Entire functions decaying rapidly on strips

Abstract

It has been known for many years that there exist non-constant entire functions f which decay to 0 along every infinite line. Recently it has been shown that if 0 < α < 1/2, then there exist entire functions f such that exp(|z| ^α )f(z) → 0 (z → ∞, z ∊ S) for every strip S; moreover there is a vector space M which consists of such functions and which is dense in the space of all entire functions with the topology of local uniform convergence. In this note the result is shown to hold for every α > 0. The proof depends upon a theorem about tangential harmonic approximation on unbounded sets. As a corollary, a new result is proved about the class of entire functions which have zero integral on every doubly infinite line.