Universal series by trigonometric system in weighted $L^1_{\mu}$ spaces

In this paper we consider the question of existence of trigonometric series universal in weighted $L^1_{\mu}[0,2\pi]$ spaces with respect to rearrangements and in usual sense.


Introduction
Let X be a Banach space.
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 f σ(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 n k such that the sequence of partial sums with numbers n k of the series (1.1) converges to f by norm of X.
Note, that many papers are devoted (see [1]- [9]) to the question on existence of variouse 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 D.E.Menshov [1] and V.Ya.Kozlov [2]. The series of the form (  [3]). He also established (see [4]), that if 1] , p > 1, then there exists a series of the form which has property: for any measurable function f (x) the members of series (1.3) can be rearranged so that the again received series converge on a measure on [0,1] to f (x). W. Orlicz [5] observed the fact that there exist functional series that are universal with respect to rearrangements in the sense of a.e. convergence in the class of a.e. finite measurable functions.
It is also useful to note that even Riemann proved that every convergent numerical series which is not absolutely convergent is universal with respect to rearrangements in the class of all real numbers.
M.G.Grigorian constructed a series of the form (see [6]), In [9] it is proved that for any given sequence of natural numbers {λ m } ∞ m=1 with λ m ր ∞ there exists a series by trigonometric system of the form so that for each ε > 0 a weighted function µ(x), can be constructed, so that the series (1.4) is universal in the weighted space L 1 µ [0, 2π] with respect simultaneously to rearrangements as well as to subseries.
In this paper we prove the following results.

BASIC LEMMA
Lemma 2.1. For any given numbers 0 < ε < 1 2 , N 0 > 2 and a step function where ∆ s is an interval of the form ∆ (i) which satisfy the conditions: for every measurable subset e of E.
Proof Let 0 < ǫ < 1 2 be an arbitrary number. Set We choose natural numbers ν 1 and N 1 so large that the following inequalities be satisfied: (By χ E (x) we denote the characteristic function of the set E.) We put Hence by (2.4),(2.5) and (2.9) we obtain Now assume that the numbers ν 1 < ν 2 < ...ν s−1 , N 1 < N 2 < ... < N s−1 , functions g 1 (x), g 2 (x), ..., g s−1 (x) and the sets E 1 , E 2 , ...., E s−1 are defined. We take sufficiently large natural numbers ν s > ν s−1 and Using the above arguments (see (2.16)-(2.18)), we conclude that the function g s (x) and the set E s satisfy the conditions: Thus, by induction we can define natural numbers We define a set E and a polynomial P (x) as follows: By Bessel's inequality and (2.3), (2.14) for all s ∈ [1, q] we get That is, the statements 1)

PROOF OF THEOREMS
Proof of Theorem 1.4 Let which satisfy the conditions: for every measurable subset e of E s .
We define a function µ(x) in the following way: Hence, obviously we have It follows from (3.8)-(3.10) that for all s ≥ n 0 and p ∈ [N s−1 , N s ) By (3.4), (3.8)-(3.10) for all s ≥ n 0 we have It is easy to see that we can choose a function f ν 1 (x) from the sequence (3.1) such that Hence, we have From (2.1), (A), (3.13) and (3.15) we obtain with m 1 = 1 Assume that numbers ν 1 < ν 2 < ... < ν q−1 ; m 1 < m 2 < ... < m q−1 are chosen in such a way that the following condition is satisfied: We choose a function f νq (x) from the sequence (3.1) such that By (3.13), (3.14) and (3.20) we obtain .
From (2.1), (A), (3.19) and (3.21) we have Thus, by induction we on q can choose from series (3.7) a sequence of members C mq e imqx , q = 1, 2, ..., and a sequence of polynomials Taking account the choice of P νq (x) and C mq e imqx (see (3.23) and (3.25)) we conclude that the series is obtained from the series (3.7) by rearrangement of members. Denote this series by C σ(k) e iσ(k)x .

The Theorem 1.4 is proved.
Proof of the Theorem 1.5 Applying Lemma consecutively, we can find a sequence {E s } ∞ s=1 of sets and a sequence of polynomials which satisfy the conditions: |C k | q < ∞, ∀q > 2.
Repeating reasoning of Theorem 1 a weighted function µ(x), 0 < µ(x) ≤ 1 can constructed so that the following condition is satisfied: Thus, the series (3.30) is universal in L 1 µ [0, 1] in the sense of usual (see Definition 1.2).
The Theorem 1.5 is proved.
The author thanks Professor M.G.Grigorian for his attention to this paper.