Uniform Convergence of Translation Operators

Abstract

We denote N, R, C the sets of natural, real and complex numbers respectively. Let ( λ n ), n ∈ N be an unbounded sequence of complex numbers. Costakis has proved the following result. There exists an entire function f with the following property: for every x, y ∈ R with 0 < x < y , every θ ∈(0,1) and every a ∈ C there is a subsequence of natural numbers ( m n ) , n ∈ N such that, for every compact subset L &#8838; C , In the present paper we show that the constant function a cannot be replaced by any non-constant entire function G . This is so even if one demands the convergence in (*) only for a single radius r and a single positive number θ . This result is related with the problem of existence of common universal vectors for an uncountable family of sequences of translation operators.