Abstract
Certain integral operators involving the Szegö, the Bergman and the Cauchy kernels are known to have the reproducing property. Both the Szegö and the Bergman kernels have series representations in terms of an orthonormal basis. In this paper we derive the Cauchy kernel by means of biorthogonality. The ideas involved are then applied to construct a non-Hermitian kernel admitting a reproducing property for a space associated with the Bergman kernel. The construction leads to a domain integral equation for the Bergman kernel.1 2
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bell, S. R. (1992): The Cauchy Transform, Potential Theory, and Conformal Mapping, CRC Press, Boca Raton.
Bergman, S. (1970): The Kernel Function and Conformai Mapping, AMS Math. Surveys Number 5, New York.
Carlson, B. C. (1977): Special Functions of Applied Mathematics, Academic Press, New York.
Davis, P. J. (1975): Interpolation and Approximations, Dover, New York.
Gradsteyn, I. S., Ryzhik I. M. (1980) Table of Integrals, Series, and Products, Academic Press, New York.
Henrici, P. (1974): Applied and Computational Complex Analysis, Vol. I, John Wiley, New York.
Henrici, P. (1986): Applied and Computational Complex Analysis, Vol. III, John Wiley, New York.
Hille, E. (1962): Analytic Function Theory, Vol. II, Ginn, Boston.
Hille, E. (1972): Introduction to the general theory of reproducing kernels. Rocky Mountain J. Math. 2, 321–368.
Kerzman, N., Stein, E. M. (1978): The Cauchy kernel, the Szegö kernel, and the Riemann mapping- function. Math. Ann. 236, 85–93.
Kerzman, N., Trümmer, M. R. (1986): Numerical conformal mapping via the Szegö kernel. J. Comp. Appl. Math. 14, 111–123.
Nehari, Z. (1952): Conformal Mapping, Dover, New York.
Papamichael, N., Warby, M. K. (1986) Stability and convergence properties of Bergman kernel methods for numerical conformai mapping. Numer. Math. 48, 639–669.
Razali, M. R. M., Nashed, M. Z., Murid, A. H. M. (1997): Numerical conformal mapping via the Bergman kernel. J. Comp. Appl. Math. 82, 333–350.
Szegö, G. (1939): Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ. 23, AMS, Providence.
Trummer, M. R. (1986): An efficient implementation of a conformal mapping method based on the Szegö kernel. SIAM J. Numer. Anal. 23(4), 853–872.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Murid, A.H.M., Nashed, M.Z. & Razali, M.R.M. A Domain Integral Equation for the Bergman Kernel. Results. Math. 35, 161–174 (1999). https://doi.org/10.1007/BF03322030
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF03322030