New criteria on positive-definite distributions

TL;DR

This paper develops new criteria to determine when certain functions, especially radial functions, define positive-definite distributions, expanding understanding of their properties in multivariate analysis.

Contribution

It introduces several sufficient conditions for functions to be positive-definite distributions, focusing on radial functions and general functions of norms.

Findings

Established new criteria for positive-definite distributions

Characterized functions of the form f(||x||) as positive-definite

Extended the class of functions known to generate positive-definite distributions

Abstract

We establish several sufficient conditions under which a locally integrable function $f:\mathbb R^n \to \mathbb R$ represents a positive-definite distribution. In particular we consider functions of the form $f(\|x\|)$ where $\|\cdot\|$ is a fixed norm in $\mathbb R^n$.