On the existence of universal series by trigonometric system

Abstract

In this paper we prove the following: let $\omega (t)$ be a continuous function, increasing in $[0,\infty )$ and $\omega (+0)=0$. Then there exists a series of the form${\sum }_{k=-\infty }^{\infty }{C}_k{e}^{ikx}$ with ${\sum }_{k=-\infty }^{\infty }{C}_k^2\omega (|C_k|)<\infty$, $C_{-k}={\bar{C}}_k$, with the following property: for each $\epsilon >0$ a weighted function $\mu (x),0<\mu (x)\le 1,|\{x\in [0,2\pi ]:\mu (x)\ne 1\}|<\epsilon$ can be constructed, so that the series is universal in the weighted space $L_{\mu }^1[0,2\pi ]$ with respect to rearrangements.