Universal series by trigonometric system in weighted Lμ1 spaces

is said to be universal in X with respect to rearrangements, if for any f ∈ X the members of (1.1) can be rearranged so that the obtained series ∑∞ k=1 fσ(k) converges to f by norm of X . Definition 1.2. The series (1.1) is said to be universal (in X) in the usual sense, if for any f ∈ X there exists a growing sequence of natural numbers nk such that the sequence of partial sums with numbers nk of the series (1.1) converges to f by norm of X . Definition 1.3. The series (1.1) is said to be universal (in X) concerning partial series, if for any f ∈ X it is possible to choose a partial series∞k=1 fnk from (1.1), which converges to the f by norm of X . Note that many papers are devoted (see [1–10]) to the question on existence of various types of universal series in the sense of convergence almost everywhere and on a measure. The first usual universal in the sense of convergence almost everywhere trigonometric series were constructed by Menshov [6] and Kozlov [5]. The series of the form