Zeros of the derivatives of Faber polynomials associated with a universal covering map

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Abstract

For a compact set E ⊂ C with connected complement, we study asymptotic behavior of normalized zero counting measures { μk} of the derivatives of Faber polynomials associated with E. For example if E has empty interior, we prove that { μk} converges in the weak-star topology to a probability measure whose support is the boundary of g (D), where g : { | z | > r } ∪ { ∞ } → over(C, -) {minus 45 degree rule} E is a universal covering map such that g (∞) = ∞ and D is the Dirichlet domain associated with g and centered at ∞. Our results are counterparts of those of Kuijlaars and Saff [Asymptotic distribution of the zeros of Faber polynomials, Math. Proc. Cambridge Philos. Soc. 118 (1995) 437-447] on zeros of Faber polynomials. © 2006 Elsevier Inc. All rights reserved.

Author-supplied keywords

  • Derivatives of Faber polynomials
  • Dirichlet domain

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Authors

  • Byung Geun Oh

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