Positive differentials, theta functions

and Hardy kernels

Author:
Akira Yamada

Journal:
Proc. Amer. Math. Soc. **127** (1999), 1399-1408

MSC (1991):
Primary 30C40; Secondary 14K25

DOI:
https://doi.org/10.1090/S0002-9939-99-04711-5

Published electronically:
January 29, 1999

MathSciNet review:
1476401

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a planar regular region whose Schottky double has genus and set . For fixed we determine the range of the function where is the Riemann theta function on . Also we introduce two weighted Hardy spaces to study the problem when the matrix is positive definite. The proof relies on new theta identities using Fay's trisecants formula.

**1.**J. D. Fay,*Theta functions on Riemann surfaces*, Lecture Notes in Mathematics**352**, Springer-Verlag, 1973. MR**49:569****2.**D. A. Hejhal,*Theta functions, kernel functions and Abelian integrals*, Amer. Math. Soc. Memoir**129**, 1972. MR**51:8403****3.**A. H. Read,*A converse of Cauchy's theorem and applications to extremal problems*, Acta Math.**160**(1959), 1-22. MR**20:4640****4.**S. Saitoh,*Theory of reproducing kernels and its applications*, Pitman Research Notes in Mathematics Series**189**, Longman Scientific & Technical, 1988. MR**90f:46045****5.**H. Widom,*Extremal polynomials associated with a system of curves in the complex plane*, Adv. in Math.**2**(1969), 127-232. MR**39:418****6.**A. Yamada,*Theta functions and domain functions*, RIMS Kokyuroku**323**(1978), 84-101 (in Japanese).

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Additional Information

**Akira Yamada**

Affiliation:
Department of Mathematics and Informatics, Tokyo Gakugei University, Koganei, Tokyo 184, Japan

Email:
yamada@u-gakugei.ac.jp

DOI:
https://doi.org/10.1090/S0002-9939-99-04711-5

Keywords:
Positive differential,
theta function,
kernel function

Received by editor(s):
June 22, 1997

Received by editor(s) in revised form:
August 18, 1997

Published electronically:
January 29, 1999

Communicated by:
Albert Baernstein II

Article copyright:
© Copyright 1999
American Mathematical Society