Bergman Orthogonal Polynomials and the Grunsky Matrix

Abstract

By exploiting a link between Bergman orthogonal polynomials and the Grunsky matrix, probably first observed by Kühnau (Ann Acad Sci Math 10:313–329, 1985), we improve on some recent results on strong asymptotics of Bergman polynomials outside the domain G of orthogonality, and on the entries of the Bergman shift operator. In our proofs, we suggest a new matrix approach involving the Grunsky matrix and use well-established results in the literature relating properties of the Grunsky matrix to the regularity of the boundary of G and the associated conformal maps. For quasiconformal boundaries, this approach allows for new insights for Bergman polynomials.

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Fig. 1

Notes

  1. 1.

    We do not need further assumptions on the boundary like \(\Gamma \) being rectifiable, compare with [29, Lemma 2.1].

  2. 2.

    Compare with [29, Theorem 2.2].

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Acknowledgements

The authors are grateful to the two referees for their helpful suggestions, especially for pointing out that the order in Theorem 1.1 is actually sharp for a specific piecewise analytic curve.

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Correspondence to Nikos Stylianopoulos.

Additional information

Bernhard Beckermann: Supported in part by the Labex CEMPI (ANR-11-LABX-0007-01).

Nikos Stylianopoulos: Supported in part by the University of Cyprus grant 3/311-21027.

Communicated by Edward B. Saff.

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Beckermann, B., Stylianopoulos, N. Bergman Orthogonal Polynomials and the Grunsky Matrix. Constr Approx 47, 211–235 (2018). https://doi.org/10.1007/s00365-017-9381-7

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Keywords

  • Bergman orthogonal polynomials
  • Faber polynomials
  • Conformal mapping
  • Grunsky matrix
  • Bergman shift
  • Quasiconformal mapping

Mathematics Subject Classification

  • 30C10
  • 30C62
  • 41A10
  • 65E05
  • 30E10