Skip to main content
Log in

Bergman Orthogonal Polynomials and the Grunsky Matrix

Constructive Approximation Aims and scope

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Notes

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

  2. Compare with [29, Theorem 2.2].

References

  1. Andrievskii, V., Blatt, H.-P.: Discrepancy of Signed Measures and Polynomial Approximation. Springer, Berlin (2002)

    Book  MATH  Google Scholar 

  2. Beckermann, B.: Complex Jacobi matrices. J. Comput. Appl. Math. 127, 17–65 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Carleman, T.: Über die Approximation analytischer Funktionen durch lineare Aggregate von vorgegebenen Potenzen. Ark. Mat. Astron. Fys. 17, 215–244 (1923)

    MATH  Google Scholar 

  4. Clunie, J.: On schlicht functions. Ann. Math. (2) 69, 511–519 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  5. Conway, J.B.: A Course in Operator Theory, Graduate Studies in Mathematics, vol. 21. AMS, Providence (2000)

    Google Scholar 

  6. Dragnev, P., Miña-Díaz, E.: On a series representation for Carleman orthogonal polynomials. Proc. Am. Math. Soc. 138(12), 4271–4279 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dragnev, P., Miña-Díaz, E.: Asymptotic behavior and zero distribution of Carleman orthogonal polynomials. J. Approx. Theory 162, 1982–2003 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dragnev, P., Miña-Díaz, E., Northington, V.M.: Asymptotics of Carleman polynomials for level curves of the inverse of a shifted Zhukovsky transformation. Comput. Methods Funct. Theory 13(1), 75–89 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dunford, N., Schwartz, J.T.: Linear Operators, Volume 2: Spectral Theory, Selfadjoint operators in Hilbert space. Wiley, Hoboken (1988)

    MATH  Google Scholar 

  10. Gaier, D.: Lectures on Complex Approximation. Birkhäuser, Boston (1987)

    Book  MATH  Google Scholar 

  11. Gaier, D.: The Faber operator and its boundedness. J. Approx. Theory 101(2), 265–277 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gustafsson, B., Putinar, M., Saff, E.B., Stylianopoulos, N.: Les polynomes orthogonaux de Bergman sur un archipel. C. R. Acad. Sci. Paris Ser. I 346(9–10), 499–502 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gustafsson, B., Putinar, M., Saff, E.B., Stylianopoulos, N.: Bergman polynomials on an archipelago: estimates, zeros and shape construction. Adv. Math. 222, 1405–1460 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Henrici, P.: Applied and Computational Complex Analysis, vol. 3. Wiley, Hoboken (1986)

    MATH  Google Scholar 

  15. Johnston, E.R.: A Study in Polynomial Approximation in the Complex Domain, Ph.D. thesis, University of Minnesota (1954)

  16. Jones, G.L.: The Grunsky operator and the Schatten ideals. Mich. Math. J. 46, 93–100 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1980)

    MATH  Google Scholar 

  18. Kühnau, R.: Entwicklung gewisser dielektrischer Grundlösungen in Orthonormalreihen. Ann. Acad. Sci. Math. 10, 313–329 (1985)

    MathSciNet  MATH  Google Scholar 

  19. Kühnau, R.: Zur Berechnung der Fredholmschen Eigenwerte ebener Kurven. ZAMM 66, 193–200 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  20. Miña-Díaz, E.: On the leading coefficient of polynomials orthogonal over domains with corners. Numer. Algorithms 70, 1–8 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Pommerenke, Ch.: Univalent Functions. Vandenhoeck and Ruprecht, Göttingen (1975)

    MATH  Google Scholar 

  22. Pommerenke, Ch.: Boundary Behaviour of Conformal Maps. Springer, Berlin (1992)

    Book  MATH  Google Scholar 

  23. Saff, E.B.: Orthogonal polynomials from a complex perspective. In: Nevai, P. (ed.) Orthogonal Polynomials: Theory and Practice, pp. 363–393. Kluwer, Dordrecht (1990)

    Chapter  Google Scholar 

  24. 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)

    Article  MathSciNet  MATH  Google Scholar 

  25. Saff, E.B., Stahl, H., Stylianopoulos, N., Totik, V.: Orthogonal polynomials for area-type measures and image recovery. SIAM J. Math. Anal. 47, 2442–2463 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  26. Shen, Y.L.: Faber polynomials with applications to univalent functions with quasiconformal extensions. Sci. China Ser. A 52(10), 2121–2131 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  27. Simon, B.: Szegő’s Theorem and Its Decendants. Princton University Press, Princton (2011)

    Google Scholar 

  28. Smirnov, V.I., Lebedev, N.A.: Functions of a Complex Variable. MIT Press, Cambrigde, MA (1968)

    MATH  Google Scholar 

  29. Stylianopoulos, N.: Strong asymptotics for Bergman orthogonal polynomials over domains with corners and applications. Constr. Approx. 38, 59–100 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  30. Stylianopoulos, N.: Boundary estimates for Bergman polynomials in domains with corners. Contemp. Math. 661, 187–198 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  31. Suetin, P.K.: Polynomials Orthogonal over a Region and Bieberbach Polynomials. American Mathematical Society, Providence (1974)

    MATH  Google Scholar 

  32. Takhtajan, L., Teo, L.P.: Weil–Petersson metric on the universal Teichmüller space. Mem. Am. Math. Soc. 861, 1–183 (2006)

    MATH  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Nikos Stylianopoulos.

Additional information

Communicated by Edward B. Saff.

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.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00365-017-9381-7

Keywords

Mathematics Subject Classification

Navigation