Positive definite functions on the unit sphere and integrals of Jacobi polynomials

TL;DR

This paper proves a conjecture regarding the positivity of integrals of Jacobi polynomials, establishing conditions for positive definite functions on spheres and other homogeneous spaces, with implications for harmonic analysis.

Contribution

It confirms a conjecture on Jacobi polynomial integrals, demonstrating positive definiteness of certain functions on spheres and homogeneous spaces, extending Polyà's criterion.

Findings

Proves positivity of Jacobi polynomial integrals for specified parameters.

Establishes positive definiteness of eltaunction on spheres for elta e rac{d}{2}e.

Extends results to compact two-point homogeneous spaces.

Abstract

It is shown that the integrals of the Jacobi polynomials \begin{equation*}%\label{eq:Fn^J} \int_0^t (t-\theta)^\delta P_n^{(\alpha-\frac12,\beta-\frac12)}(\cos \theta) \left(\sin \tfrac{\theta}2\right)^{2 \alpha} \left(\cos \tfrac{\theta}2\right)^{2 \beta} d\theta > 0 \end{equation*} for all $t \in (0,\pi]$ and $n \in \mathbb{N}$ if $\delta \ge \alpha + 1$ for $\alpha,\beta \in \mathbb{N}_0$ and $\max\{\alpha,\beta\} > 0$. This proves a conjecture on the integral of the Gegenbauer polynomials in \cite{BCX} that implies the strictly positive definiteness of the function $\theta \mapsto (t - \theta)_+^\delta$ on the unit sphere $\mathbb{S}^{d-1}$ for $\delta \ge \lceil \frac{d}{2}\rceil$ and the Poly\`a criterion for positive definite functions on the sphere for all dimensions. Moreover, the positive definiteness of the function $\theta \mapsto (t - \theta)_+^\delta$ is also established on the compact two-point homogeneous spaces.