Computational Methods and Function Theory 5 (2005), No. 1, 185--221
Copyright Heldermann Verlag 2005
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.
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.
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