Zero Distributions for Polynomials Orthogonal with Weights over Certain Planar Regions

Abstract

Let G be a bounded Jordan domain in ℂ and let w n = 0 be an analytic function on G such that tS 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 Kω(·, 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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Correspondence to Erwin Miña-Díaz.

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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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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