Permissible growth rates for Birkhoff type universal harmonic functions

Abstract

A harmonic function H on R n (n 2) is said to be universal (in the sense of Birkhoff) if its set of translates {x → H (a + x): a ∈ R n } is dense in the space of all harmonic functions on R n with the topology of local uniform convergence. The main theorem includes the result that such functions, H, can have any prescribed order and type. The growth result is compared with a similar known theorem for G.D. Birkhoff's universal holomorphic functions and contrasted with known growth theorems for MacLane-type universal harmonic and holomorphic functions.

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

“…Existence of Birkhoff-universal entire functions enjoying some kind of controlled overall growth (but not necessary bounded on prescribed sets) was obtained in the analytic case by Duios-Ruis [21], Chan and Shapiro [18], Arakelian and Hakobian [2], and in the harmonic case by Armitage [3]. 6.…”

Rate of growth of frequently hypercyclic functions

“…Armitage [5] has proved that if φ : [0, ∞) → (0, ∞) is a continuous increasing function such that (log φ(r))/(log r) 2 → ∞ as r → ∞, then there exists a harmonic function f on R N that is hypercyclic for T a such that |f (x)| ≤ φ( x ) for x ∈ R N , and lim sup x→∞ f (x) φ( x ) = 1.…”

“…In particular, there are hypercyclic harmonic functions of order 0. However, the methods of [5] do not seem to be adaptable to the study of frequent hypercyclicity of T a . (2b) Does there exist a ∂ ∂x k -frequently hypercyclic harmonic function h on R N such that M 2 (h, r) ≤ C e r r N/2−3/4 for r > 0 ?…”

“…Armitage [5] In particular, there are hypercyclic harmonic functions of order 0. However, the methods of [5] do not seem to be adaptable to the study of frequent hypercyclicity of T a .…”

“…Analogous investigations have been carried out for harmonic functions. Dzagnidze [20] obtained the analogue of Birkhoff's result, see also Armitage and Gauthier [6]; Armitage [5] recently studied corresponding rates of growth. Hypercyclic partial differentiation operators and related growth rates were investigated by Aldred and Armitage [1], [2] and [3].…”