Positive definite functions on a regular domain

TL;DR

This paper characterizes positive definite functions on various regular domains such as the unit ball, hyperbolic surface, and simplex, using distance-preserving maps to the sphere.

Contribution

It introduces a unified approach to characterize positive definite functions on multiple regular domains via embeddings into the sphere.

Findings

Characterizations of positive definite functions on the listed domains.

Embedding methods relate these functions to sphere-based functions.

Provides a framework for analyzing functions on complex geometric domains.

Abstract

We define positive and strictly positive definite functions on a domain and study these functions on a list of regular domains. The list includes the unit ball, conic surface, hyperbolic surface, solid hyperboloid, and simplex. Each of these domains is embedded in a quadrant or a union of quadrants of the unit sphere by a distance preserving map, from which characterizations of positive definite and strictly positive definite functions are derived for these regular domains.