Discrete sufficient sets for some spaces of entire functions
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- by B. A. Taylor PDF
- Trans. Amer. Math. Soc. 163 (1972), 207-214 Request permission
Abstract:
Let $E$ denote the space of all entire functions $f$ of exponential type (i.e. $|f(z)| = O(\exp (B|z|))$) for some $B > 0$). Let $\mathcal {K}$ denote the space of all positive continuous functions $k$ on the complex plane $C$ with $\exp (B|z|) = O(k(z))$ for each $B > 0$. For $k \in \mathcal {K}$ and $S \subset C$, let $||f|{|_{k,s}} = \sup \{ |f(z)|/k(z):z \in S\}$. We prove that the two families of seminorms ${\{ |||{|_{k,C}}\} _{k \in \mathcal {K}}}$ and ${\{ |||{|_{k,s}}\} _{k \in \mathcal {K}}}$, where \[ S = \{ n + im: - \infty < n,m < + \infty \} \], determine the same topology on $E$.References
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Additional Information
- © Copyright 1972 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 163 (1972), 207-214
- MSC: Primary 46.30; Secondary 30.00
- DOI: https://doi.org/10.1090/S0002-9947-1972-0290084-3
- MathSciNet review: 0290084