On some positive definite functions

TL;DR

This paper investigates specific functions involving Euclidean norms to determine their positive definiteness, conditional positive definiteness, and infinite divisibility properties across various dimensions.

Contribution

It provides new insights into the positive definiteness and related properties of a class of functions involving Euclidean norms, expanding understanding in harmonic analysis.

Findings

Identifies conditions under which the functions are positive definite.

Establishes criteria for conditional positive definiteness.

Analyzes infinite divisibility of the functions.

Abstract

We study the function $(1 - \|x\|)\slash (1 - \|x\|^r),$ and its reciprocal, on the Euclidean space $\mathbb{R}^n,$ with respect to properties like being positive definite, conditionally positive definite, and infinitely divisible.