Asymptotics for Hessenberg Matrices for the Bergman Shift Operator on Jordan Regions

Abstract

Let \(G\) be a bounded Jordan domain in the complex plane. The Bergman polynomials \(\{p_n\}_{n=0}^\infty \) of \(G\) are the orthonormal polynomials with respect to the area measure over \(G\). They are uniquely defined by the entries of an infinite upper Hessenberg matrix \(M\). This matrix represents the Bergman shift operator of \(G\). The main purpose of the paper is to describe and analyze a close relation between \(M\) and the Toeplitz matrix with symbol the normalized conformal map of the exterior of the unit circle onto the complement of \(\overline{G}\). Our results are based on the strong asymptotics of \(p_n\). As an application, we describe and analyze an algorithm for recovering the shape of \(G\) from its area moments.

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Notes

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    This theorem, along with a sketch of its proof given in Sect. 5.3, was presented by the first author at the Joint Meeting of the AMS and MAA in Phoenix, January 2004.

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Correspondence to Edward B. Saff.

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The research of E. B. Saff was supported, in part, by U.S. National Science Foundation grants DMS-0808093 and DMS-1109266. The research of N. Stylianopoulos was supported by the University of Cyprus research grant 21027.

Communicated by Mihai Putinar.

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Saff, E.B., Stylianopoulos, N. Asymptotics for Hessenberg Matrices for the Bergman Shift Operator on Jordan Regions. Complex Anal. Oper. Theory 8, 1–24 (2014). https://doi.org/10.1007/s11785-012-0252-8

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Keywords

  • Bergman orthogonal polynomials
  • Faber polynomials
  • Bergman shift operator
  • Toeplitz matrix
  • Strong asymptotics
  • Conformal mapping

Mathematics Subject Classification (2010)

  • 47B35
  • 30C10
  • 30C30
  • 41A10
  • 45Q05