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 
of the form
that possess the following property: for any function 
there exists a growing sequence of natural numbers 
such that the subseries 
converges to 
in the 
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.
There are many other works (see [2-11]) 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 
there exists a measurable function 
with
such that for any 
and any function 
there exists a series in the Walsh generalized system 
of the following form
(1)
which converges to 
in the
—metric and almost everywhere.
Note that there exist functions in the space 
that can not be represented by series in the Walsh system 
(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 
there exists a measurable function 
with
and a series in the Walsh generalized system 
of the form
that possess the following property: for any function 
there exists a growing sequence of natural numbers 
such that the subseries
converges to 
in the 
-norm and a.e.
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 
which converges to 
in 
norm .
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 
in order to construct a Walsh series 
to be universal in the space with respect to subseries?
2. Proofs of Main Lemmas
Let 
be a fixed integer and
. Recall the following definitions.
The Rademacher system of order 
is defined inductively as follows. For 
let
and for 
let
The Walsh generalized system (see [3] and [13,14]) of order 
is defined by
and if
, where
, 
then
.
We denote the generalized Walsh system of order 
by
. Note that 
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.
• Each 
-th Rademacher function has period
.
• 
, 
, and 
(mod
).
• 
is a finite product of Rademacher functions with values in
.
• 
if
.
• 
, 
is a complete orthonormal system in
and it is basic in 
for
.
We put
(2)
and periodically extend these functions on 
with period 1.
By 
we denote the characteristic function of the set
, i.e.
(3)
Then, clearly
(4)
and let for the natural numbers 
(5)
(6)
Hence
(7)
. (8)
Lemma 1. Let dyadic interval
and numbers 
be given. Then there exists a measurable set 
and a polynomial 
in the Walsh generalized system 
of the following form
which satisfy the following conditions:
1) the coefficients 
are 
or
2)
3)
4) 
where 
is a constant5)
.
Proof. Let
(9)
We define the polynomial 
and the numbers
, 
and 
in the following form:
(10)
(11)
(12)
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:
(13)
where
(14)
Then let
Clearly that (see (2) and (10)),
(15)
(16)
Hence
where 
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
, one can find a set 
and a polynomial in the Walsh generalized system
satisfying the following conditions:
and the non-zero coefficients in
are in decreasing order2)
3) 
4)
for every measurable subset e of E5)
.
Proof. We choose some non-overlapping binary intervals 
and a step function
(17)
satisfying the conditions
(18)
(19)
(20)
Successively applying Lemma 1, we determine some sets 
and polynomials
(21)
where 
or
, if
,
(22)
, (23)
, (24)
Then let
(25)
(26)
>From (19), (21), (22) and (25) follows, that
and 
and the non-zero coefficients in 
are in decreasing order, i.e. the statements 1) - 3) of Lemma 2 are valid.
To verify the statement 4), for any 
determine 
from the condition
. Then by (21) and (26)
(27)
Since for any point
, 
(see (17), (23) and (26)), then from the conditions (18), (24), and (27) for every measurable subset e of E.
We have
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.
Lemma 3. Let 
the Walsh generalized system, then for any 
there exist a weight function
with 
such that for any numbers 
, and evry function
, one can find polynomial in the Walsh generalized system
satisfying the following conditions:
2) 
3)
.
Proof of Lemma 3
Let
(28)
be the sequence of all algebraic polynomials with rational coefficients. Applying repeatedly Lemma 2, we obtain sequences of 
sets and polynomials in the Walsh systems 
(29)
where
which satisfy the following conditions:
(30)
(31)
(32)
for every measurable subset e of 
(33)
Setting
(34)
It is clear (see (33), (34))
We define a function 
in the following way:
(35)
where
(36)
It follows from (34)-(36) that for all 
(37)
In a similar way for all 
we have
(38)
By the conditions (31), (35)-(38) for all 
we obtain
(39)
Taking relations (32), (34)-(36) into account we obtain that for all
, and 
(40)
From the sequence (28) we choose a function 
such that
(41)
. (42)
Then, we set
,
.
Now, it is not difficult to verify (see (30), (39)-(42)) that the function 
and the polynomials 
satisfy the requirements of Lemma 3.
Remark: In Lemma 3 polynom 
can be chosen such that
Lemma 3 is proved.
Proof Theorem 2
Let 
and let
(43)
be the sequence of all algebraic polynomials with rational coefficients. Applying repeatedly Lemma 3, we obtain a weight function 
with 
and
, a sequences of polynomials in the Walsh generalized systems 
(44)
where
which satisfy the following conditions:
(45)
(46)
(47)
Consider a series
(48)
Clearly (see (45), (48))
let 
and let
. We choose some 
from sequence (43), to have
Suppose that the numbers 
and polynomials 
are already determined satisfying to the following conditions:
(49)
(50)
Let a function
, 
be chosen from the sequence (43) such that
(51)
Hence by (49) we obtain
(52)
From the conditions(46) (47), (52) follows that
(53)
(54)
Then we obtain that the series
where
converges to 
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.
Theorem 2 is proved.
NOTES