Skip to main content
SpringerLink
Log in

Computation of Analytic Capacity and Applications to the Subadditivity Problem

  • Published:
Computational Methods and Function Theory Aims and scope Submit manuscript

Abstract

We develop a least-squares method for computing the analytic capacity of compact plane sets with piecewise-analytic boundary. The method furnishes rigorous upper and lower bounds which converge to the true value of the capacity. Several illustrative examples are presented. We are led to formulate a conjecture which, if true, would imply that analytic capacity is subadditive. The conjecture is proved in a special case.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Similar content being viewed by others

References

  1. Ahlfors, L.: Bounded analytic functions. Duke Math. J. 14, 1–11 (1947)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ahlfors, L.: Complex Analysis. McGraw-Hill Book Co., New York (1978)

  3. Bell, S.: The Cauchy transform, potential theory, and conformal mapping studies. In: Advanced Mathematics (1992)

  4. Davie, A.M.: Analytic capacity and approximation problems. Trans. Am. Math. Soc. 171, 409–444 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dudziak, J.: Vitushkin’s Conjecture for Removable Sets. Springer, New York (2010)

    Book  MATH  Google Scholar 

  6. Duren, P.: Theory of \(H^{p}\) Spaces. Academic Press, New York (1970)

    MATH  Google Scholar 

  7. Garabedian, P.R.: Schwarz’s lemma and the Szegö kernel function. Trans. Am. Math. Soc. 67, 1–35 (1949)

    MathSciNet  MATH  Google Scholar 

  8. Garnett, J.: Analytic Capacity and Measure. Springer, Berlin (1972)

    Book  MATH  Google Scholar 

  9. Ja, S.: Havinson, Analytic capacity of sets, joint non-triviality of various classes of analytic functions and the Schwarz lemma in arbitrary domains (Russian) Mat. Sb. (N.S.) 54(96), 3–50 (1961). Translation in Am. Math. Soc. Transl. (2) 43, 215–266 (1964)

  10. Melnikov, M., Analytic capacity: a discrete approach and the curvature of measure (Russian) Mat. Sb. 186, 57–76. Translation in Sb. Math. 186, 827–846 (1995)

  11. Murai, T.: Analytic capacity (a theory of the Szegő kernel function). Am. Math. Soc. Transl. Ser. 2(161), 51–74 (1994)

    MathSciNet  Google Scholar 

  12. Pommerenke, C.: Boundary Behaviour of Conformal Maps. Springer, Berlin (1992)

    Book  MATH  Google Scholar 

  13. Ransford, T.: Computation of logarithmic capacity. Comput. Methods Funct. Theory 10, 555–578 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Rostand, J.: Computing logarithmic capacity with linear programming. Exp. Math. 6, 221–238 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  15. Rudin, W.: Real and complex analysis. McGraw-Hill, New York (1987)

    MATH  Google Scholar 

  16. Smirnov, V., Lebedev, N.A.: Functions of a Complex Variable: Constructive theory. M.I.T. Press, Cambridge (1968)

    MATH  Google Scholar 

  17. Smith, E.P.: The Garabedian function of an arbitrary compact set. Pac. J. Math. 51, 289–300 (1974)

    Article  MATH  Google Scholar 

  18. Suita, N.: On a metric induced by analytic capacity. Kōdai Math. Sem. Rep. 25, 215–218 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  19. Suita, N.: On subadditivity of analytic capacity for two continua. Kōdai Math. J. 7, 73–75 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  20. Tolsa, X.: Painlevé’s problem and the semiadditivity of analytic capacity. Acta Math. 190, 105–149 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  21. Tolsa, X.: Painlevé’s problem and analytic capacity. Collect. Math. Vol. Extra 89–125 (2006)

  22. Tumarkin, G., Havinson, S.: An expansion theorem for analytic functions of class \(E_{p}\) in multiply connected domains (Russian). Uspehi Mat. Nauk (N.S.) 13, 223–228 (1958)

    MathSciNet  MATH  Google Scholar 

  23. Vituškin, A.: Analytic capacity of sets in problems of approximation theory (Russian). Uspehi Mat. Nauk 22, 141–199 (1967)

    MathSciNet  Google Scholar 

  24. Whittaker, E.T., Watson, G.N.: A course of modern analysis. An introduction to the general theory of infinite processes and of analytic functions: with an account of the principal transcendental functions. Cambridge University Press, New York (1962)

  25. Zalcman, L.: Analytic capacity and rational approximation. Springer, Berlin (1986)

    Google Scholar 

Download references

Acknowledgments

The authors thank Vladimir Andrievskii, Juan Arias de Reyna, Dmitry Khavinson, Tony O’Farrell and Nikos Stylianopoulos for helpful discussions.

Author information

Authors and Affiliations

  1. Département de Mathématiques et de Statistique, Pavillon Alexandre–Vachon, Université Laval, 1045 av. de la Médecine, Québec, QC, G1V 0A6, Canada

    Malik Younsi & Thomas Ransford

Authors
  1. Malik Younsi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Thomas Ransford
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Thomas Ransford.

Additional information

Communicated by Edward B. Saff.

M. Younsi was supported by the Vanier Canada Graduate Scholarships program. T. Ransford was supported by grants from NSERC and the Canada Research Chairs program.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Younsi, M., Ransford, T. Computation of Analytic Capacity and Applications to the Subadditivity Problem . Comput. Methods Funct. Theory 13, 337–382 (2013). https://doi.org/10.1007/s40315-013-0026-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40315-013-0026-y

Keywords

Mathematics Subject Classification (1991) 

Search

Navigation