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

 

Geometric condition for universal interpolation in $ \hat{\mathcal{E}}'$


Author: William A. Squires
Journal: Trans. Amer. Math. Soc. 280 (1983), 401-413
MSC: Primary 30E05; Secondary 30D15, 42A38
MathSciNet review: 712268
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Abstract: It is known that if $ h$ is an entire function of exponential type and $ Z(h) = {\{ {z_k}\} _{k = 1}}$ with $ \vert h^{\prime}({z_k})\vert \geqslant \varepsilon \exp (- c\vert{z_k}\vert)$ for constants $ \epsilon$, $ C$ independent of $ k$, then $ \{ {z_k}\} _{k = 1}^\infty $ is a universal interpolation sequence. That is, given any sequence of complex numbers $ \{ {a_k}\} _{k = 1}^\infty $ such that $ \vert{a_k}\vert \leqslant A\,\exp (B\vert{z_k}\vert)$ for constants $ A,B$ independent of $ K$ then there exists $ g$ of exponential type such that $ g({z_k}) = {a_k}$. This note is concerned with finding geometric conditions which make $ \{ {z_k}\} _{k = 1}^\infty $ a universal interpolation sequence for various spaces of entire functions. For the space of entire functions of exponential type a necessary and sufficient condition for $ \{ {z_k}\} _{k = 1}^\infty $ to be a universal interpolation sequence is that $ \int_0^{\vert{z_k}\vert} {n({z_k},t)\,dt/t \leqslant C\vert{z_k}\vert + D,k = 1} , 2,\ldots$, where $ n({z_k},t)$ is the number of points of $ \{ {z_k}\} _{k = 1}^\infty $ in the disc of radius $ t$ about $ {z_k}$, excluding $ {z_k}$, and $ C,D$ are constants independent of $ k$. Results for the space $ \hat{\mathcal{E}}^\prime= \{ f\;{\text{entire}}\vert\vert f(z)\vert \leqslant A\;\exp [B\vert\operatorname{Im} z\vert + B\log (1 + \vert z\vert^{2})]\}$ are given but the theory is not as complete as for the above example.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1983-0712268-7
PII: S 0002-9947(1983)0712268-7
Article copyright: © Copyright 1983 American Mathematical Society