Positive differentials, theta functions and $H^2$ Hardy kernels
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- by Akira Yamada PDF
- Proc. Amer. Math. Soc. 127 (1999), 1399-1408 Request permission
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
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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
- 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
- © Copyright 1999 American Mathematical Society
- 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
- MathSciNet review: 1476401