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


Some inequalities for entire functions

Author: Saburou Saitoh
Journal: Proc. Amer. Math. Soc. 80 (1980), 254-258
MSC: Primary 30D20; Secondary 15A69
MathSciNet review: 577754
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Abstract: For any entire functions $ \varphi (z)$ and $ \psi (z)$ on C with finite norm

$\displaystyle {\left\{ {\frac{1}{\pi }\int {\int\limits_{\mathbf{C}} {\vert f(z){\vert^2}\exp ( - \vert z{\vert^2})dx\;dy} } } \right\}^{1/2}} < \infty ,$

we show that the inequality

\begin{displaymath}\begin{array}{*{20}{c}} {\frac{2}{\pi }\int {\int\limits_{\ma... ...p ( - \vert z{\vert^2})\;dx\;dy} } } } } \hfill \\ \end{array} \end{displaymath}

holds. This inequality is obtained as a special case of a general result. We also refer to some properties of a tensor product of spaces of entire functions.

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Additional Information

PII: S 0002-9939(1980)0577754-4
Keywords: Space of entire functions, Fischer space, inequality for entire functions, tensor product of Hilbert spaces, general theory of reproducing kernels
Article copyright: © Copyright 1980 American Mathematical Society

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