note on n-divisible positive definite functions

TL;DR

This paper investigates the properties and cardinality of the class of positive definite functions on real numbers that are n-th roots of a given n-divisible positive definite function, highlighting differences from infinite divisibility.

Contribution

It provides a detailed analysis of the structure and size of the set of n-th roots of n-divisible positive definite functions, including precise estimates of its cardinality.

Findings

The class of n-th roots can be very rich and non-unique.

Infinite divisibility implies unique roots, unlike n-divisible functions.

The paper offers explicit bounds on the number of roots for n-divisible 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.