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

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.**John D. Fay,*Theta functions on Riemann surfaces*, Lecture Notes in Mathematics, Vol. 352, Springer-Verlag, Berlin-New York, 1973. MR**0335789****2.**Dennis A. Hejhal,*Theta functions, kernel functions, and Abelian integrals*, American Mathematical Society, Providence, R.I., 1972. Memoirs of the American Mathematical Society, No. 129. MR**0372187****3.**Arthur H. Read,*A converse of Cauchy’s theorem and applications to extremal problems.*, Acta Math.**100**(1958), 1–22. MR**0098178****4.**Saburou Saitoh,*Theory of reproducing kernels and its applications*, Pitman Research Notes in Mathematics Series, vol. 189, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1988. MR**983117****5.**Harold Widom,*Extremal polynomials associated with a system of curves in the complex plane*, Advances in Math.**3**(1969), 127–232. MR**0239059****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