Conformal Models and Fingerprints of Pseudo-lemniscates

Abstract

We prove that every meromorphic function on the closure of an analytic Jordan domain that is sufficiently well-behaved on the boundary is conformally equivalent to a rational map whose degree is smallest possible. We also show that the minimality of the degree fails in general without the boundary assumptions. As an application, we generalize a theorem of Ebenfelt, Khavinson, and Shapiro by characterizing fingerprints of polynomial pseudo-lemniscates.

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Acknowledgments

The authors thank the anonymous referees for helpful suggestions.

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Correspondence to Malik Younsi.

Additional information

Malik Younsi supported by NSERC.

Communicated by Edward B. Saff.

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Richards, T., Younsi, M. Conformal Models and Fingerprints of Pseudo-lemniscates. Constr Approx 45, 129–141 (2017). https://doi.org/10.1007/s00365-016-9348-0

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Keywords

  • Meromorphic functions
  • Conformal welding
  • Conformal models
  • Pseudo-lemniscates
  • Fingerprints

Mathematics Subject Classification

  • Primary 30C35
  • Secondary 37E10