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

Non-Existence of Common Hypercyclic Entire Functions for Certain Families of Translation Operators Let $${\mathcal {H}}({\mathbb {C}})$$ H ( C ) be the set of entire functions endowed with the topology of local uniform convergence. Fix a sequence of non-zero complex numbers $$({\lambda }_n)$$ ( λ n ) with $$\liminf _{n}\frac{|{\lambda }_{n+1}|}{|{\lambda }_n|}>2.$$ lim inf n | λ n + 1 | | λ n | > 2 . We prove that there exists no entire function $$f$$ f such that for every $$b\in \mathbb {C}\setminus \{ 0\}$$ b ∈ C \ { 0 } the set $$\{ f(z+{\lambda }_nb): n=1,2,\ldots \}$$ { f ( z + λ n b ) : n = 1 , 2 , … } is dense in $${\mathcal {H}}({\mathbb {C}}).$$ 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

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

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References (36)

Publisher
Springer Journals
Copyright
Copyright © 2015 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/s40315-015-0107-1
Publisher site
See Article on Publisher Site

Abstract

Let $${\mathcal {H}}({\mathbb {C}})$$ H ( C ) be the set of entire functions endowed with the topology of local uniform convergence. Fix a sequence of non-zero complex numbers $$({\lambda }_n)$$ ( λ n ) with $$\liminf _{n}\frac{|{\lambda }_{n+1}|}{|{\lambda }_n|}>2.$$ lim inf n | λ n + 1 | | λ n | > 2 . We prove that there exists no entire function $$f$$ f such that for every $$b\in \mathbb {C}\setminus \{ 0\}$$ b ∈ C \ { 0 } the set $$\{ f(z+{\lambda }_nb): n=1,2,\ldots \}$$ { f ( z + λ n b ) : n = 1 , 2 , … } is dense in $${\mathcal {H}}({\mathbb {C}}).$$ 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.

Journal

Computational Methods and Function TheorySpringer Journals

Published: Jan 29, 2015

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