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 ⊂ C n S \subset {{\mathbf {C}}^n} such that d ( z , S ) ≤ const ⁡ | z | 1 − ρ / 2 d(z,S) \leq \operatorname {const}|z{|^{1 - \rho /2}} for all z ∈ C n z \in {{\mathbf {C}}^n} , then S is a sufficient set for the space of entire functions on C n {{\mathbf {C}}^n} of order less than ρ \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.