Sufficient sets for some spaces of entire functions
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- by Dennis M. Schneider PDF
- Trans. Amer. Math. Soc. 197 (1974), 161-180 Request permission
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 $d(z,S) \leq \operatorname {const}|z{|^{1 - \rho /2}}$ 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.References
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Additional Information
- © Copyright 1974 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 197 (1974), 161-180
- MSC: Primary 32A15; Secondary 46E10
- DOI: https://doi.org/10.1090/S0002-9947-1974-0357835-2
- MathSciNet review: 0357835