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 non-zero 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.

Lineability criteria, with applications

“…Dirichlet spaces. In fact, the dense-lineability of E ϕ has already been established, even with several additional properties (boundedness on large sets, vanishing on large sets as z → ∞, universality in the sense of Birkhoff, action of certain operators, etc), see for instance [4,16,17,28,36,44]. As Theorem 4.17 below shows, E ϕ enjoys stronger lineability properties.…”

A dynamical characterization of sub-Arakelian subsets

“…Here ρ(A) = sup{r > 0 : there exists a closed ball B of radius r with B ⊂ A} is the inscribed radius of a subset A ⊂ C. In 2010, Bernal, Luh and the first author [9] stated the following: If F ⊂ C is an unbounded Arakelian set with ρ(C \ F ) = +∞, then there is a dense linear manifold M of entire functions all of whose nonzero members are Birkhoff-universal and exp(|z|…”

Compositionally Universal Entire Functions on the Plane and the Punctured Plane

“…F 0 is closed and C ∞ \ F 0 is connected and locally connected at ∞, where C ∞ is the onepoint compactification of C) such that ρ i (C \ F 0 ) = +∞ (here ρ i (A) = sup{r > 0 : there exists a closed ball B of radius r with B ⊂ A}, the inscribed radius of a subset A ⊂ C). In 2010, Calderón, Luh and the author [6] stated the following: if A ⊂ C is an unbounded Arakelian set with ρ i (C \ A) = +∞, there is a dense linear manifold M of entire functions all of whose nonzero members are Birkhoffuniversal and exp(|z| α )f (z) → 0 (z → ∞, z ∈ A) for all α < 1/2 and f ∈ M . Passing to C * , A. Vogt [27] has recently constructed a multiplicative universal entire function ϕ with respect to a given unbounded sequence (a n ) such that ϕ is bounded on some curve Γ tending to ∞.…”