Representation of Functions in 1 μ L Weighted Spaces by Series with Monotone Coefficients in the Walsh Genrealized System *

Let    n  x  be the Walsh generalized system. In the paper constructed a weighted space 1 L , and series n n a   in the Walsh generalized system with monotonically decreasing coefficient 0 n a  such that for each function   1 f x L  in the space one can find a subseries   k k n n a  x  that converges to   f x in the weighted 1 L and almost everywhere on   0,1 .


Introduction
In the present paper we study the following natural question: does there exist a weighted space , with , such that for every function in the space one can find a series in the Walsh generalized system   n  of the form 1 , with 0, that possess the following property: for any function there exists a growing sequence of natural numbers such that the subseries  norm and a.e.Note that the problem of representing a function by a series in classical and general orthonormal systems has a long history.Of course the problem of the representation of functions was studied before Luzin's work.It goes back to D. Bernoulli, L. Euler and many others.A question posed by Lusin in 1915 asks whether it is possible to find for every measurable function a trigonometric series, with coefficient sequence converging to zero, that converges to the function almost everywhere.For real-valued functions, this question was given an affirmative answer by Men'shov [1] in 1941.[0,2π] There are many other works (see [2][3][4][5][6][7][8][9][10][11]) devoted to representations of functions by series in classical and general orthonormal systems and the existence of different types of universal series in the sense of convergence almost is everywhere and by measure.
Since the trigonometric and Walsh systems have many properties in common, one would think that there should be a corresponding result for the Walsh system.This is, indeed, the case, and, in fact, the same sort of result holding for a multitude of Walsh subsystems, many of them are quite sparse and far from complete.
In this paper we prove the following theorem: Theorem 1.For any 0 < < 1  there exists a such that for any and any function there exists a series in the Walsh generalized system   n  of the following form , where 0 and , which converges to in the -metric and almost everywhere.
Note that there exist functions in the space   1 0,1 L that can not be represented by series in the Walsh system   k  (see [8], pp.124-125).Theorem 1 is a consequence of the more general Theorem 2, which is stated as follows: Theorem 2. For any 0 < < 1  there exists a mea- Recall the following definition: a series  is said to be universal with respect to subseries in the space , if for each function , one can select a subseries   1 0,1 L  The above-mentioned definitions are given not in the most general form and only in the generality, in which they will be applied in the present paper.
Note that the result of the Theorem 2 is definitive in a certain sense: one can not replace by  because no orthonormal system of bounded functions does there exist a series universal in with respect to subseries.This is almost obvious.
The following problems remain open.Question 1. Are the theorems 1 and 2 true for the trigonometric system?Question 2. What kind of necessary and sufficient conditions should be imposed on the weight function   x  n in order to construct a Walsh series to be universal in the space with respect to subseries? .Recall the following definitions.

Proofs of Main Lemmas
The Rademacher system of order is defined inductively as follows.For let The Walsh generalized system (see [3] and [13,14]) of order is defined by a We denote the generalized Walsh system of order by a a  .Note that 2  is the classical Walsh system.The basic properties of the generalized Walsh system of order have been obtained by H. E. Chrestenson, J. Fine, C. Vateri, W. Young, N. Vilenkin and others.Next we list some properties of , which will be useful later.
and it is basic in and periodically extend these functions on with period 1.   Then, clearly and let for the natural numbers 1 , and and numbers be given.Then there exists a measurable set which satisfy the following conditions: 1) the coefficients are 0 or We define the polynomial   Taking into consideration the following equation and having the following relations ( 5)-( 8) and ( 10)-( 12), we obtain that the polynomial has the following form: where Then let Clearly that (see ( 2) and (10)), Repeating the arguments in the proof of Lemma 1, we get a proof of the last statement of Lemma 1. Lemma 1 is proved.
Lemma 2. Let given the numbers .Then for any function and a polynomial in the Walsh generalized system Proof.We choose some non-overlapping binary intervals and a step function satisfying the conditions Successively applying Lemma 1, we determine some sets where or 0   21), ( 22) and (25) follows, that  .
Repeating the arguments in the proof of Lemma 2, we get a proof of the last statement of Lemma 2. Lemma 2 is proved.
The main tool in the proof of Theorem 2 is the following result.
the Walsh generalized system, then for any  there exist a weight function such that for any numbers , and , one can find polynomial in the Walsh generalized system satisfying the following conditions: 1) be the sequence of all algebraic polynomials with rational coefficients.Applying repeatedly Lemma 2, we sets and polynomials in the Walsh systems where Open Access AM M. GRIGORYAN, A. MINASYAN 10 which satisfy the following conditions: for every measurable subset e of k E We define a function   x  in the following way: where In a similar way for all we have By the conditions (31), ( 35)-( 38) for all we obtain Taking relations (32), ( 34)-( 36) into account we obtain that for all From the sequence (28) we choose a function Then, we set Now, it is not difficult to verify (see (30), (39)-( 42)) that the function   Remark: In Lemma 3 polynom can be chosen such that be the sequence of all algebraic polynomials with rational coefficients.Applying repeatedly Lemma 3, we obtain a weight function   where which satisfy the following conditions: , where if , see 30 .   be chosen from the sequence (43) such that Hence by (49) we obtain From the conditions(46) (47), (52) follows that     Then we obtain that the series   , where , , 0, otherwise f x in the -norm.Repeating the arguments in the proof of Theorem 2 and using Lemma 1, Lemma 2 and remark of Lemma 3 we get the proof of the second statement of Theorem 2. f characteristic function of the set , i.e.
the following form: are in decreasing order, i.e. the statements 1) -3) of Lemma 2 are valid.M a To verify the statement 4), for any determine < N m M    from the condition 1 m m< m     .Then by (21) and (26)