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

 

Approximation of the Bergman norm by the norms of the direct product of two Szegő spaces


Author: Saburou Saitoh
Journal: Proc. Amer. Math. Soc. 75 (1979), 226-230
MSC: Primary 30C40; Secondary 30E10
MathSciNet review: 532141
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Abstract: Let G be a bounded regular region in the plane. Let $ H_2^{1/2}(G)$ denote the Szegő space of G composed of analytic functions on G with finite norm

$\displaystyle {\left\{ {\frac{1}{{2\pi }}\int_{\partial G} {\vert f(z){\vert^2}\vert dz\vert} } \right\}^{1/2}} < \infty .$

We set $ f(z) = \Sigma _{j = 1}^\infty {\varphi _j}(z){\psi _j}(z)\;({\varphi _j},{\psi _j} \in H_2^{1/2}(G))$. Then, we determine a necessary and sufficient condition for $ f(z)$ to make the equality

$\displaystyle \frac{1}{\pi }\int {\int\limits_G {\vert f(z){\vert^2}\;dx\;dy = ... ... _j}({z_2}} )\overline {{\psi _k}({z_2})} \vert d{z_2}\vert} } } } \right\}} } $

hold. The minimum is taken here over all analytic functions $ \Sigma _{j = 1}^\infty {\varphi _j}({z_1}){\psi _j}({z_2})$ on $ G \times G$ satisfying $ f(z) = \Sigma _{j = 1}^\infty {\varphi _j}(z){\psi _j}(z)$ on G.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1979-0532141-1
PII: S 0002-9939(1979)0532141-1
Keywords: Bergman and Szegő kernels, the direct product of two Szegő spaces, Dirichlet norm, approximation of the Bergman norm, generalized isoperimetric inequalities
Article copyright: © Copyright 1979 American Mathematical Society