Non-Existence of Common Hypercyclic Entire Functions for Certain Families of Translation Operators

• Published in: Computational methods and function theory 2015-09, Vol.15 (3), p.393-401
• Main Authors: Costakis, George, Tsirivas, Nikos, Vlachou, Vagia
• Format: Article
• Language: English
• Subjects: Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable; Mathematics; Mathematics and Statistics
• ISSN: 1617-9447; 2195-3724
• DOI: 10.1007/s40315-015-0107-1

Let H ( C ) be the set of entire functions endowed with the topology of local uniform convergence. Fix a sequence of non-zero complex numbers ( λ n ) with lim inf n | λ n + 1 | | λ n | > 2 . We prove that there exists no entire function f such that for every b ∈ C \ { 0 } the set { f ( z + λ n b ) : n = 1 , 2 , … } is dense in H ( C ) . This, on one hand gives a negative answer to Costakis (Approximation by translates of entire functions, Complex and harmonic analysis. Destech Publ., Inc., Lancaster, pp 213–219 2007 , Question 2) and on the other hand shows that certain results from Tsirivas (Existence of common hypercyclic vectors for translation operators, 2014 ; Common hypercyclic functions for translation operators with large gaps, 2014 ) are sharp.