Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Next Article Computational Methods and Function Theory 5 (2005), No. 1, 185--221 Copyright Heldermann Verlag 2005 Zero Distributions for Polynomials Orthogonal with Weights over Certain Planar Regions Erwin Miña-Díaz emina@math.vanderbilt.edu , Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, 37240 Nashville, U.S.A. Edward B. Saff esaff@math.vanderbilt.edu , Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, 37240 Nashville, U.S.A. Nikos S. Stylianopoulos nikos@ucy.ac.cy , Department of Mathematics and Statistics, University of Cyprus, P.O. Box 20537, 1678 Nicosia, Cyprus. [Abstract-pdf] [Abstract-ps] Let $G$ be a bounded Jordan domain in $\mathbb{C}$ and let $w\not\equiv 0$ be an analytic function on $G$ such that $\int_{G}|w|^2dm<\infty$, where $dm$ is the area measure. We investigate the zero distribution of the sequence of polynomials that are orthogonal on $G$ with respect to $|w|^2dm$. We find that such a distribution depends on the location of the singularities of the reproducing kernel $K_w(z,\zeta)$ of the space $\mathcal{L}^2_w(G):=\left\{ f\ \mbox{analytic on } G:\int_G|f|^2|w|^2\,dm<\infty \right\}$. A fundamental theorem is given for the case when $K_w(\cdot,\zeta)$ has a singularity on $\partial G$ for at least some $\zeta\in G$. To investigate the opposite case, we consider two examples in detail: first when $G$ is the unit disk and $w$ is meromorphic, and second when~$G$ is a lens-shaped domain and $w$ is entire. Our analysis can also be applied for $w\equiv 1$ in the case when $G$ is a rectangle or a special triangle. We also provide formulas for $K_w(\cdot,\zeta)$ that are of help for the determination of~its~singularities. Keywords: Orthogonal polynomials, zeros of polynomials, kernel function, logarithmic potential, equilibrium measure. MSC 2000: Primary 30C10; Secondary 30C15, 30C40, 31A05, 31A15. [FullText-pdf (856 K)] [FullText-ps (6946 K)]