New criteria on positive-definite distributions

TL;DR

The paper addresses when a locally integrable function $f:\mathbb{R}^n\to\mathbb{R}$ represents a positive-definite distribution, extending radial forms $f(|x|)$ and norm-based forms $f(\|x\|_K)$ beyond classical criteria. It develops new sufficient conditions by exploiting Radon-transform connections and monotonicity properties of radial profiles, proving that $|x|^{-n+2}f(|x|)$ is positive-definite under specific decay/integrability constraints and that $f(\|x\|_K)$ is positive-definite for any symmetric convex body $K$ under derivative-monotonicity assumptions. Among the contributions, it shows $g_{n,p}(x)=|x|^{1-n}e^{-|x|^p}$ is positive-definite for all $p>0$ when $n\ge3$, and that both $g_{n,p}$ and its Fourier transform are non-negative in appropriate regimes, while also clarifying non-converse aspects in subspace cases. These results broaden the toolkit for certifying positive-definite distributions, with implications for convex geometry, probability, and integral equations, and provide a bridge between one-dimensional criteria (e.g., Pólya) and higher-dimensional radial/norm-based constructions.

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 $\mathbb R^n$.

Theorems & Definitions (8)

• Theorem 1: koldobsky2002derivatives
• Theorem 2
• Theorem 4
• Lemma 1
• proof
• proof : Proof of Theorem \ref{['thm_decreasing_dimn']}
• proof : Proof of Theorem \ref{['thm_omega']}
• proof : Proof of Theorem \ref{['thm_convex_curiosity']}