Abstract
We give a new proof of Dennis Hejhal’s theorem on the nondegeneracy of the matrix that appears in the identity relating the Bergman and Szegő kernels of a smoothly bounded finitely connected domain in the plane. Mergelyan’s theorem is at the heart of the argument. We explore connections of Hejhal’s theorem to properties of the zeroes of the Szegő kernel and propose some ideas to better understand Hejhal’s original theorem.
Similar content being viewed by others
Data availability statement
This research does not depend on data.
References
Aharonov, D., Shapiro, H.S.: Domains on which analytic functions satisfy quadrature identities. J. D’Analyse Mathématique 30, 39–73 (1976)
Avci, Y.: Quadrature identities and the Schwarz function, Stanford University PhD thesis, (1977)
Barker, W.H., II.: Kernel functions on domains with hyperelliptic double. Trans. Am. Math. Soc. 231, 339–347 (1977)
Bell, S.: The Cauchy transform, potential theory, and conformal mapping, 2nd edn. CRC Press, Boca Raton (2016)
Bell, S.: The Szegő projection and the classical objects of potential theory in the plane. Duke Math. J. 64, 1–26 (1991)
Bell, S., Gustafsson, B., Sylvan, Z.: Szegő coordinates, quadrature domains, and double quadrature domains. Comput. Methods Funct. Theory 11(1), 25–44 (2011)
Bergman, S.: The kernel function and conformal mapping, Mathematical Surveys, Vol.5 (second edition), American Math. Society, Providence, (1970)
Greene, R., Krantz, S.: Function theory of one complex variable. Wiley, New York (1997)
Grunsky, H.: Lectures on Theory of Functions in Multiply Connected Domains. Vandenhoeck and Ruprecht, Göttingen, (1978)
Hawley, N.S., Schiffer, M.: Half-order differentials on Riemann surfaces. Acta Math. 115, 199–236 (1966)
Hejhal, D.: Theta functions, kernel functions, and integrals, Memoirs of the Amer. Math. Soc., Number 129, Amer. Math. Soc., Providence, (1972)
Nehari, Z.: Conformal Mapping. Dover, New York (1952)
Rudin, W.: Real and Complex Analysis. McGraw Hill, New York (1987)
Shapiro, H.S.: The Schwarz function and its generalization to higher dimensions. Univ. of Arkansas Lecture Notes in the Mathematical Sciences, Wiley, New York (1992)
Shapiro, H., Ullemar,C.: Conformal mappings satisfying certain extremal properties and associated quadrature identities, Research Report TRITA-MAT-1986-6, Royal Inst. of Technology, 40pp., (1981)
Schiffer, M., Spencer, D.: Functionals of finite Riemann surfaces. Princeton Univ. Press, Princeton (1954)
Stein, E.M., Shakarchi, R.: Complex analysis, Princeton Lectures in Analysis II. Princeton Univ. Press, Princeton (2003)
Suita, N.: Capacities and kernels on Riemann surfaces. Arch. Rational Mech. Anal. 46, 212–217 (1972)
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
All authors certify that they have no affiliations with or involvement in any organization or entity with any financial interest or non-financial interest in the subject matter discussed in this manuscript.
Additional information
To celebrate the legacy of Harold S. Shapiro.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The original version of this article was revised to move the dedication line.
Rights and permissions
About this article
Cite this article
Bell, S.R., Gustafsson, B. Ruminations on Hejhal’s theorem about the Bergman and Szegő kernels. Anal.Math.Phys. 12, 24 (2022). https://doi.org/10.1007/s13324-021-00634-w
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s13324-021-00634-w