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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Positive differentials, theta functions
and Hardy ${H^2}$ 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: Let $R$ be a planar regular region whose Schottky double ${\hat{R}}$ has genus $g$ and set ${\hat{T}_0}=\{z\in \mathbb C^g|\sqrt{-1}\,z\in \mathbb R^g \}$. For fixed $a\in R$ we determine the range of the function $F(e)=\theta(a-\bar{a}+e)/\theta(e)\ (e\in {\hat{T}_0})$ where $\theta(z)$ is the Riemann theta function on ${\hat{R}}$. Also we introduce two weighted Hardy spaces to study the problem when the matrix $(\frac{\partial^2\log F}{\partial z_i\partial z_j}(e))$ is positive definite. The proof relies on new theta identities using Fay's trisecants formula.

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

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Additional Information

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

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