Dense-lineability of sets of Birkhoff-universal functions with rapid decay

Abstract

Let A be an unbounded Arakelian set in the complex plane whose complement has infinite inscribed radius, and ψ be an increasing positive function on the positive real numbers. We prove the existence of a dense linear manifold M of entire functions all of whose nonzero members are Birkhoff-universal, such that each function in M has overall growth faster than ψ and, in addition, exp(|z|α)f(z) → 0 (z → ∞, z ∈ A) for all α < 1/2 and f ∈ M. With slightly more restrictive conditions on A, we get that the last property also holds for the action T f of certain holomorphic operators T. Our results unify, extend and complete recent work by several authors.