Skip to main content
SpringerLink
Log in

Generalized Fourier coefficients of a quasisymmetric homeomorphism and the Fredholm eigenvalue

  • Published:
Journal d'Analyse Mathématique Aims and scope

Abstract

Recently, Reiner Kühnau gave some new characterizations of the Fredholm eigenvalue by means of generalized Fourier coefficients. In this note we give a fast approach to Kühnau’s results and answer some questions posed by Kühnau as well.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. L.V. Ahlfors, Remarks on the Neumann-Poincaré integral equations, Pacific J. Math. 2 (1952), 271–280.

    MATH  MathSciNet  Google Scholar 

  2. L. V. Ahlfors, Conformal Invariants, Topics in Geometric Function Theory, McGraw-Hill, New York, 1973.

    MATH  Google Scholar 

  3. A. Beurling and L. V. Ahlfors, The boundary correspondence under quasiconformal mappings, Acta Math. 96 (1956), 125–142.

    Article  MATH  MathSciNet  Google Scholar 

  4. S. Bergman and M. M. Schiffer, Kernel functions and conformal mappings, Compositio. Math. 8 (1951), 205–249.

    MATH  MathSciNet  Google Scholar 

  5. F. P. Gardiner and D. Sullivan, Symmetric structures on a closed curve, Amer. J. Math. 114 (1992), 683–736.

    Article  MATH  MathSciNet  Google Scholar 

  6. J. B. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981.

    MATH  Google Scholar 

  7. J. G. Krzyż and D. Partyka, Generalized Neumann-Poincaré operator, chord-arc curves and Fredholm eigenvalues, Complex Variables Theory Appl. 21 (1993), 253–263.

    MATH  MathSciNet  Google Scholar 

  8. R. Kühnau, Koeffizientenbedingungen vom Grunskyschen Typ und Fredholmsche Eigenwerte, Ann. Univ. Mariae Curie-Skłlodowska Lublin Sect. A 58 (2004), 79–87.

    MATH  Google Scholar 

  9. R. Kühnau, New characterizations of Fredholm eigenvalues of quasicircles, Rev. Roumaine Math. Pures Appl. 51 (2006), 295–307.

    Google Scholar 

  10. R. Kühnau, A new matrix characterization of Fredholm eigenvalues of quasicircles, J. Anal. Math. 99 (2006), 295–307.

    Article  MATH  MathSciNet  Google Scholar 

  11. R. Kühnau, Quadratic Forms in Geometric Function Theory, Quasiconformal Extensions, Fredholm Eigenvalues in Complex Analysis and Dynamical Systems III, Amer. Math. Soc. Providence, RI, 2008, pp. 237–256.

  12. S. Nag and D. Sullivan, Teichmüller theory and the universal period mapping via quantum calculus and the H½ space on the circle, Osaka J. Math. 32 (1995), 1–34.

    MATH  MathSciNet  Google Scholar 

  13. D. Partyka, The Generalized Neumann-Poincaré Operator and its Spectrum, Dissertationes Math. (Rozprawy Mat) 366 1997.

  14. D. Partyka, The Grunsky type inequalties for quasisymmetric automorphism of the unit circle, Bull. Soc. Sci. Lett. Lódź, Sér. Rech. Déform. 31 (2000), 135–142.

    MathSciNet  Google Scholar 

  15. M. Schiffer, Fredholm eigenvalues of plane domains, Pacific J. Math. 7 (1957), 1187–1225.

    MATH  MathSciNet  Google Scholar 

  16. M. Schiffer, Fredholm eigenvalues and Grunsky matrices, Ann. Polon. Math. 39 (1981), 149–164.

    MathSciNet  Google Scholar 

  17. G. Schober, Continuity of curve functionals and a technique involving quasiconformal mappings, Arch. Rational Mech. Anal. 29 (1968), 378–389.

    Article  MATH  MathSciNet  Google Scholar 

  18. G. Schober, Semicontinuity of curve functionals, Arch. Rational Mech. Anal. 33 (1969), 374–376.

    Article  MATH  MathSciNet  Google Scholar 

  19. Y. Shen, Quasiconformal mappings and harmonic functions, Adv. Math. (China) 28 (1999), 347–357.

    MATH  MathSciNet  Google Scholar 

  20. Y. Shen, Pull-back operators by quasisymmetric functions and invariant metrics on Teichmüller spaces, Complex Variables Theory Appl. 42 (2000), 289–307.

    MATH  MathSciNet  Google Scholar 

  21. Y. Shen, Notes on pull-back operators by quasisymmetric homeomorphisms with applications to Schober’s functionals, Chinese Ann.Math. Ser. A 24 (2003), 209–218; translation in Chinese J. Contemp. Math. 24 (2003), 187–196.

    MATH  Google Scholar 

  22. Y. Shen, On Grunsky operator, Sci. China Ser. A 50 (2007), 1805–1817.

    Article  MATH  MathSciNet  Google Scholar 

  23. G. Springer, Fredholm eigenvalues and quasiconformal mapping, Acta Math. 111 (1964), 121–141.

    Article  MATH  MathSciNet  Google Scholar 

  24. K. Strebel, Zur Frage der Eindeutigkeit extremaler quasikonformer Abbildungen des Einheitskreises II, Comment. Math. Helv. 39 (1964), 77–89.

    Article  MATH  MathSciNet  Google Scholar 

  25. K. Strebel, Extremal quasiconformal mappings, Result. Math. 10 (1986), 169–209.

    MathSciNet  Google Scholar 

  26. L. Takhtajan and Lee-Peng Teo, Weil-Petersson metric on the universal Teichmüller space, Mem. Amer. Math. Soc. 183 (2006), no. 861.

  27. A. Zygmund, Trigonometric Series, Cambridge University Press, Cambridge, 1979.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Soochow University, Suzhou, 215006, P.R. China

    Shen Yuliang

Authors
  1. Shen Yuliang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shen Yuliang.

Additional information

Research supported by Program for New Century Excellent Talents in University and the National Natural Science Foundation of China.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yuliang, S. Generalized Fourier coefficients of a quasisymmetric homeomorphism and the Fredholm eigenvalue. JAMA 112, 33–48 (2010). https://doi.org/10.1007/s11854-010-0024-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11854-010-0024-1

Search

Navigation