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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Asymptotics of reproducing kernels on a plane domain


Author: Miroslav Engliš
Journal: Proc. Amer. Math. Soc. 123 (1995), 3157-3160
MSC: Primary 30C40; Secondary 30E15, 46E22
MathSciNet review: 1277107
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Abstract: Let $ \Omega $. be a plane domain of hyperbolic type, $ \vert dz\vert/w(z)$ the Poincaré metric on $ \Omega $, and $ {K_{\Omega ,q}}(x,\bar y)$ the reproducing kernel for the Hilbert space $ \mathcal{A}_q^2(\Omega )$ of all holomorphic functions on $ \Omega $ square-integrable with respect to the measure $ w{(z)^{2q - 2}}\vert dz \wedge d\bar z\vert$. It is proved that

$\displaystyle \mathop {\lim }\limits_{q \to + \infty } \frac{{{K_{\Omega ,q}}(z,\bar z)w{{(z)}^{2q}}}}{{2q}} = \frac{1}{\pi }.$


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1995-1277107-0
PII: S 0002-9939(1995)1277107-0
Keywords: Bergman kernel, automorphic functions
Article copyright: © Copyright 1995 American Mathematical Society