On the existence of universal series by trigonometric system

In this paper we prove the following: let omega(t) be a continuous function, increasing in [0, 00) and omega(+0) = 0. Then there exists a series of the form Sigma(infinity)(k=-infinity) C(k)e(ikx) with Sigma(infinity)(k=-infinity) C-k(2) omega(vertical bar C-k vertical bar) < infinity, C-k = <(C)over bar>(k) with the following property: for each epsilon > 0 a weighted function mu (x), 0 < mu(x) <= 1, vertical bar{x is an element of [0,2 pi] : mu(x) not equal 1}vertical bar < epsilon can be constructed, so that the series is universal in the weighted space L-mu(1)[0,2 pi] with respect to rearrangements. (c) 2005 Elsevier Inc. All rights reserved.