On the existence of universal series by trigonometric system

In this paper we prove the following: let ω(t)ω(t) be a continuous function, increasing in [0,∞)[0,∞) and ω(+0)=0ω(+0)=0. Then there exists a series of the form∑k=-∞∞Ckeikxwith∑k=-∞∞Ck2ω(|Ck|)<∞,C-k=C¯kwith the following property: for each ɛ>0ɛ>0 a weighted function μ(x),0<μ(x)⩽1,|{x∈[0,2π]:μ(x),0<μ(x)⩽1,{x∈[0,2π]:μ(x)≠1}|<ɛμ(x)≠1}<ɛ can be constructed, so that the series is universal in the weighted space Lμ1[0,2π] with respect to rearrangements.