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Transactions of the American Mathematical Society

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Sufficient sets for some spaces of entire functions

Author: Dennis M. Schneider
Journal: Trans. Amer. Math. Soc. 197 (1974), 161-180
MSC: Primary 32A15; Secondary 46E10
MathSciNet review: 0357835
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Abstract: B. A. Taylor [13] has shown that the lattice points in the plane form a sufficient set for the space of entire functions of order less than two. We obtain a generalization of this result to functions of several variables and to more general spaces of entire functions. For example, we prove that if $ S \subset {{\mathbf{C}}^n}$ such that $ d(z,S) \leq \operatorname{const}\vert z{\vert^{1 - \rho /2}}$ for all $ z \in {{\mathbf{C}}^n}$, then S is a sufficient set for the space of entire functions on $ {{\mathbf{C}}^n}$ of order less than $ \rho $. The proof involves estimating the growth rate of an entire function from its growth rate on S. We also introduce the concept of a weakly sufficient set and obtain sufficient conditions for a set to be weakly sufficient. We prove that sufficient sets are weakly sufficient and that certain types of effective sets [8] are weakly sufficient.

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Keywords: Entire function, sufficient set, weakly sufficient set, effective set, analytic variety, subharmonic function, locally convex space
Article copyright: © Copyright 1974 American Mathematical Society

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