Sufficient sets for some spaces of entire functions

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.