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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Discrete sufficient sets for some spaces of entire functions


Author: B. A. Taylor
Journal: Trans. Amer. Math. Soc. 163 (1972), 207-214
MSC: Primary 46.30; Secondary 30.00
MathSciNet review: 0290084
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Abstract: Let $ E$ denote the space of all entire functions $ f$ of exponential type (i.e. $ \vert f(z)\vert = O(\exp (B\vert z\vert))$) for some $ B > 0$). Let $ \mathcal{K}$ denote the space of all positive continuous functions $ k$ on the complex plane $ C$ with $ \exp (B\vert z\vert) = O(k(z))$ for each $ B > 0$. For $ k \in \mathcal{K}$ and $ S \subset C$, let $ \vert\vert f\vert{\vert _{k,s}} = \sup \{ \vert f(z)\vert/k(z):z \in S\}$. We prove that the two families of seminorms $ {\{ \vert\vert\vert{\vert _{k,C}}\} _{k \in \mathcal{K}}}$ and $ {\{ \vert\vert\vert{\vert _{k,s}}\} _{k \in \mathcal{K}}}$, where

$\displaystyle S = \{ n + im: - \infty < n,m < + \infty \} $

, determine the same topology on $ E$.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1972-0290084-3
PII: S 0002-9947(1972)0290084-3
Keywords: Entire function, sufficient set, Fourier transform, subharmonic function
Article copyright: © Copyright 1972 American Mathematical Society