On the existence of universal series by trigonometric system

In this paper we prove the following: let (t) be a continuous function, increasing in [0,∞) and (+0) = 0. Then there exists a series of the form ∞ k=−∞ Ckeikx with ∞ k=−∞ C2 k (|Ck|)<∞, C−k = Ck with the following property: for each ε>0 a weighted function (x), 0< (x) 1, |{x ∈ [0, 2 ] : (x) = 1}|<ε can be constructed, so that the series is universal in the weighted space L1 [0, 2 ] with respect to rearrangements. © 2005 Elsevier Inc. All rights reserved.