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Positive differentials, theta functions and Hardy kernels

Authors:	Akira Yamada

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

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

Keywords:	Positive differential theta function kernel function

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