Positive definite collections of disks
| dc.creator | Tkachev, Vladimir | |
| dc.date | 2007-09-27 | |
| dc.date.accessioned | 2026-07-07T08:32:39Z | |
| dc.date.available | 2026-07-07T08:32:39Z | |
| dc.description | 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 $R<ρ_n$, where $z=2ρ_n^2-1$ is the smallest $\ne-1$ zero of the Jacobi polynomial $\mathcal{P}^{n-2ν,-1}_ν(z)$, $ν=[n/2]$. | |
| dc.description | 24 pages | |
| dc.identifier | https://arxiv.org/abs/0709.4460 | |
| dc.identifier | http://arxiv.org/abs/0709.4460 | |
| dc.identifier | Indiana Univ. Math. J., 55 (2006), no. 6, 1907-1934 | |
| dc.identifier | doi:10.1512/iumj.2006.55.3004 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/138860 | |
| dc.subject | Complex Variables | |
| dc.subject | Metric Geometry | |
| dc.subject | 30E05; 15A48; 33C10; 26C10 | |
| dc.title | Positive definite collections of disks | |
| dc.type | text |