Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Kernel functions on domains with hyperelliptic double


Author: William H. Barker
Journal: Trans. Amer. Math. Soc. 231 (1977), 339-347
MSC: Primary 30A31; Secondary 30A24, 30A42, 30A46
DOI: https://doi.org/10.1090/S0002-9947-1977-0466517-0
MathSciNet review: 0466517
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we show that the structure of the Bergman and Szegö kernel functions is especially simple on domains with hyperelliptic double. Each such domain is conformally equivalent to the exterior of a system of slits taken from the real axis, and on such domains the Bergman kernel function and its adjoint are essentially the same, while the Szegö kernel function and its adjoint are elementary and can be written in a closed form involving nothing worse than fourth roots of polynomials. Additionally, a number of applications of these results are obtained.


References [Enhancements On Off] (What's this?)

  • [1] L. V Ahlfors, Bounded analytic functions, Duke Math. J. 14 (1947), 1-11. MR 9, 24. MR 0021108 (9:24a)
  • [2] H. F. Baker, Abel's theorem and the Allud theory, including the theory of theta functions, Cambridge Univ. Press, London and New York, 1897.
  • [3] W. H. Barker, Plane domains with hyperelliptic double, Dissertation, Stanford Univ., 1975.
  • [4] S. Bergman, The kernel function and conformai mappings, 2nd ed., Math. Survey, no. 5, Amer. Math. Soc., Providence, R. I., 1970. (1st ed., 1950; MR 12, 402.)
  • [5] P. R. Garabedian, Schwarz's lemma and the Szegö kernel function, Trans. Amer. Math. Soc. 67 (1949), 1-35. MR 11, 340. MR 0032747 (11:340f)
  • [6] N. S. Hawley and M. M. Schiffer, Half order differentials on Riemann surfaces, Acta Math. 115 (1966), 199-236. MR 32 #7739. MR 0190326 (32:7739)
  • [7] M. Heins, A lemma on positive harmonic functions, Ann. of Math. (2) 52 (1950), 568-573. MR 12, 259. MR 0037420 (12:259b)
  • [8] D. A. Hejhal, Theta functions, kernel functions, and Abelian differentials, Mem Amer. Math. Soc., no. 129, 1972. MR 51 #8403. MR 0372187 (51:8403)
  • [9] Z. Nehari, Conformal mapping, McGraw-Hill, New York, 1952. MR 13, 640. MR 0045823 (13:640h)
  • [10] M. M. Schiffer, Various types of orthogonalization, Duke Math. J. 17 (1950), 329-366. MR 12, 491. MR 0039071 (12:491g)
  • [11] M. M. Schiffer and N. S. Hawley, Connections and conformal mappings, Acta Math. 107 (1962), 175-274. MR 32 #202. MR 0182720 (32:202)
  • [12] M. M. Schiffer and D. C. Spencer, Functionals of finite Riemann surfaces, Princeton Univ. Press, Princeton, N. J., 1954. MR 16, 461. MR 0065652 (16:461g)
  • [13] F. Schottky, Über die konforme Abbildung mehrfach zusammenhängender ebener Flächen, Crelles J. 83 (1877), 300-351.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 30A31, 30A24, 30A42, 30A46

Retrieve articles in all journals with MSC: 30A31, 30A24, 30A42, 30A46


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1977-0466517-0
Keywords: Hyperelliptic double, kernel function, plane domains
Article copyright: © Copyright 1977 American Mathematical Society

American Mathematical Society