## Abstract

Let *G* be a bounded Jordan domain in ℂ and let *w* n = 0 be an analytic function on *G* such that t*S*
_{G}
*s*¦ω¦^{2}
*dm* < ∞, where *dm* is the area measure. We investigate the zero distribution of the sequence of polynomials that are orthogonal on *G* with respect to ¦ω¦^{2}
*dm*. We find that such a distribution depends on the location of the singularities of the reproducing kernel *K*
_{w}(*z*, ζ) of the space *L*
_{ω}
^{2}(*G*):= *f* analytic on *G*: ∫_{G}
*¦ f* ¦^{2}¦ω¦^{2}
*dm* < ∞. A fundamental theorem is given for the case when K_{ω}(*·*, ζ) has a singularity on ∂*G* for at least some ζ ∈ *G*. To investigate the opposite case, we consider two examples in detail: first when *G* is the unit disk and ω is meromorphic, and second when *G* is a lens-shaped domain and ω is entire. Our analysis can also be applied for ω *≡* 1 in the case when *G* is a rectangle or a special triangle. We also provide formulas for *K*
_{ω}(*·*, ζ) that are of help for the determination of its singularities.

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## Additional information

The research of E. B. Saff was supported, in part, by the U. S. National Science Foundation under grant DMS-0296026.

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### Cite this article

Miña-Díaz, E., Saff, E.B. & Stylianopoulos, N.S. Zero Distributions for Polynomials Orthogonal with Weights over Certain Planar Regions.
*Comput. Methods Funct. Theory* **5, **185–221 (2005). https://doi.org/10.1007/BF03321094

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### Keywords

- Orthogonal polynomials
- zeros of polynomials
- kernel function
- logarithmic potential
- equilibrium measure

### 2000 MSC

- 30C10
- 30C15
- 30C40
- 31A05
- 31A15