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Positive definite functions on a regular domain

Martin Buhmann, Yuan Xu

TL;DR

The paper develops a unified framework for positive definite and strictly positive definite functions on several regular domains by leveraging distance-preserving mappings to quadrants of the unit sphere. It deploys Fourier-Gegenbauer expansions and reproducing-kernel methods to derive explicit characterizations and addition-formula–style kernels for PDFs on the unit sphere, unit ball, hyperbolic surface, solid hyperboloid, conic surface, and simplex. Key contributions include precise conditions on the Fourier-Gegenbauer coefficients, parity-based kernel constructions, and distance-preserving correspondences that reduce domain-specific questions to sphere-based harmonic analysis. These results provide principled criteria for kernel-based interpolation and approximation on a broad class of regular domains, with direct implications for radial-basis function methods and multivariate approximation in constrained geometries.

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.

Positive definite functions on a regular domain

TL;DR

The paper develops a unified framework for positive definite and strictly positive definite functions on several regular domains by leveraging distance-preserving mappings to quadrants of the unit sphere. It deploys Fourier-Gegenbauer expansions and reproducing-kernel methods to derive explicit characterizations and addition-formula–style kernels for PDFs on the unit sphere, unit ball, hyperbolic surface, solid hyperboloid, conic surface, and simplex. Key contributions include precise conditions on the Fourier-Gegenbauer coefficients, parity-based kernel constructions, and distance-preserving correspondences that reduce domain-specific questions to sphere-based harmonic analysis. These results provide principled criteria for kernel-based interpolation and approximation on a broad class of regular domains, with direct implications for radial-basis function methods and multivariate approximation in constrained geometries.

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.
Paper Structure (12 sections, 10 theorems, 89 equations)

This paper contains 12 sections, 10 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. PDFs on the unit sphere
  4. Reproducing kernels for orthogonal polynomials
  5. PDFs on a domain
  6. Positive definite functions on quadrants of the unit sphere
  7. Positive definite functions on several regular domains
  8. PDF on the unit ball
  9. PDF on the hyperbolic surface
  10. PDF on the solid hyperboloid
  11. PDF on the conic surface
  12. PDF on the simplex

Key Result

Theorem 2.1

Let $d\ge 2$ and ${\lambda} = \frac{d-1}{2}$. A continuous function $f: [-1,1]\mapsto {\mathbb R}$ is positive definite on ${\mathbb S}^d$ if and only if $\hat{f}_n^{\lambda} \ge 0$ for all $n$, in which case the series eq:FourierGegen converges absolutely and uniformly to $f(\cos{\theta})$ on $[0,\

Theorems & Definitions (15)

  • Definition 1.1
  • Theorem 2.1
  • Theorem 2.2
  • Definition 2.3
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 4.1
  • Theorem 4.2
  • ...and 5 more