IUMJ

Title: Positive definite collections of disks

Authors: Vladimir Tkachev

Issue: Volume 55 (2006), Issue 6, 1907-1934

Abstract:

Let $Q(z,w) = -\prod_{k=1}^{n}[(z - a_k)(\bar{w} - \bar{a}_{k}) - R^{2}]$. The main result of the paper states that in the case when the nodes $a_{j}$ are situated at the vertices of a regular $n$-gon inscribed in the unit circle, 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 $\mathcal{P}^{n-2\nu, {-}1}_{\nu}(z)$, $\nu=[n/2]$.