Positive definite collections of disks

TL;DR

This paper extends previous results on positive definiteness of matrices derived from collections of disks, showing that for symmetric, overlapping disks arranged at the vertices of a regular polygon, positive definiteness depends on a specific radius bound related to Jacobi polynomial zeros.

Contribution

The authors generalize the positive definiteness criterion to symmetric overlapping disks arranged in a regular polygon, linking it to zeros of Jacobi polynomials.

Findings

Positive definiteness depends on the radius being less than a specific threshold.

The threshold radius is related to the smallest zero of a Jacobi polynomial.

The result applies to symmetric collections of overlapping disks at polygon vertices.

Abstract

Let $Q(z,w)=-\prod_{k=1}^n [(z-a_k)(\bar{w}-\bar{a}_k)-R_k^2]$. M. Putinar and B. Gustafsson proved recently that the matrix $Q(a_i,a_j)$, $1\leq i,j\leq n$, is positive definite if disks $|z-a_i|<R_i$ form a disjoint collection. We extend this result on symmetric collections of discs with overlapping. More precisely, we show that in the case when the nodes $a_j$ are situated at the vertices of a regular $n$-gon inscribed in the unit circle and $\forall i: R_i\equiv R$, the matrix $Q(a_i,a_j)$ is positive definite if and only if $R<\rho_n$, where $z=2\rho_n^2-1$ is the smallest $\ne-1$ zero of the Jacobi polynomial $\mathcal{P}^{n-2\nu,-1}_\nu(z)$, $\nu=[n/2]$.