[Paper Review] A Pólya criterion for (strict) positive definiteness on the sphere

R. K. Beatson, Wolfgang zu Castell+1 more|arXiv (Cornell University)|Oct 11, 2011

Mathematical functions and polynomials3 references4 citations

TL;DR

This paper proposes a Pólya-type criterion for (strict) positive definiteness of zonal functions on the sphere $\mathbb{S}^{d-1}$, based on the non-negativity of an integral involving Gegenbauer polynomials. It proves the conjecture for dimensions $d = 3$ to $8$, providing a practical sufficient condition to verify strict positive definiteness without computing all Gegenbauer coefficients, thus enabling reliable kernel-based interpolation.

ABSTRACT

Positive definite functions are very important in both theory and applications of approximation theory, probability and statistics. In particular, identifying strictly positive definite kernels is of great interest as interpolation problems corresponding to these kernels are guaranteed to be poised. A Bochner type result of Schoenberg characterises continuous positive definite zonal functions, $f(\cos \cdot)$, on the sphere $\Sdmone$, as those with nonnegative Gegenbauer coefficients. More recent results characterise strictly positive definite functions on $\Sdmone$ by stronger conditions on the signs of the Gegenbauer coefficients. Unfortunately, given a function $f$, checking the signs of all the Gegenbauer coefficients can be an onerous, or impossible, task. Therefore, it is natural to seek simpler sufficient conditions which guarantee (strict) positive definiteness. We state a conjecture which leads to a Pólya type criterion for functions to be (strictly) positive definite on the sphere $\Sdmone$. In analogy to the case of the Euclidean space, the conjecture claims positivity of a certain integral involving Gegenbauer polynomials. We provide a proof of the conjecture for $d$ from 3 to 8.

Motivation & Objective

• To develop a simple, verifiable sufficient condition for (strict) positive definiteness of zonal functions on the sphere $\mathbb{S}^{d-1}$, avoiding the need to compute all Gegenbauer coefficients.
• To extend the classical Pólya criterion from Euclidean space to the spherical setting, leveraging integral conditions on Gegenbauer polynomials.
• To establish a conjecture involving the non-negativity of a specific integral transform as a sufficient condition for (strict) positive definiteness.
• To prove the conjecture for dimensions $d = 3$ to $8$, providing a constructive framework for identifying strictly positive definite kernels.
• To support the construction of stable and poised interpolation schemes by identifying kernels with guaranteed strict positive definiteness.

Proposed method

• Formulate a conjecture that a function $f(\cos \theta)$ is (strictly) positive definite on $\mathbb{S}^{d-1}$ if a certain integral involving Gegenbauer polynomials is non-negative.
• Use the Gegenbauer expansion of $f(\cos \theta)$ to express the kernel in terms of orthogonal polynomials on $[-1,1]$.
• Define an auxiliary function $H_n(u)$ based on derivatives of a generating function $h(u)$, and analyze its sign to verify positivity of the kernel's integral condition.
• Employ asymptotic analysis and numerical computation to determine intervals where $H_n(u) < 0$, ensuring the positivity of the kernel's transform $F_n^3(t)$.
• Establish overlap between regions of positivity of $F_n^3(t)$ using bounds on the largest zero of $C_n^3(\cos \theta)$ and asymptotic estimates of $\sin t_n^*$.
• Complete the proof for $n \leq 14$ by direct numerical plotting of the explicit expression for $F_n^3(t)$ on $[0, \pi]$.

Experimental results

Research questions

• RQ1 Can a Pólya-type sufficient condition for (strict) positive definiteness on the sphere be formulated using integral conditions on Gegenbauer polynomials?
• RQ2 Is the proposed conjecture—non-negativity of a specific integral involving Gegenbauer polynomials—valid for dimensions $d = 3$ to $8$?
• RQ3 Can the conjecture be proven without computing all Gegenbauer coefficients, thus enabling practical verification of strict positive definiteness?
• RQ4 What is the behavior of the kernel transform $F_n^3(t)$ across $[0, \pi]$ for large and small $n$, and does it remain positive?
• RQ5 How do the asymptotic estimates of the largest zero of $C_n^3(\cos \theta)$ and the positivity regions of $F_n^3(t)$ interact to ensure global positivity?

Key findings

• The conjecture for a Pólya-type criterion for (strict) positive definiteness on $\mathbb{S}^{d-1}$ is proven true for all dimensions $d = 3$ to $8$.
• The function $h(u) = \int_0^1 (1-s)^4 \cos(su) \, ds$ is used to construct a test function whose derivatives are analyzed to verify sign conditions on $H_n(u)$.
• It is shown that $H_n(u) < 0$ for $u > u^* \approx 3.63661$ and $n \geq 9$, implying $\frac{d}{dn}G_n^{2,3}(t) > 0$ for $t > u^*/n$, which supports positivity of the kernel transform.
• The positivity of $F_n^3(t)$ is established on $(0, \sin t_n^*)$ via bounds on the largest zero of $C_n^3(\cos \theta)$, with $\sin t_n^* \geq \frac{\sqrt{6 + \pi^2}}{n+1} + \mathcal{O}(n^{-2})$.
• For $n > 14$, the positivity regions of $F_n^3(t)$ from asymptotic and numerical analysis overlap and cover the entire interval $(0, \pi)$, ensuring global positivity.
• The proof is completed for $n \leq 14$ by direct numerical evaluation of $F_n^3(t)$, confirming positivity on $[0, \pi]$.