Positive definite collections of disks

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 $Q(a_i,a_j)$, $1\le i,j\le n$, 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 ${\mathcal{P}}_{\nu }^{n-2\nu ,-1}(z)$, $\nu =[n/2]$.

24 pages